electronics notes 2016

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Electronics Notes-2014 Imperial College London page 2 of 72

Contents

1 Introduction to fundamentals ............................................................................................................ 4

2 Basic AC/DC Circuits ....................................................................................................................... 4

2.1 Electrostatic Forces, Field and Potential Energy ..................................................................... 4

2.2 Definition of current ................................................................................................................. 5

2.3 Resistance ............................................................................................................................... 6

2.3.1

Metallic conduction .............................................................................................................. 6

2.3.2 Drift velocity and the origins of resistance ........................................................................... 6

2.4

Power dissipated in a resistor .................................................................................................. 8

2.5 Ohm’s law ................................................................................................................................ 9

2.6 Resistors in series ................................................................................................................. 10

2.7 Resistors in parallel ............................................................................................................... 10

2.8 The voltage divider circuit ...................................................................................................... 11

2.9 Voltage/Current Sources ....................................................................................................... 12

2.9.1 An ideal voltage source ..................................................................................................... 12

2.9.2 An ideal current source ...................................................................................................... 12

2.9.3 Batteries ............................................................................................................................. 12

2.9.4 Real voltage and current sources ...................................................................................... 13

2.9.5

Multiple sources ................................................................................................................. 15

2.10

Equivalent Circuits ................................................................................................................. 16

2.10.1 Finding equivalent circuits ............................................................................................. 17

2.11 Impedance matching ............................................................................................................. 21

2.12 Alternating Current ................................................................................................................ 21

2.12.1

AC signals ...................................................................................................................... 22

2.12.2 Electrical Power in AC circuits ....................................................................................... 22

3 RC Filters ....................................................................................................................................... 24

3.1 Capacitors .............................................................................................................................. 24

3.1.1 Capacitors: design and construction ................................................................................. 24

3.2 Capacitors in series and parallel ........................................................................................... 26

3.3 Energy stored in a capacitor .................................................................................................. 27

3.4 Charging/discharging behaviour ............................................................................................ 28

3.5

The RC filter circuit using a time domain analysis ................................................................. 31

3.6

Complex Voltages and Currents............................................................................................ 33

3.7 Resistance becomes impedance........................................................................................... 34

3.8 Complex impedance .............................................................................................................. 35

3.9

Complex analysis of the filter circuit ................................................................................ 35

3.10 Bode Plot - power ratios and decibels ................................................................................... 36

3.11

Frequency dependence of A and : Bode plots ................................................................... 37

3.12 The high pass filter ................................................................................................................ 40

4 LCR tuned circuits .......................................................................................................................... 41

4.1 Inductance ............................................................................................................................. 41

4.1.1 Electromagnetic Induction ................................................................................................. 41

4.1.2 Self Induction: Inductors .................................................................................................... 42

4.2 Energy stored in an inductor .................................................................................................. 43

4.3

Inductors in series and parallel .............................................................................................. 44

4.4

Transformers ......................................................................................................................... 44

4.5 Transient Response – the RL circuit ..................................................................................... 46

Appendix (to Section 4.5) ............................................................................................................... 47

4.6 Inductive reactance and the circuit .................................................................................. 48

4.7 Frequency response of an circuit ..................................................................................... 48

4.8 Cascaded low pass filters ...................................................................................................... 49

4.9 Resonant circuits ................................................................................................................... 51

4.10 The circuit ...................................................................................................................... 51

4.11 Energy considerations ........................................................................................................... 55

5 Digital Circuits ................................................................................................................................ 57

5.1 Binary Arithmetic.................................................................................................................... 57

5.2 Boolean Algebra .................................................................................................................... 58

5.3

Gates ..................................................................................................................................... 59 5.4 Truth Tables........................................................................................................................... 59

5.5 Basic Electronic Building Blocks............................................................................................ 61




5.6 Higher Level Functions .......................................................................................................... 62

6

Operational Amplifiers .................................................................................................................... 62

6.1 Differential Amplifiers ............................................................................................................. 63

6.2 Negative feedback ................................................................................................................. 64

6.3 Unity Gain buffer .................................................................................................................... 65

6.4 Frequency dependent gain .................................................................................................... 66

6.5

Gain-bandwidth product ........................................................................................................ 67 6.6 Control of gain - the non-inverting amplifier .......................................................................... 68

6.7

The inverting amplifier ........................................................................................................... 69

6.8 Differential feedback amplifier ............................................................................................... 70

6.9 Other op-amp operations ....................................................................................................... 70

6.9.1 A summing amplifier – inverting adder .............................................................................. 71

6.9.2 Integrators/differentiators ................................................................................................... 71

6.10 Positive feedback .................................................................................................................. 72




1 Introduction to fundamentals

Electronics is all about the motion of electrons.

These tiny particles have a property called charge which gives rise to some of the most dramatic andfar reaching behaviour.

Protons also have a charge, equal and opposite to that of the electron, but, by-and-large, they tend tobe bound up inside materials and are not mobile.

Electrons, however, can often find themselves liberated from their parent atoms and they can thenrespond much more freely to electric and magnetic field.

A current, , is then just the movement of charge, .

In circuit theory the electrons move along wires and the current then has a magnitude equal to the

amount of charge per second passing a given point, , with units of Coulombs/sec or Amp(ere)s.

It also has a direction along the wire.

Historically, unfortunately, the direction has been defined as positive for + charges and negative for –charges. Hence the electrons themselves actually travel in the opposite direction to the current.

The movement of the electrons is caused by the presence of forces acting upon them.

Usually these are electrostatic forces due to electric fields set up by the presence of other charges.

A charge, , in an electric field , will experience a force .

Sometimes they are magnetic forces

A charge, , moving with velocity in a magnetic field with experience a Lorentz force

Note: Faraday induction arises through the combined action of these two forces. The chargeswithin a conductor which is formed into a loop of area, , through which a time dependent magneticfield is present will separate causing a potential difference, (sometimes misleadingly called an

electromotive force), which has units of Volts (and not force!).

2 Basic AC/DC Circuits

2.1 Electrostatic Forces, Field and Potential Energy

Coulomb first established that the electrostatic force, , between two charges and separated by

a distance, , follows an inverse square law relationship

|| 2-1

Where is a fundamental constant called the (electrical) permittivity of free space. In SI units the

charges should be in coulombs (1 electron has a charge of 1.60210 C), and

9.0 10Nm2/C2. The concept of a ‘field’ mediating the force can be expressed by saying that

the charge experiences the electric field, , due to . Alternatively it can be said that the charge




experiences the electric field, , due to . These must be equivalent statements and it follows

that the electric field, , due to a charge is

|| 2-2

and that the force on another charge, , which sits in this field, is

2-3

Another useful pair of concepts is that of potential energy and potential. The force equation 1-1 can

be integrated from ∞ to to find out the potential energy, , between the two charges to give

∙

2-4

Potential energy exists between two (or more) charges due to their relative positioning.

However one can also talk about the potential due to a single charge. This is similar to the use of

field (equation 2-2) rather than force, and we can say that the potential energy in equation 2-4 is dueto the presence of charge, , siting in the potential of charge , and vice-versa. This implies the

potential, , due to a charge, , is given by

2-5

and the potential energy due to anther charge, , which sits in this potential is

2-6

Note and are vector quantities, whereas and are scalars. The relationships between them

can be written and V . The potential, , is often referred to as voltage.

2.2 Definition of current

Current, , is the rate of flow of electric charge, :

dt

dq I Ampères (= Coulombs/s) 2-7

The charges can be ions in an electrolyte (like a battery) or a gas discharge (like a fluorescent strip

light) but for most solids it is the movement of electrons that generates current. Each electron carries

a negative charge of 1.60210-19

coulombs and the unit of current, the ampere, is defined as the

passage of 1 coulomb of charge (about 61018

electrons) per second along a metal wire. Such largenumbers are difficult to comprehend but to give you an idea, this number is greater than the age of

the universe in seconds.




2.3 Resistance

2.3.1 Metallic conduction

Figure 2-1: Metallic periodic lattice

Metals are crucial to electronics; they are conducting. Conducting means that some of the electrons

in the material are mobile and are able to respond to an electric field. The material itself is, of course,

still electrically neutral. Solids are held together by bonds linking the atoms and many solids adopt a

periodic spacing or lattice to accommodate the atoms (shown as small black dots in Figure 2-1). One

of the most important metals used in electric circuits is copper. Its lattice is called face-centred-cubic

(fcc) with an atom at each corner of the cube and one in the middle of each face. The fcc lattice

constant, , (edge length of the cube) for Cu is 0.361 nm. Each Cu atom has 29 electrons arranged in

shells (or orbits) and designated by quantum numbers. The first 28 electrons occupy complete shells

which are strongly bound to the copper nucleus. The outermost “valence” electron is not so strongly

bound; it is shielded from the nucleus by all the other electrons. In solid Cu the atoms are so close

together that the outer valence orbits between neighbouring atoms begin to overlap and the valence

electrons cannot be considered to belong to an individual atom. Consequently they are free to move

around and are available for conduction. They are said to be delocalised. These are the electrons that

will move in response to an applied electric field. This is an example of metallic bonding where

positively charged Cu ions are immersed in a sea of electrons, the solid remaining overall electrically

neutral. Using the lattice constant it is possible to calculate the number of free electrons in one m3 of

Cu to be ~ 81028

.

2.3.2 Drift velocity and the origins of resistance

Figure 2-2: Electron motion through conductors

Figure 2-2 shows a length of metal wire with voltage applied between the ends, generating a

uniform electric field along the wire. An electron in the conductor is subject to a force of (where

V

E

q A

vd




is the charge on the electron) where /. is a vector and lies along the direction shown in the

diagram from the positive potential (anode) of the battery to the negative potential (cathode). The

convention for the direction of current is that positive charges move along the direction of . Electrons

have a negative charge and will travel in the opposite direction but the direction of current is the

same.

Applying Newton’s law of motion, , the electron should accelerate in the electric field, but just

as an object falling in a gravitational field reaches a terminal velocity (due to a drag force – air

resistance), so an electron attains a drift velocity, . What causes the drag force which limits the

velocity here? If we were able to see inside the metal the movement of the electrons would be rapid

and apparently random. This is due to them having a relatively large thermal energy, ~, which

manifests itself as a large thermal speed, from , of typically 10

6 m/s. Superimposed upon

this motion is the drift of electrons due to the applied electric field. Since the density of electrons is

very high they will collide with each other or with the Cu nuclei/atoms that make up the wire. Thesecollisions randomise the direction of the electrons and limit the net speed in the direction of the

electric field. This is the origin of Resistance. The energy lost in electron collisions ends up as heat in

the wire.

Figure 2-2 (right) shows a short section of copper wire of cross-sectional area, . When current flows

in the wire, electrons travel along the wire with a drift velocity ms-1

. All the electrons within the

shaded cylinder of length will pass through the wire in one second. The volume of the shaded

cylinder is and the amount of charge contained within it is coulombs where is the number

density of free electrons (per m3) in copper (given previously as 8.510

28 m

-3). Since current is the

rate of flow of charge C/s or amperes (amps). We can use this information to estimate

for a specific case as follows; suppose a bicycle lamp connected to a battery by copper wires 1.5 mm

thick carries a current of 350 mA. Then 350 10 8.5 10 1.6 10 0.75 10

which gives: 1.46 10 ms-1

or 0.0146 mms-1

. This is a surprisingly small value especially when

compared with the electron’s thermal velocity.

Even conductors have some resistance and the value depends on the geometrical shape of the

conductor. The material itself has an intrinsic property called resistivity, .

A wire with a large cross-sectional area has more electrons to carry the current whilst for longer wires

there will be more scattering events as the electron travels through the material. Thus we expect the

resistance of a wire to vary directly with length , and inversely with area , so that /. The

constant of proportionality is the resistivity, , and thus

A

L R

2-8

where the units of are m. The resistivity of some metals, semiconductors and insulators is shown

in the table. The range is huge and reflects the ease with which electrons become delocalised (or how

tightly bound they are to their parent atoms). Copper is the most commonly used highly conducting




metal; we can estimate the resistance of a 1 m length of copper wire of diameter 1.5 mm, using

equation 2-9, to be 0.009. This is a very small value and indicates that copper wire provides very

little opposition to the flow of current and explains why it is commonly used to connect components in

electric circuits.

Electronic circuits make use of resistors with values ranging from ~0 Ω up to 10Ω. These are made

using a number of different technologies and come in various sizes according to how much current

(power) handling is required for the specific application.

Material

(Ω·m)

Silver 1.59×10−

Copper 1.72×10−

Gold 2.44×10−

Aluminium 2.82×10−8

Tungsten 5.60×10−8

Zinc 5.90×10−

Nickel 6.99×10−

Iron 1.0×10−7

Tin 1.09×10−

Platinum 1.06×10−

Lead 2.2×10−

Manganin 4.82×10

−7

Constantan 4.9×10

−7

Mercury 9.8×10−

Nichrome 1.10×10−

Carbon 3.5×10−5

Germanium 4.6×10−1

Silicon 6.40×10

Quartz 751016

Table 2-1: Resistivities for some common materials

2.4 Power dissipated in a resistor

The model developed in section 2.3.2 allows us to work out how much power is dissipated in aresistor.

Power is work done per unit time, and work done is simply force times distance. For each electronpassing through the resistor the work done is

where we have used the fact that the electric field is voltage/distance.

The number of electrons passing through the resistor per second is simply the current divided by thecharge on the electron, /.

Hence power is then




2-9

2.5 Ohm’s lawIn 1827 Georg Ohm showed empirically that for a resistance, , there is a linear relationship between

the current flowing through it, , and the applied potential difference across it, .

This is expressed as

2-10

Ohm’s law (equation 2-10) is probably the most important law in electronics and the one you are most

likely to use. Resistance is measured in and a 1 resistor will carry a current of 1A when a voltage

of 1V is applied across its ends.

Current will only flow if there is a connection around a complete closed circuit, including a battery (or

power supply) as shown in Figure 2-3.

Figure 2-3: A closed circuit for measurement of Ohm's law

The battery has a potential difference between it two ends referred to as its voltage, . Voltage is a

relative quantity; in principle its absolute value is determined by the relative position of all the charges

in the Universe. Hence we always define a zero reference potential point within a circuit and use

symbols like

These are often referred to as ‘ground’ or ‘earth’ points and often in real circuits where it is critical to

properly control the reference point a real physical conducting connection to ground is made so that

the Earth itself provides the reference potential.

Another critical point to realise is that charge is conserved in any circuit. In particular any charge

which flows out of one side of the battery must be able to flow back into the other side. This is true

no matter how complex the other circuitry is.

Charge/current conservation allows us to work out how combinations of resistors behave.




2.6 Resistors in series

R1

V i

0 V

I

R2

Figure 2-4: Simple circuit with two resistors in series

Consider the circuit shown in Figure 2-4. This circuit draws a current from the +ve terminal of thebattery. The same current flows back into the –ve terminal having passed through both resistorsone after the other.

By Ohm’s Law (equation 2-10) the voltage across resistor 1 must be equal to 1.

Similarly the voltage across resistor 2 must be equal to 2.

However the sum of the two voltages across the two resistors must add up to the voltage provided bythe battery as there is no other source of voltage in the circuit; i.e. the battery voltage alone is whatdrives the current.

On the left-hand side is the driving voltage whilst on the right-hand are individual voltages developedacross resistors around a closed loop. In a generalised form to include an arbitrary number of

resistors this is Kirchhoff’s Voltage Law (KVL)

Hence 12 12 2-11

Equation 2-11 just looks like Ohm’s Law again with where is effectively the total resistancepresented by the resistors in series.

We now have 1 2 which can be generalised to any number of resistors in series to give

1 2 ⋯ 2-12

2.7 Resistors in parallel

R1

V i

0 V

I

R2

I

Figure 2-5: Simple circuit with two resistors in parallel

Consider the circuit shown in Figure 2-5. This circuit draws a current from the +ve terminal of thebattery. The same current flows back into the –ve terminal. However on its outward journey at point

A

B




A it has a choice of two paths and so it divides without loss. At point B the two currents mustrecombine. We can express this as

at point A at point B

Where the left-hand side has current flowing into the node and the right-hand side current flowing out.

This is an example of Kirchhoff’s Current Law (KCL) which can be generalised to any number ofconnections to a single node.

By Ohm’s Law (equation 2-9) the current through resistor 1 must be equal to /1.

Similarly the current through resistor 2 must be equal to /2.

However the sum of the two currents through the two resistors must add up to the current drawn fromthe battery as there is no other source of current in the circuit.

Hence

/1/2 2-13

Equation 2-13 can be rearranged to look like Ohm’s Law with

where is effectively the total resistance presented by the resistors in

parallel.

We now have

which can be generalised to any number of resistors in parallel to give

⋯ 2-14

The parallel arrangement of resistors in the circuit in Figure 2-5 is sometimes referred to as a currentdivider in that a fraction of the current goes through each resistor. Combining equation 2-13 with theKCL relation at node A (or B) gives

2 1 2

and 2-15 1

1 2

2.8 The voltage divider circuit

In Figure 2-4 we saw a circuit with two dissimilar resistors. This is redrawn in Figure 2-6 to emphasise

its use as a voltage divider in which the voltage across R2 is a fraction of that produced by the driving

source.

KVL gives:

– – 0 2-16




Figure 2-6: Potential divider with unequal resistors

We can find the current I from Ohm’s law since we know that the total resistance of the circuit from is

21 R R Rtotal

So

21 R R I

and the voltage across can be found by substituting for in to get

21

12

R R

R IR

= 21

2

R R

R

2-17

This circuit is a voltage divider circuit since an appropriate value of can be used to provide any

voltage from 0 to the battery voltage .

2.9 Voltage/Current Sources

2.9.1 An ideal voltage source

An ideal voltage source, as already assumed in Figure 2-3, produces a fixed output voltage

independent of the ‘load’ resistor which is drawing current from it.

This implies an ability to provide an infinite amount of current if the load resistor is reduced to zero Ω.

In practice this is not realistic

2.9.2 An ideal current source

In some circumstances it is useful to drive a circuit with a constant current source rather than a

constant voltage source. An ideal current source would drive the same current through the load

regardless of its resistance. If the resistance is arbitrarily large this implies that the voltage across the

load will become arbitrarily high. In practice this again is not realistic

2.9.3 Batteries

Alessandro Volta showed that a potential difference was created across a series of copper and zinc

plates separated by cardboard soaked in brine (an ionic solution). Ion migration is the basis of most

modern batteries. The potential difference is generated as the metal ions are released or collected

into the ionic solution causing a separation of + and - charges. A car battery consists of lead/lead




oxide electrodes immersed in H2SO4. These batteries are able to deliver 12V at very large currents to

power the car starter motor, lights and ignition plugs. Such a battery cannot last forever and the

electrodes deteriorate due to the chemical reactions. Fortunately this can be reversed by continuously

recharging using an alternator driven by the fan belt. A modern ‘zinc carbon’ battery or “dry cell”

consists of an anode (carbon), a cathode (zinc) and an acidic electrolyte manganese oxide (chloride)paste. These batteries are gradually being replaced by alkaline types which use potassium hydroxide,

KOH as the electrolyte and deliver 1.5V. Lithium-ion batteries are more sophisticated (they use

organic electrolytes) and are used to power laptops, mobile telephones, kitchen scales, watches and

many other devices. None of these is rechargeable.

To maintain a potential difference at the battery terminals requires the movement of ions within the

electrolyte. This comes at a cost: there is opposition to the ion flow which manifests itself as an

internal resistance.

2.9.4 Real voltage and current sources

2.9.4.1 Voltage sourcesIn section 2.9.3 the concept of an internal resistance was introduced. This turns out to be a very good

way of representing the way in which real power sources differ from the ideal. The concept for a

voltage source is illustrated in Figure 2-7. The voltage source is enclosed in the dashed box and has

an internal resistance in series with the voltage, V, and it has a value, RS. The presence of this

internal resistance means that even if the load resistance, RL goes to zero, the current drawn is

limited to /. The load resistor RL represents a generic way of representing any resistor or

circuitry attached to the power supply.

Another, unfortunate effect of the internal resistance is that the circuit now looks like that of a potential

divider in Figure 2-6, with and which means that the voltage which actually is

presented across the load is reduced to

. Not only is this lower than V but it also depends on

the load resistance.

Figure 2-7: A real voltage source with an internal resistance, RS




2.9.4.2 Current sourcesIn section 2.9.3 the concept of sources having an internal resistance was introduced. This turns out

to be a very good way of representing the way in which real power sources differ from the ideal. The

concept is illustrated for a current source in Figure 2-8. The current source is enclosed in the dashed

box and has an internal resistance in series with the ideal current source, V, and it has a value, RS.

The presence of this internal resistance means that even if the load resistance, RL goes to infinity, the

current can still be driven through RS with a voltage limited to . The load resistor RL

represents a generic way of representing any resistor or circuitry attached to the power supply.

Another, unfortunate effect of the internal resistance is that the circuit now looks like that of a current

divider as shown in Figure 2-5, with and which means that the current which actually

flows through the load is reduced to

. Not only is this lower than I but it also depends on the

load resistance.

Figure 2-8: A real current source with an internal resistance, RS. Alternative symbols forcurrent sources are shown in the lower panel.




2.9.5 Multiple sources

Figure 2-9: Multiple voltage sourcesFigure 2-9 shows a circuit containing two voltage sources connected in series. Recalling that the +

sign indicates the anode of the voltage source; the conventional current direction is clockwise for

and anticlockwise for . Thus the net current in the circuit will depend on the relative values of

and . Let us assume . We can apply KVL to this circuit:

0)( 2121 R R I V V

This shows that voltage sources in opposition subtract to produce a net voltage ( ). The

corollary is that voltage sources acting in the same direction add. Essentially this analysis is using the

principle of superposition. This method can be extended to mixed sources (voltage and current) and

calculates the effect of each source (voltage or current) one at a time, replacing the voltage (current)

sources by a short (open) circuit and analyses the circuit using KVL and KCL. The contributions from

the sources are then added. The principle of superposition is not limited to electric circuits but applies

to many other areas of physics: optics, acoustics, vector fields and is the basis of Fourier analysis.

Example

Here we give a worked example of a circuit with one voltage and one current source as shown in

Figure 2-10

Figure 2-10: Superposition of voltage and current sources




We shall analyse the circuit in Figure 2-10 to find the voltage across the 2k resistor and the current

through it. We begin by separating the two sources and replacing the voltage source with a short

circuit and the current source with an open circuit as shown in Figure 2-11.

Figure 2-11: Separating the circuits

The circuit shown on the left is a current divider so we see that:

mA13

21

1'

I

The circuit on the right is a voltage divider and we can simply calculate the voltage drop across the

2k resistor to be

V4621

2

out V . Note that this is the contribution to Vout from the voltage

source. We can add to this the contribution to Vout from the 1mA current supplied by the current

source of 1mA2k = 2V. Note that the current travels from the top to the bottom of the 2k resistor

so we add to find 24 6V. Alternatively we can calculate the current I in the right hand

circuit. This is found from Ohm’s law to be

mA210

21

6 3

. Both sources drive current in the

same direction so that the voltage drop across the 2k resistor is (I´+I´´)2103

= 6V and I=(I´+I´´) =

3mA.

2.10 Equivalent Circuits

Electronic circuits are frequently considered to be “black boxes” which means we can avoid the

details of the circuit and characterise it based on measurements that we can make at the input or

output terminals. When this is done we find that most circuits can be reduced to a combination of an

ideal voltage (current) source in series (parallel) with a resistor called a source resistance. These are

called equivalent circuits.

An equivalent circuit comprising an ideal voltage source in series with a source resistance is called a

Thévenin equivalent circuit. The Thévenin equivalent circuit has /

Thévenin’s theorem states: From the point of view of measurements made at its terminals a linear

circuit consisting of voltage sources, current sources and resistances is equivalent to an ideal voltage

source equal to the open circuit voltage of the linear circuit, in series with an resistance (source or

output resistance) whose value equals the open-circuit voltage divided by the short-circuit current.




An equivalent circuit comprising an ideal current source in parallel with a source resistance is called a

Norton equivalent circuit. The Norton equivalent circuit has (same as for Thévenin)

Norton’s theorem states: From the point of view of measurements made at its terminals a linear

circuit consisting of voltage sources, current sources and impedances is equivalent to an ideal current

source equal to the short-circuit current in parallel with an impedance (source or output impedance)

whose value is equal to the open-circuit voltage divided by the short-circuit current.

2.10.1 Finding equivalent circuits

This section will show how equivalent circuits can be deduced for relatively simple circuits. Our first

example is shown in Figure 2-12.out V indicates the voltage at the output terminals of the circuit. Since

this is a voltage divider circuit we can write:

inout V R R

RV

21

2

Figure 2-12: Potential divider circuit

If we make a measurement ofout V using a high resistance (>10 M) voltmeter (equivalent to using a

large load resistance) this would yield the open circuit voltage,OC V ,

inOC V R R

RV

21

2

2-18

In order to complete our Thévenin equivalent we need to find

. Imagine placing a wire across the

output terminals: no current would flow through and will flow along the short circuit as indicated

in Figure 2-12:

Then

1 R

V I in

SC 2-19

Using / and substituting from equations 2-18 and 2-19 gives:

21

211

21

2

R R

R R

V

RV

R R

R

I

V R

in

in

SC

OC S

2-20

Then the Thévenin equivalent looks like this:




Figure 2-13: Thévenin Equivalent of Figure 2-12

Although this method is relatively straightforward there is another way to determine : find the value

of the resistance that would be seen if you were to ‘look’ into the output terminals. An ideal voltage

source must have zero internal resistance (we saw this when discussing superposition) so is

replaced by a short circuit and looking into the output terminals we see that is the parallel

combination of and :

21

111

R R RS

giving a source resistance of

21

21

R R

R R RS

in agreement with the earlier value calculated using

and

The circuit shown in Figure 2-14 is a little more complicated but the method is the same:

Figure 2-14: Current source with multiple resistors

First find using : note that the circuit consists of a parallel combination of and




Apply the current divider rule: the current through the combination is

in R R I R R R

R I

321

1

32

and the voltage drop across the resistor is

in R R I R R R

R R

R I 321

13

332 This is .

Now find : short circuit the output terminals; this means that no current will flow through

and the circuit now simplifies to a parallel combination of and

Apply the current divider rule to obtain the current flowing through : this is and it equals

in I R R

R

21

1

Finally, recalling / gives:

321

213 )(

R R R

R R R RS

And the Thévenin equivalent circuit is shown in Figure 2-15.

Figure 2-15: Thevenin Equivalent to Figure 2-14

Again, it is possible to use the alternative method of determining by evaluating the total resistance

looking in from the output terminals. In this case since we have a current source, we must replace it

with an open circuit. Then the total resistance is derived from a parallel combination of and

which gives

321

213 )(

R R R

R R R RS

as before. In solving this problem it may seem more sensible

to derive the Norton equivalent circuit given that we started with a circuit that contained a current

source. The result is shown in Figure 2-16.




Figure 2-16: Norton Equivalent of Error! Reference source not found.

A more complicated circuit containing two sources is shown in Figure 2-17.

Figure 2-17: Circuit with multiple sources

This is best approached using the principle of superposition to evaluate the contributions to the open

circuit voltage, and the short circuit current from each source in turn. The diagram below

shows this:

Figure 2-18: Multiple sources separated

For the circuit containing only the voltage source the circuit is reduced to a voltage divider. Then

V618

3015

15

OC V

Turning to the circuit on the right containing only the current source this drives 1mA through the

parallel combination of the 30k and 15k resistors which is equivalent to a single 10 k resistor.




Then the voltage across AB, now labelled OC V is given by 1mA 10k =10V. Accordingly the value

of the open circuit voltage including both sources is given by V16 OC OC OC V V V .

To find the source resistance we have two options: find the short circuit current in each case above

and apply (5.2) or adopt the method presented in the previous example of replacing each source by

an open circuit (current source) or a short circuit (voltage source) and calculating the total resistance.

This is the easier method and the resulting circuit is shown above. is the resistance you would

measure by placing an ohmmeter across points A and B. This is the parallel combination of 30k and

15k resistors giving the source resistance as 10k.

2.11 Impedance matching

When connecting different electrical circuits the equivalent circuit concept is very useful. For instance,

consider an audio system for a concert hall or recording studio. A microphone generates a voltage

that is proportional to the sound produced by instruments/voices. The measured voltage is usuallyvery small and must be amplified before connecting the output of the amplifier to a loudspeaker which

then transforms the electrical signal back into one that we can hear. Connecting the

microphone/amplifier/loudspeaker to make a complete audio system is made much easier if each

individual sub-system can be considered as an equivalent circuit.

The main problem is “impedance matching” (we will discover more about impedance later –

essentially impedance takes account of capacitances and inductances which make the “resistance”

change with frequency). This involves designing the input impedance of a subsystem and the output

impedance of the signal source in order to achieve maximum power or maximum voltage transfer. We

have just seen that equivalent circuits are an elegant way of simplifying circuits to voltage (current)

sources and series (parallel) source resistances. The microphone can be considered as a Thévenin

circuit comprising an ideal voltage source in series with a source resistance . A high quality

microphone will have an output impedance (or source resistance) of 600. The output voltage of a

microphone is usually very small (0-100 V) and this has to be amplified by about 70dB before it can

be output to a loudspeaker. In this case the object is to maximise voltage (and not power) transfer

from the microphone to the amplifier. If we consider the amplifier to be a “black box” with input and

output terminals, maximum voltage transfer is achieved by having an amplifier input impedance that is

much larger than the output impedance of the microphone (think voltage divider). At the output sideof the amplifier, loudspeakers typically have input impedances of 8 and in this case it is desirable to

have maximum power transfer from amplifier to loudspeaker. To cover both requirements the audio

amplifier will be designed to have a very large input impedance (M) and a relatively small output

impedance (a few ) to fulfil the criteria specified above. We will return to this later when we look at

operational amplifiers.

2.12 Alternating Current

So far we have looked at direct current (DC) circuits where the driving voltage is produced by

chemical means in a battery or a mains-operated DC voltage generator using rectification. Alternating




current (AC) is used in a wide range of applications. For many applications we can consider the

voltages and currents to vary sinusoidally.

2.12.1 AC signals

Figure 2-19 (left) shows a voltage waveform

sin where

is the amplitude

is the period of the wave measured in seconds

is the frequency (=1/T) and is measured in s-1

(Hz)

Figure 2-19: Alternating voltages

The graph can also be drawn as a function of angle rather than time, in which case the wave

repeats every 2 (Figure 2-19 (right)), is the angular frequency equal to 2 / (= 2 ) measured in

rad/s.

A second voltage sin ∅ is represented by the dashed line in Figure 2-19 (right). The

negative sign indicates that this wave is retarded in phase relative to the original sine wave, so the

peaks occur later in time. A negative value of ∅ therefore corresponds to a phase LAG. Conversely a

positive value of ∅ indicates a phase LEAD. A cosine wave drawn on the same graph would reach its

peak value /2 radians before the sine wave so cos sin /2. Note that phase is relative;

we could have shifted the origin to coincide with the zero of the dashed line wave in which case the

full line wave would then be given by sin ∅. Consequently when analysing circuits we

can choose any voltage or current as our reference.

For circuits which only contain resistive elements with only one sinusoidal drive voltage all

phase values are zero and currents are in phase with voltages.

2.12.2 Electrical Power in AC circuits

In section 2.4 it was shown that the electrical power dissipated as heat when a direct current flows in

a resistor is / or and this value is independent of time. Since an AC wave is

constantly changing in magnitude (its amplitude is a fixed quantity) the instantaneous power in a

resistor carrying AC current is:




R

V P

2

or R I P 2

where the current is cos and the voltage is cos. The average power measured

over many cycles is:

t R I t R I Pav 22

0

22

0 coscos

where the bracket signifies time averaged (or mean) value: t I 22

0 cos is the mean square

current. and are amplitudes and have no time dependence. Since the waveform repeats every

period, integrating over the period and dividing the answer by gives the time averaged or mean

square value of t 2cos over one cycle:

2

12cos1

2

11cos

1cos

0 0

22 dt t T

dt t T

t

T T

and so:

2

2

02 R I R I Pav

This looks like the DC case divided by a factor of two. We can also write R I R I

rms

22

0

2 where the

square root of the time averaged current yields the root mean square (rms) value of , where

0

2

0 7071.02

I I I rms

Similar arguments can be applied to the AC voltage to give /√ 2 and we have a new definition

for the average power of . AC voltmeters and ammeters measure rms values

(otherwise the values would oscillate wildly). An alternative and more intuitive method is to compare

graphs of the cos and cos functions as shown in Figure 2-20: Time average of ac waveforms;

cos averages to zero over a period but cos has twice the frequency and is always positive. The

dotted line shows that the time averaged value is equal to half the amplitude.

Figure 2-20: Time average of ac waveforms

The UK domestic supply is 240V: this is actually the rms value; the amplitude is 340 volts or 680 V

peak-to-peak.




3 RC Filters At the end of the last section we noted that for resistors AC signals can be treated more or less like

DC signals and that there is no frequency dependent behaviour involved. However, there are two

other electronic components that do exhibit a frequency dependent behaviour and one of the mostuseful applications of this is in filter circuits. These can be used to enhance otherwise weak signals

whose frequency is known but would remain buried in noise without the use of filters. In this sectin we

will look at the first of these components, the capacitor

3.1 Capacitors

Capacitors are components that can be charged by connecting them to a voltage source such as a

battery and they are able to store electrostatic energy. This energy can be released rapidly as

happens in a camera flash or pulsed laser applications. The time taken to fully charge and discharge

depends on the size of the capacitor and any resistance in the circuit. This chapter will describe theconstruction of capacitors and their transient response during charging and discharging. Later we will

describe the response of capacitors to alternating current, AC waveforms and show that they can be

described with a complex impedance as the voltage and current are not in phase.

3.1.1 Capacitors: design and construction

A capacitor consists of two conducting (metal or metal foil) electrodes with air or a dielectric

(insulating) material separating the two.

Figure 3-1: Parallel plate capacitors with air (left) and with dielectric (middle)

The left part of Figure 3-1 shows a schematic of an air filled capacitor indicating the electric field

which is generated by the presence of charge or – on the electrodes (plates). It can be shown

that the charge on the plates is proportional to the voltage applied and hence:

3-1

The proportionality constant is called the capacitance and is measured in Farads (or Coulomb2/

Joule) in honour of Michael Faraday. In the E&M course you will see how to apply Gauss’s law to

show that the electric field between the plates is given by




0

E 3-2

where is the permittivity of free space and has the value 8.8510-12

Fm-1

(a constant that is related

to the speed of light); is the charge density on each plate and is equal to

/ where

is the area of

the plates (the use of for charge density is unfortunate and should not be confused with

conductivity). Combining equations 3-1 and 3-2 and recalling that the electric field between the plates

isd

V E

gives an expression for the capacitance:

d

AC 0 3-3

The Farad turns out to be a very large unit. For instance, metal plates having an area of 1m2 and a

separation of 1mm would have a capacitance of only 8.85 nF. Although this is not an atypical value of

the inclusion of plates 1m2 in a circuit would be inconvenient. The photograph on the right of Figure

3-1 shows a much larger capacitor having a value of 150F yet with an area of only a few cm2. The

plates have been replaced with thin aluminium foil separated by a dielectric material rolled up into a

compact geometry. The middle panel of Figure 3-1 shows the situation when the capacitor is filled

with such a dielectric material. Dielectrics are insulating materials such as mica, plastics or glass

which do not conduct electricity (thus maintaining the charge on each plate). These materials are at

the opposite end of the scale to metals in terms of their resistivity and this is a consequence of the

electrons being tightly bound to their parent atoms. So there can be no conduction through the

dielectric material (as indeed there could not be across a capacitor filled with air). However, when a

dielectric material is subjected to a large electric field there is a separation of the outer electrons

relative to the nucleus forming a dipole. This is indicated in Figure 3-1; note that the ions are fixed and

only the outer electrons move towards the positively charged plate. Since the dielectric consists of

many atoms, the net effect is to produce a layer of positive and negative charge at each surface. The

dielectric therefore sets up an internal electric field which opposes that generated by the plates. Thus

the amount of charge that can be stored on the plates for the same applied voltage is increased by

the presence of the dielectric. This results in an increase in the value of by a factor known as the

dielectric constant (also known as the relative permittivity) so that equation 3-3 becomes

d

AC 0

You will also find the product referred to as the permittivity of the material. Values of vary

from 2-3 for glass and many plastics, 13 for Si and 88 for water. Thus the way to obtain reasonable

values of capacitance is to have large area plates separated by a very thin dielectric layer having a

high dielectric constant.

Aside: dielectrics and light

The dielectric constant has a greater significance than its ability to increase capacitance: it is related

to the refractive index of a material. When an atom in glass interacts with light the sinusoidally varying




electric field of the light wave produces a time varying dipole moment. The magnitude of the dipole

moment will depend on the detailed nature of the bond between the electron and its parent atom and

also the frequency of the light, since the bonds will have a resonant frequency; for instance at very

high light frequencies the electron may not be able to follow the rapidly changing electric field. When

the frequency of the light matches the resonant frequency of the atom/electron bond the displacementis large; when an electron oscillates with large amplitude it emits light and so certain light frequencies

are able to travel through the glass. We know that this happens in the visible part of the

electromagnetic spectrum. This is the origin of the refractive index of a material and for a material

such as glass the refractive index √ . At non-resonant frequencies there will be absorption of the

light and the refractive index can be represented as a complex quantity.

3.2 Capacitors in series and parallel

Figure 3-2: Capacitors in series (left) and parallel (right)

Figure 3-2 (left) shows two capacitors in series. The voltage across is

1

11

C

QV whilst that across

is

2

22

C

QV . Now consider the wire that joins and . At one end there is a charge and at the

other end a charge . Since the wire connecting and has a high conductivity, charge will flow

freely until . Kirchhoff’s voltage law tells us that the total voltage across the two capactors is:

Assuming that the two capacitors can be replaced by a single capacitor of value , where

total

totaltotal

C

QV and substituting for the voltages gives:

2

2

1

1

C

Q

C

Q

C

Q

total

total

but giving:

21

111

C C C total

3-4

Figure 3-2 (right) shows a parallel combination of the capacitors. In this case the voltage across both

components is the same which means that the charge on each must be different; so

;




and the total capacitance . Since we must also have this leads to the

rule for capacitors in parallel as

3-5

3.3 Energy stored in a capacitorWhen a capacitor is connected to a battery, charge is transferred to the plates (current flows). This

charge generates a voltage between the plates which eventually becomes equal and opposite to the

battery voltage and no further current flows. At this point the capacitor is fully charged and the work

done in transferring the charge to the plates is stored as electrostatic energy in the capacitor.

Work Done =

where is the final charge on the plates. Now consider the situation when the capacitor is partially

charged: a small amount of work is required to add a small amount of charge to the plates

according to:

Since / gives dqC

qdw . To find the total work needed to charge the capacitor completely

from its uncharged state we integrate:

f Q

f

C

Qdqq

C W

0

2

2

1

This work is equal to the change in PE (equals the PE of charged state minus PE of uncharged state)

and defining the PE of the uncharged capacitor to be zero gives

22

22CV

C

QU F 3-6

where is the final voltage across the capacitor when fully charged. Using equation 3-3 we obtain

d

Ed AU

2

)( 2

0

where the electric field between the plates isd

V . The electric field is contained within the volume

and so the energy density within the capacitor is2

2

0 E . This gives some insight into the way a

capacitor operates in a circuit. A frequently used analogy is to compare the charging with the

compression of a spring. When the force is removed the stored PE is converted to KE as the spring

expands. In a capacitor the PE is stored in the electric field. When the voltage source that pushes the

charge on to the capacitor plates is removed the capacitor will discharge and the stored PE is

converted back to a current flowing in the opposite direction. If the circuit contains a resistor the

electrostatic PE will be dissipated as heat as discussed in the following section.




3.4 Charging/discharging behaviour

Figure 3-3: Capacitor circuit with switch

For the circuit shown in Figure 3-3, switch is open and there is no voltage across or charge on the

capacitor plates. When is closed a current (lower case symbols indicate time-varying quantities)

starts to flow around the circuit charging the capacitor. Before analysing this in detail some predictions

about the way the circuit behaves can be made:

Immediately after is closed the capacitor is uncharged and sinceC

qV C the battery

voltage is dropped across and the current /.

After S has been closed for some time, the capacitor will be fully charged (negative charge

on the lower plate and positive charge on the upper plate) and will be equal to but

opposite in direction, and the current will reduce to zero.

To find out what happens at intermediate times we apply Kirchhoff’s voltage law:

– – 0 3-7

WhereC

qV C and iRV R are the instantaneous values of voltage across the capacitor and

resistor. Rewriting for the instantaneous current i

RC

q

R

V

i

0

3-8

At time 0, 0 and the initial current is / (this is the value that we would obtain if the

capacitor were absent). As current begins to flow around the circuit the capacitor begins to charge

(you can think of the capacitor as a battery whose emf gradually increases until it matches ). Now

0 and

RC

Q

R

V F 0




where is the final charge on the capacitor. Note that does not depend on . Returning to

equation 3-8 and writingdt

dqi we have:

RC

q

R

V

dt

dq

0

3-9

This differential equation can be solved when we apply the initial and final conditions just discussed to

give:

RC

t

F RC

t

eQeCV q 110 3-10

Recalling that / yields an expression for the voltage across the capacitor as a function of time:

RC

t

C eV V

C

q10

Alternatively we can find an expression for the current in the circuit by combining equations 3-9 and 3-

10 to get:

RC

t

RC

t

e I e R

V i

dt

dq 0

0

Figure 3-4: Voltage and current as function of time

Figure 3-4 plots and as functions of time. Since and are fixed values for the circuit, the

product is known as the time constant of the circuit. When ,




1

0 1 eV V C

which corresponds to 63% of the applied voltage. Since the increase in capacitor voltage is

exponential never quite reaches but the capacitor is almost fully charged after ~5 and current

flow becomes extremely small. A useful yardstick for circuits like this is the time taken for to

increase from 10% to 90% of . This is called the rise time and it corresponds to 2.2 (check this

with equation 3-10). Thus the flow of current in an circuit is a short transient and a capacitor will

effectively (after it is charged) block a DC signal.

Now consider the case where is opened, the battery removed and replaced with a short circuit and

reopened. Charge can now redistribute throughout the circuit and the capacitor will discharge.

Current flows through in the opposite direction and the electrostatic energy stored in the capacitor

will be dissipated as heat generated in the resistor. Initially the capacitor is fully charged, the voltage

between the plates is , and this value will decrease as charge leaks away. Returning to equation 3-

8 and setting to zero (the battery has been removed) results in RC

qdt dqi which has a

solution: RC

t

F eQq

. This leads to an expression for the capacitor voltage RC

t

C eV V

0 and the

current:

RC

t

e R

V i

0

Figure 3-5: Capacitor charging/discharging

Figure 3-5 shows the behaviour of the capacitor voltage and circuit current when is closed

(charging) and then reopened with the battery removed (discharging). Note that the current is in the

opposite direction to that shown in Figure 3-4 and zero time has been redefined. After seconds

has reduced to 37% of its initial value whilst has increased to 63% of its final value.

This analysis was performed entirely as a function of time and is known as time domain analysis. In

the coming lectures we shall look at frequency domain analysis where the behaviour of circuits is




analysed over a wide range of frequencies. Time domain analysis and frequency domain analysis are

not independent and Fourier analysis allows us to find the behaviour of a circuit in the time domain,

for instance, using knowledge of the behaviour in the frequency domain.

3.5 The RC filter circuit using a time domain analysis

Figure 3-6: The circuit of Figure 3-3 driven by AC

We have earlier analysed the transient behaviour of an circuit. Figure 3-6 shows this circuit again

but now it is driven by an AC voltage, . First we will look at the response to an AC voltage from

each component. A resistor limits the current flow depending on the applied voltage. It does not

distinguish between the directions of current flow and we can write Ohm’s law as: I

V R R where

and are the instantaneous values of voltage and current for the resistor. Despite the constantly

changing values of and the ratio remains fixed since the two always remain in phase. Thus

I

V R R is a fixed value equal to the ratio of the amplitudes of the voltage and current and we ignore

the cosine term. Note that in a series circuit the current through all components must be the same and

so we do not require a subscript for the current through the resistor.

The defining equation for a capacitor is (where and are the instantaneous values of

charge and voltage associated with the capacitor). Differentiating this equation defines the current

flowing to and from the capacitor during each AC cycle as:

dt

dV C

dt

dqi C

C

If the current in the circuit is t I iC cos the voltage across the capacitor is:




2cossin

cos

t

C

I t

C

I

C

t I V C

This shows that the capacitor current and voltage are not in phase: the voltage LAGs the current by

/2 (equivalently, the current LEADs the voltage by /2). In analogy to Ohm’s law we can write

|||| which defines the equivalent of the ‘resistance’ for the capacitor:

C C I

I

I

V X

C

C

C

C

c

1

is known as the capacitive reactance; has units of . Reactance is a measure of the opposition

to the flow of AC by the capacitor and is frequency dependent

for high frequencies , is small and the capacitor acts like a short circuit

at low frequencies 0, tends to infinity and the capacitor acts like an open circuit (it does

not pass DC current).Figure 3-7 shows a plot of the voltage across the resistor, , the voltage across the capacitor

2

cos

t C

I V C and (Kirchhoff’s law). is in phase with the current

t I I cos0 , lags the current by /2 and lags the current by .

The value of will depend on the values of and and the latter will vary with the frequency of the

driving voltage. Applying a graphical method such as this to determine voltages and phase shifts is

extremely tedious. In fact it is much more likely that a problem will specify and ask for the value of

the current.

Figure 3-7: Voltage-current relationship across capacitor




3.6 Complex Voltages and Currents

Previously we showed that for AC analysis we need to represent sinusoidal voltages and currents

with arbitrary phase relationships between them and find mathematical rules for allowing a relatively

simple method of analysing AC circuits. Each signal will have a magnitude and a phase. This

suggests representations either in phasor form or as a complex number in rectangular

form . The complex form turns out to be particularly elegant and powerful and this will

be developed here.

[Note: it is conventional to use j rather than I when complex analysis is used in electronics]

Consider a voltage represented by . The projection of onto the Real axis is

cos. This is shown in Figure 3-8. A second voltage is also shown which has the

same amplitude but is rotated anticlockwise about the origin by a phase difference of /2. This

would correspond to a voltage which leads by /2. On the Argand diagram this rotation is

equivalent to multiplication by where 1. Then /2 or

–. Alternatively we can use cosV a and sinV b and we

now have a third way of representing the vector: )sin(cos jV ; this is the trigonometric

form. Finally, invoking Euler’s theorem jeV we have the exponential form which completes

the different ways of representing the voltage .

Figure 3-8: Sinusoidal voltages with different phases shown on an Argand diagram

The projection of onto the Real axis is cos as before. A second phasor is

shown which has the same amplitude but is rotated anticlockwise about the origin by /2. This




would correspond to a voltage which leads by /2. On the Argand diagram this rotation is

equivalent to multiplication by where 1. Then /2 or

–. Alternatively we can use cosV a and sinV b and we

now have a third way of representing the vector: )sin(cos jV ; this is the trigonometric

form. Finally, invoking Euler’s theorem jeV we have the exponential form which completes

the different ways of representing the voltage . In the following rectangular representation will be

used.

3.7 Resistance becomes impedance

Figure 3-9: Phase diagrams showing the relationship between voltage and current for varioustypes of component. The left shows resistance. The right-hand diagram is for a capacitor.The centre diagram will be discussed in detail in the next section.

For an AC voltage, resistance is given by the instantaneous value of I

V . For instance, if

t V V cos0 and t I I cos0 then / where the phasors 0 and / 0 both

lie along the real axis as shown in Figure 3-9 (left). Thus / is a fixed value since and

are always in phase.

Figure 3-9 (middle) shows the current and voltage phases for an inductor. If the current flowing

through the inductor is t I I cos0 then the voltage leads the current through the inductor by /2

and is given by

2

cos0

t L I V

Las will be shown later. A phase lead of

/2 means the

voltage lies along the positive imaginary axis and has an amplitude . Taking the

complex ratio of the amplitudes of the / we obtain . This has the magnitude of the

inductive reactance but the presence of the term tells us that the voltage and current are not in

phase. is a phasor quantity and is called impedance.

Finally Figure 3-9 (right) shows the phase corresponding to the voltage and current across a

capacitor. Following the earlier reasoning if a current t I I cos0 flows to and from the capacitor

terminals then the voltage across the capacitor is

2cos0

t C

I

V C and the voltage lags the




current. In Figure 3-8 this is shown as the capacitive reactance multiplied by – to take account of the

phase shift. Then 1/ .

Using to write the impedance of a capacitor or inductor in a compact way may not seem particularly

useful but this apparently trivial step leads to a method where we can calculate voltages, currents,

impedances and circuit gain algebraically in a very elegant and straightforward way.

3.8 Complex impedance

From the preceding discussion resistance is a special case of impedance in that voltage and current

are always in phase and its value is independent of frequency. However we must not overlook the fact

that the values of capacitive and inductive reactance are frequency dependent and hence their

impedance will vary with frequency. Earlier we derived the rules for finding the total resistance for

series and parallel of resistors. In fact the same rules apply to impedances so that:

For series impedances ⋯

For parallel combinations ....111

21 Z Z Z total

As an example a parallel combination of capacitors will have:

total

total

C jC C j

C jC j Z Z Z

21

2121

1

1

1

1111

3.9 Complex analysis of the filter circuit

The circuit shown in Figure 3-10 should be very familiar to you. The only change is that we label the

voltage across the capacitor as . We will analyse this circuit in the frequency domain, that is,

determine (or strictly /) over a wide range of frequencies. Provided the frequency of is

not too high (say in the GHz range) the resistor will behave as an ideal linear component with a

constant value of . Previously we showed that the impedance of a capacitor is 1/ . Treating

the circuit as an impedance divider

in

C

C out

V Z R

Z V

Figure 3-10: The circuit again




When dealing with passive circuits like this it is common to use the voltage ratio /. This is also

known as the voltage transfer ratio or more generally, the gain, . It may seem odd to use the term

“gain” since we know that can never exceed making less than unity but in electronics the

term is used quite generally for all circuits. Later we will discuss amplifiers and here / can be

much greater than unity. Note also that is a complex operator which acts on the complex

representations of and , i.e. and so is a quantity having amplitude and phase.

Substituting for :CR jC j R

C j A

V

V

in

out

1

1

/1

/1 3-11

Where the presence of the term indicates a phase change between and . We can rationalise

equation 3-11 by multiplying the numerator and denominator with the complex conjugate of the

denominator 1 to give:

222222

11

1

RC

CR j

RC V

V

in

out

3-12

which gives the rectangular form of with2221

1

RC ja

and

2221 RC j

CRb

. If we

were performing an experiment in the laboratory we would measure and and any phase

change between them using an oscilloscope. In doing so we would be measuring the modulus and

angle of equation 3-12. These are given by:

2/1222

2/122

)1(

1)(

RC ba

V

V

in

out

and CRab /tan 3-13

So we have achieved in three lines of algebra as much as we could by drawing a graph (only for a

single frequency) or a phasor diagram.

3.10 Bode Plot - power ratios and decibels

Our ears and eyes do not have a linear response to sound or light intensity. This is expressed in the

Weber-Fechner law which states that human perception has a response which is logarithmic.

Consequently it is common to measure the power or intensity of sound or light sources relative to a

prescribed level and express this as a logarithm using units called “bels”. This originated from a need

to quantify the reduction in audio levels in telephone circuits. The link with telephony or sound

intensity resulted in these dimensionless units being named in honour of Alexander Graham Bell,

inventor of the telephone. As an example Figure 3-11 shows a schematic of an optical fibre used to

carry digital telephone signals. On the left hand side is a “bit” of information consisting of an optical

pulse of intensity Pin. As the light pulse travels along the fibre it is absorbed and scattered losing

intensity (this occurs over hundreds of km) so that the emerging light pulse may have a much smaller

intensity, Pout shown on the right hand side.




Figure 3-11: Optical pulse attenuation along a fibre

This reduction can be expressed as the dimensionless ratio Pout/Pin but engineers prefer to express

such ratios as bels:

belsP

PratioPower

in

out

10log 3-14

The bel turns out to be a fairly large unit and decibels (=0.1 bel, abbreviated dB) are more commonly

used so the power ratio is dB

P

P

in

out

10log10 3-15

Decibels can be applied to power or intensity ratios of any physical quantity and are frequently used in

electronic engineering. We showed in the previous lecture that the electrical power dissipated in a

resistor is given by /. If the voltage across the resistor is increased from V1 to V2

then the increase in dissipated power will be:

dBV

V

V

V ratioPower

1

2102

1

2

210 log20log10 3-16

Note that if V1 < V2 the result will be a positive number, if V1 > V2 it will be negative whilst if V1 = V2 it

will be zero. The benefits of these units are:

(i) a large range of values can be represented by small numbers

(ii) systems which consist of many components such as amplifiers may increase the voltage

with each stage of amplification (positive values of dB) and the total amplification is the

sum of all the individual amplifier gains expressed in dB

We will use dB units when we look at the frequency response of filter circuits.

3.11 Frequency dependence of A and : Bode plots

The modulus of / or gain is plotted in Figure 3-12 over a wide frequency range for values

166 and 300nF. As predicted, the gain is close to unity for low frequencies dropping to

about 0.2 at 105 rad/s. This circuit is known as a low pass filter . It is used in many applications to limit

the bandwidth of for instance telephone signals. Although the human hearing range can extend from

20Hz to 20kHz telephone companies have found that limiting the bandwidth to just 3kHz leads to an

imperceptible difference in quality. However nearly seven simultaneous telephone calls can be carried

by the same wire connection using a technique called multiplexing thereby making more money for

the same investment by the telephone company.




Figure 3-12: Bode plot

The data for Figure 3-12 are plotted using the more usual logarithmic axes in Figure 3-13 where the

ordinate is given in dB units.

Figure 3-13: Logarithmic Bode plot

This plot provides us at a glance with the important aspects of the circuit, namely that the gain is 1 (or

0 dB) up to a value slightly greater than 104 rad/s above which it drops rapidly. Following on from our

time domain analysis of a capacitor we introduced the time constant ; we do something similar

here and assign 1/. So that




2/1

2

0

2

1

1

in

out

V

V and

0

tan

3-17

When = 0 this becomes:

7071.02

1

1

12/1

2

0

2

in

out

V

V

Recalling that

in

out

V

V dBGain 10log20)( we see that the gain = – 3dB at as indicated on the

graph. The graph can be approximately fitted by two asymptotes which meet at . For this reason

is called the – 3dB point or the “corner frequency”. Below the gain is unity (0 dB) whilst above

it decreases with a constant slope called the “roll off”. The value of the roll-off can be found by

considering the change in the gain at high frequencies where ≫ where we can approximate

equation 10-4 by

0in

out

V

V so for every doubling of the frequency (octave) the gain reduces by a

factor of 2. A reduction by a factor of 2 means a drop of -6dB in the gain and the roll-off has a value of

-6dB per octave. Similar reasoning gives the value -20dB per decade.

Figure 3-14: Phase change as function of frequency

Equation 3-17 shows that the phase change between and is also dependent on frequency

and Figure 3-14 shows a plot of against log. Three regions of the graph are identified:

Up to ~0.1 and are in phase above ~10 lags by /2




for intermediate values there is an almost linear shift and at , lags by /4

Amplitude and phase graphs are known collectively as Bode plots. Their great advantage is that only

the value of need be calculated from the values of and and the form of the plot is easy to draw.

3.12 The high pass filter

Figure 3-15: A high-pass filter

If we interchange the capacitor and resistor as shown in Figure 3-15 it can be shown that:

01

1

11

1

jCR j

AV

V

in

out

3-18

Rationalising this complex expression gives

2/1

2

2

01

1

in

out

V

V 3-19

At low frequencies where the gain ~ / , is small and increases with increasing

At high frequencies the gain ~ 1.

This behaviour is the opposite of that found for the low pass filter; this circuit is called a high pass filter

and the Bode plots are sketched in Figure 3-16.

Figure 3-16: Bode plots for high-pass filter

The phase shift is given by

0arctan . At low frequency /10 leads by /2; at

high frequencies 10 and are in phase. At and leads by /4.




4 LCR tuned circuits

4.1 Inductance

Inductors, like capacitors and resistors are ‘passive’ components commonly used in electronic

circuits. Whereas capacitors store electrostatic energy in the electric field between the plates,

inductors store magnetic energy in coils which controls the rate of change of current through them.

Inductors have a variety of practical uses: they prevent voltage surges, and are used in

integrating/differentiating circuits, filters, spark plugs for cars and, perhaps most importantly, in

transformers.

4.1.1 Electromagnetic Induction

A length of wire carrying a current is surrounded by concentric magnetic field lines as shown in the

left hand part of Figure 4-1 according to the right hand rule (thumb of right hand along direction of

current, fingers give the direction of the magnetic field).

Figure 4-1: Magnetic fields generated by current

If the wire is wound into a coil as shown in the right hand part of Figure 4-1 the magnetic field

generated by each turn adds to produce a dense concentration of magnetic field lines within the coil,

creating an electromagnet. The field lines point from the magnetic “north” to the magnetic “south”.

The strength of the magnetic field is proportional to current and is given by Ampère’s law:

I l

N B 10 Tesla (=Weber/m

2) 4-1

where is the number of turns in the coil, is the length of the coil and is the permeability of free

space, the magnetic counterpart of (you will cover this in much more detail in the E&M course). The

magnetic flux, ,through the coil is defined as:

I l

r N BA

2

10 Weber 4-2




where represents the area of a surface intersected by the magnetic field lines. Inside the coil or

close to the end of the coil, where is the radius of the coil.

If another short coil having turns (coil 2), which is not initially carrying current, is moved into place

next to the first coil (coil 1) then every turn of coil 2 becomes intersected by the flux 1 from the coil 1

and a voltage:

dt

dI M

dt

dI

l

r N N

dt

d N 12

2

21012

4-3

is induced in coil 2. This is Faraday’s law of induction. is called the mutual inductance between

the coils. The induced emf will drive current around coil 2 which generates a magnetic field from coil

2 as well. The minus sign shows that this magnetic field is in the opposite direction to that produced

by coil 1 and opposes the motion towards coil 1. This behaviour is referred to as Lenz’s law which

states:

"An induced current is always in such a direction as to oppose the motion or change causing it"

Moving coil 2 away from coil 1 also generates an induced voltage but now in the opposite direction so

that the coils attract. This experiment can also be performed by moving coil 2 towards or away from a

bar magnet. The magnet supplies the field and the motion generates a changing magnetic flux.

A changing magnetic flux can also be produced by passing a time varying current (AC) through coil 1.

This would induce a voltage in coil 2 which will depend on the rate of change of the current (the AC

frequency). This is the basis of a transformer which enables alternating voltages to be amplified

(stepped up or stepped down). Transformers are crucial for efficient electrical power distribution and

in combination with diodes are used to produce DC voltages from a mains source.

4.1.2 Self Induction: Inductors

Now imagine coil 1 to be part of an electric circuit. If the current in the circuit is the coil will generate

a magnetic field and magnetic flux according to equations 4-1 and 4-2. If the solenoid current changes

with time then:

dt

dI

l

N r

dt

d 1

2

0

4-4

Following the argument in the previous section the changing magnetic flux will induce an emf in each

turn of the coil according to:

dt

dI L

dt

dI

l

N r

dt

d N 1

2

1

2

01

4-5

Wherel

N r L

2

1

2

01

4-6

is known as the self-inductance and depends only on the geometry of the coil. Comparing equation 4-

6 with equation 4-3 we see that the changing magnetic flux associated with coil 1 causes the coil toact upon itself. Lenz’s law requires that any change in the current in the circuit will be opposed by the




self-inductance of the coil. When included in a circuit such coils are known as inductors and they act

in a way as to oppose any change to the current in that circuit. From equation 4-6 the unit of

inductance is the Henry (in recognition of Joseph Henry who discovered electromagnetic induction

around the same time as Faraday). The Henry (like the Farad) is a very large unit and typical circuit

inductors will be in the H or nH range. Larger values can be obtained by winding the coil around a

magnetic material such as iron or ferrite. These materials increase the strength of the magnetic field

by a factor the relative permeability. Values of can be many tens of thousands. This is the

magnetic analogue of the relative permittivity discussed previously for capacitors.

4.2 Energy stored in an inductor

Inductors, like capacitors which store electrostatic energy, store magnetic energy which can later be

released. An example is the spark produced in a spark plug of a car engine from a charged ignition

coil. As the coil is charged work has to be done to increase the current in the inductor. The rate of

electrical work (electrical power) is given by:

dt

di LiiP 4-7

where is the induced emf when the instantaneous current in the inductor is (equation 4-5). The

electrical work done creates a magnetic field within the inductor. If the current reaches a constant

value the magnetic field has the potential to do work. The magnetic PE is the product of power and

time and the total PE stored in taking the inductor current from zero current to a steady state value of

is

max

0

2

max

2

I

LI di LiU 4-8

Substituting for from (7.5) gives:

2

0

0

22

0

2

1

2

l

NI Al I

l

A N U

Where

2

0

l

NI is the square of the magnetic field strength (see equation 4-1). Dividing by the

volume occupied by the magnetic field inside the coil gives the magnetic energy density

0

2

2

BJm

-3. In

chapter 2 we discussed the dissipation of energy in a resistor whereby collisions of the flowing

electrons caused heating. Note that if we unwound each coil there would be no voltage drop

(assuming a negligible resistance for the wire) and so there is no mechanism by which an ideal

inductor can dissipate the stored magnetic energy.




4.3 Inductors in series and parallel

Figure 4-2: Inductors in series and parallel

Figure 4-2 shows two inductors and in series. If the current is increasing with time so that there

is an increasing magnetic flux this will induce voltages and across and which oppose the

increase in current flow and the total voltage drop across and is where:

dt

di L

dt

di LV V V 2121

and this must equaldt

di Ltotal where is the equivalent series inductance. This leads to

4-9

so the rule for inductors is the same as that for resistors. For inductors in parallel, shown in the lower

part of the diagram, the voltage across and is the same but the current is different. In this case

we have:

vdt

L

vdt

L

vdt

L

iii

total

111

21

21

where the inductance of the pair is . This gives the rule for inductors in parallel as:

21

111

L L Ltotal

4-10

the same as that for resistors.

4.4 Transformers

In section 4.2 we discussed the interaction between two coils one of which carried a current. In that

case a changing flux was created by the movement of the second coil. Now we consider passing a

time varying current through coil 1 (now called the primary) to produce a changing magnetic flux. This

will induce a voltage in coil 2 (called the secondary) according to Faraday’s law. The magnitude of

the induced voltage will depend on the flux linkage with coil 1.




Figure 4-3: Two inductors used as a transformer

This can be maximised if the coils are wound on a magnetic former made from ferrite or iron indicated

by the lines between the coils in Figure 4-3. This also gives the transformer rigidity. If the primary coil

has windings and carries a time varying current then the induced emf measured across the

terminals of the primary coil is:

dt

di L N

dt

d N

p

PPPP

4-11

where is the self inductance of the primary coil. If the changing magnetic flux from the primary is

transferred efficiently to the secondary coil its terminal voltage will be given by:

dt

di L N

dt

d N S

S S S S

4-12

Since the rate of change of fluxdt

d is the same for both primary and secondary then:

P

S

P

S

P

S

V

V

N

N

4-13

where , are the amplitudes of the voltages. The secondary terminal voltage can therefore be

made larger (step-up transformer) or smaller (step down transformer) by suitable choice of the

number of turns on each coil. For an ideal transformer with zero losses the electrical power in equals

the electrical power out:

PPS S iV iV 4-14

and so

S

P

P

S

P

S

I

I

V

V

N

N 4-15

and an increase in secondary voltage must be accompanied by a decrease in secondary current. The

physical size of a transformer will be dictated by the maximum electrical input power. This can be the

size of a house for a step up/down transformer forming part of the national grid to compact versions

found in most electrical equipment.




4.5 Transient Response – the RL circuit

In chapter 6 we looked at the transient (time domain) response of an circuit. Now we repeat this

for the case of an circuit shown in Figure 4-4.

Figure 4-4: An circuit with switch

Based on our knowledge of inductors we can make predictions about the behaviour of this circuit.

When the switch, , is first closed no current can pass through the inductor since there is a

large induced voltage generated to oppose it (the inductor therefore acts like an open circuit

when / is large).

Gradually the circuit current grows at a rate determined by and reaches a steady maximum

value / at which point / 0. Then there is no opposition to the current

(inductor behaves like a short circuit) and the applied voltage is dropped entirely across the

resistor.

Figure 4-5: Current through inductor

Figure 4-5 shows the anticipated growth in current in the circuit for different values of . For very small

there is no opposition to the growth of current and is reached almost instantaneously. With

increasing values of it takes correspondingly longer times to reach .

A mathematical solution for the circuit is given in the appendix and results in the following expression

for the current:




t

L

R

R

V i exp1 4-16

which is the form of the curves shown in Figure 4-5. Comparing this with the result we obtained for

the circuit we identify the time constant in the circuit as /. Thus the current rises to0.630 after a time ; 0.861 after 2 and 0.99 after 5 . After a suitably long time → , / = 0 and there will be no induced voltage at the inductor

terminals. However there will be a constant magnetic field within the coil. Now imagine replacing the

battery with a short circuit; the energy associated with the magnetic field will be converted into a

current which will flow through the resistor and be dissipated as heat. The rate at which the current

decays is given by solving the differential equation a-1 given in the appendix to this chapter, but

reducing to zero:

t

L

R

e I I

max 4-17

This is shown in Figure 4-6. At 0, and at long times 0. After one time constant R

L ,

will have dropped to 0.37

Figure 4-6: Current decaying after switch is opened

Appendix (to Section 4.5)

Equation 4-16 is derived from* 0dt

di LiRV (a-1)

Rearranging dt iRV Ldi )( (a-2)

and in terms of , dt L

Ri

R

V di

(a-3)




to give finally dt L

R

i R

V

di

(a-4)

When is closed the inductor will initially limit the growth in current but as this builds up the opposition

(induced voltage) will reduce allowing eventually a current to flow which is given by /. So we

integrate (a-4) with the limits shown:

i t

dt L

R

i R

V

di

0 0

(a-5)

To give: t L

R

R

V i

i

0

ln (a-6)

Applying the limits t L R

RV

RV i lnln (a-7)

And rearranging t L

R

RV

RV i

/

/ln (a-8)

Which gives us the result

t

e R

V i 1 (a-7)

Where / is the time constant for the circuit.

* Although this equation appears to be a statement of Kirchhoff’s voltage law the situation is more

complicated when dealing with induced emf – see discussion in section 30.1 of Young & Freedman.

4.6 Inductive reactance and the circuit

The reactance associated with an inductor can be found by considering its response to a current

t I I cos0 . The voltage across the terminals whendt

dI is increasing is

dt

dI LV L (the negative of

the induced emf). Substituting for gives

2

cossincos 000

t L I t L I t I

dt

d LV L

and the inductor voltage LEADs the inductor current and has amplitude . The inductive reactance

is then defined as the ratio of the amplitudes of voltage and current as

4.7 Frequency response of an circuit

The combination of an inductor and resistor also results in a filter circuit. Figure 4-7 shows an

circuit driven by an ac voltage source with the output voltage taken across the inductor .




Figure 4-7: Filter with circuit

The impedance of the inductor is and the gain is:

L j R L j

R Z Z

V V

L

L

in

out

4-18

Rationalising this expression we obtain:

2

2

01

1

in

out

V

V 4-19

where /. Looking at the form of this equation we conclude that:

At low frequencies the gain ~ / and is small

at the gain=2

1or -3 dB

at high frequencies ≫ the gain 1.

The Bode plot for the gain is the same as that shown in Figure 3-16 and this is a high pass filter.

Although an analysis of this circuit taking the output voltage across the resistor will not be given we

can predict that it will be a low pass filter.

There are many Java applets available on the web that allows you to perform virtual experiments. For

version of the low and high pass filters see:

https://www.st-andrews.ac.uk/~www_pa/Scots_Guide/experiment/lowpass/lpf.html

4.8 Cascaded low pass filters

If two low pass filters are cascaded the output voltage is not simply the product of two transfer

functions. This is because the second filter acts as a load for the first.




Figure 4-8: Cascaded low pass filters

Although slightly more involved than anything presented until now it is possible to show that the gain

is given by:

CR j RC V

V

in

out

3)1(

1222

When 1/,3

1

jV

V

in

out and the amplitude of the gain is 1/3 and the phase is /2.

However, we shall see later that a simple operational amplifier circuit called the unity gain buffer acts

as an impedance matcher so that loading of the first filter by the second does not occur. For such an

arrangement employing different values of and corresponding to different values of ( and

):

)/1)(/1(

1

21

j jV

V

in

out

where and are the corner frequencies of the two circuits. The figure shows the resulting Bode

plot and also indicates a useful property the log-log plots, namely that they can be added.

Figure 4-9: Bode plot for cascaded filter

Above the roll-off is twice that of the single circuit i.e. -12 dB per octave. This is an advantage if the

circuit is used to reject signals above .




4.9 Resonant circuits

Simple pendula, water waves and bungee jumpers are examples of oscillatory motion. In the absence

of any frictional effects the motion would continue without loss of amplitude, but in practice the

oscillations diminish and the mass will eventually come to rest unless it is given a periodic push.

When the push is applied at the appropriate time (frequency) the amplitude of the motion can bemaintained with the minimum of force. In some cases the amplitude of motion can increase

dramatically such as happened in the Tacoma Narrows Bridge:

see http://www.youtube.com/watch?v=j-zczJXSxnw

This effect is known as resonance. An example on a microscopic scale is the excitation of atoms in

solids by light which creates oscillating dipoles. Electrons bound to atoms have “natural” or resonant

frequencies which, when excited by a light wave of the correct frequency, will induce large amplitudes

of the electrons. Subsequent collisions with adjacent atoms results in absorption of the light. At

frequencies away from resonance there are phase changes between the electron motion and the light

wave which leads to changes in the speed of light in the solid given by the refractive index, as

described in chapter 6. In this lecture we’ll see how the frequency dependent impedances of

capacitors and inductors combine to produce resonance in electrical circuits. Such a selective

frequency response makes these “tuned” circuits useful in radio and TV transmission.

4.10 The circuit

Figure 4-10: circuit

Figure 4-10 shows an circuit driven with an AC voltage source cos. We can analyse

this circuit by applying Faraday’s law:

dt

dI LV V IRV C Lin

where is the current in the circuit. Since the inductor is only a wire there will be a negligible voltage

drop between the ends of the wire and 0; / and the

dt

dI L term on the right hand side is




the induced voltage across the inductor. Rearranging and writing everything in terms of this

equation then becomes:

t V C

q

dt

dq R

dt

qd L cos02

2

This second order differential equation can be solved (but is not done here) to give the current as:

)(cos/1 22

0

t C L R

V I 4-20

This equation allows a calculation of the amplitude of the current , but we would have to plot the

resistance and reactances on a phasor diagram to determine the phase difference between the

current and voltage in the circuit for a given frequency . We have seen in previous chapters the

shortcomings of this approach and we will instead find the complex impedance of the circuit. This

turns out to be much simpler and gives the same result. Since this is a series circuit

C j L j R Z

1 and the complex form of Ohm’s law gives I

V Z . Rearranging for the current:

)/1(/1 C L j R

V

C j L j R

V I inin

To rationalise this expression multiply the numerator and denominator by C L j R /1 to

obtain the current in form where 22 /1 C L R

Ra

and

22

/1

/1

C L R

C Lb

and calculate the modulus of the current 22 ba :

22

0

/1 C L R

V I

4-21

This is identical to the amplitude of given in (11.1). Using the values of and the phase angle is:

R

C L

a

b

/1tan

4-22

Equations 4-21 and 4-22 provide us with all the information required to describe the behaviour of the

LCR circuit to a driving voltage covering a wide frequency range and we can make the following

predictions:

The current will have a maximum value when 1/ since this minimises the

denominator in (4-21) and becomes purely resistive. Then / at the resonant

frequency LC 10 . In addition equation 4-22 shows that tan 0 (so 0) at the

resonant frequency current and voltage are in phase.

For low frequencies well below the denominator of equation 4-21 is dominated by the

reactance of the capacitor and is small. The phase angle is positive showing that the

current leads voltage as expected for a capacitor. The actual value will depend on the relative

values of and .




At high frequencies the denominator of equation 4-21 is dominated by the reactance of the

inductor and is again small. From equation 4-22 the phase angle is negative showing that

the voltage leads current as expected for an inductor. The actual value will depend on the

relative values of and .

Figure 4-11: Resonant behaviour of circuit

Figure 4-11 shows a sketch of the circuit current amplitude as a function of frequency as just

described. / at and tends to zero at lower and higher frequencies: this is electrical

resonance. An important parameter is the sharpness of the resonance or “” factor defined as /

where is the width of the resonance curve at the half power points. When using tuned

circuits for radio or TV applications electrical power and not amplitude is the important factor and

since power is proportional to the square of the amplitude the half power points correspond to2

max I

or 0.7071. It can be shown that / so the width of the resonance curve is proportional to

(and inversely proportional to ). Then equalsC

L

R

1.




Figure 4-12: Resonant responses as function of factor

Figure 4-12 shows a series of resonance curves corresponding to different values of and a constant

resonant frequency of 1 MHz. Radio reception requires a value of 10 kHz consistent with a of

100. This is sufficiently narrow to be able to reject adjacent radio stations but wide enough for high

quality sound.

Figure 4-13: Phase angle variation for the circuit

Figure 4-13 shows the variation of the phase angle , that is, the angle between the circuit current and

the driving voltage as a function of log . In reality this would be a smooth curve but are here

shown as Bode plots. For frequencies well below where the capacitor dominates the impedance




LC

1

, equation 4-22 shows that → /2 and the circuit current leads the driving voltage as

expected. At frequencies much higher than ,C

L

1

,and the inductor dominates the

impedance with → /2 and the driving voltage leads the circuit current.

Our analysis has shown that the series circuit can sustain a large current at the resonant

frequency and the circuit impedance is real and consists only of resistance. In fact if the circuit is

very large the current at resonance may be enough to significantly raise the temperature of the

resistor or even damage it. But what about the voltages across the capacitor and inductor at

resonance? To investigate this we shall use component values typical of those used in a radio

frequency circuit: 10, 60H and 30pF. This gives a resonant frequency of

2.37 10 rad/s or 3.7 MHz. If we consider the voltage across the capacitor, and treat the

circuit as a voltage divider:CR j LC C j L j R

C jV V

in

C

)1(

1/1

/12

At resonance 1 andCR jV

V

in

C

0

1

. This equation tells us that:

(i) lags by /2 (because of the term in the denominator) and

(ii) the modulus isCRV

V

in

C

0

1

. Inserting the component values and

rad/s1045.7

1 5

0 LC gives inC V V 141 . Including the phase information gives

the phasor representing the capacitor voltage as 141. Thus there is a very large

voltage across the capacitor which lags the input voltage by /2. Repeating these

calculations for the inductor gives 141 (you should check this). This means that

leads by /2 and is out of phase with so that their sum is zero and only appears

across . These large voltages ≫ are evidence of electrical resonance.

It is surprising that such large voltages can exist across the capacitor and inductor simultaneously but

the important difference is their phase. The next section considers the energy stored in and and

shows that it is constant with time.

4.11 Energy considerations

We will assume that any damping in the circuit is negligible and that a steady AC current

cos exists in the circuit. Then the magnetic energy stored in the inductor is

t LI

LI U L 2

2

02cos

22

1




Figure 4-14: Energy balance within the circuit

To find the energy stored in the capacitor we have to find . Recalling / we have:

t

o

t

C C t C

I dt t I

C dt I

C V

0 0

0 sincos11

So: )cos1(2

sin22

1 2

2

2

02

2

2

02t

C

I t

C

I CV U C C

giving the total energy (magnetic and electrostatic) stored is:

C

Lt I

C

I U

total 2

2

2

0

2

2

0 1cos

22

4-23

which contains a time-independent part and another that varies with twice the driving frequency (note

that the resistor cannot store energy, only dissipate it). Figure 4-14 shows plots of and

according to the equations above. Note that they are in anti-phase implying that energy is transferred

from to and vice versa with time. Since LC 10 then

22

2

02

0

2

0 L I C

I U total

showing that at the resonant frequency the second term in equation 4-23 becomes zero and the total

energy stored by and in their magnetic and electric fields is constant in time.




5 Digital CircuitsDigital circuit are used in two main situations; control and computing. In control applications they are

often referred to as ‘logic’ circuits which are used to make true-false type decisions. A typicalsituation would be in the control of industrial processes which require particular conditions to be

fulfilled before a particular action is taken. In computing the current preference is still for arithmetic

using binary representations. In both cases the key point is that only two ‘values’ are used which are

referred to as high-low, true-false, 0-1. The electronics which is used to implement this is thus

designed to only have two possible states for inputs and outputs. Although this may seem restrictive

it gives rise to an enormous richness in possible applications and implementation.

5.1 Binary Arithmetic

Our everyday arithmetic is based on a decimal system (base 10) and numbers are represented bystrings of digits with values between 0 and 9. Each digit then represents a multiplier of 10 where

is position of the digit counting from the right-hand end of the string.

For example the decimal number 1234 can be written as 1 1 0 2 10 3 10 4 10

A binary number (base 2) can only use digits with one of two values (0 and 1) and each digit is then a

multiplier of 2 where is position of the digit again counting from the right-hand end of the string.

For example the binary number 1011 is 1 2 0 2 1 2 1 2 (= decimal 11)

Although binary is amenable to easy electronic implementation it is immediately apparent that a major

disadvantage is that numbers begin to need a large number of digits to represent even modest

decimal numbers. For this reason it is common to combine single digits (or bits) into larger units.

Historically the first level bunching was to put 4 digits (bits) together into ‘nibbles’. Each nibble can

then have a value between decimal 0 and 15 and can be though of as a hexadecimal octal (base 16)

value. It is usual to express hexadecimal values with digits between 0 and 9 followed by A to F cover

decimal 10 to 15.

Hexadecimal notation is still commonly used. Less common nowadays is the use of octal numbers

(base 8). These require 3 binary bits to cover the decimal numbers between 0 and 7

Two nibbles together provide a ‘byte’ (covering decimal numbers 0 to 255). Next is a ‘word’ with 16

bits. Beyond that the units become more standardised in terms of ‘bytes’ as shown in Table 5-1.




Table 5-1: Commonly used notation for collections of bits and bytes

Number of bits Common terminology Decimal value range

1 (2 Bit 0 or 1 only

4 (2 Nibble 0 to 15

8 (2 Byte 0 to 255

16 (2 Word 0 to 65535

1024 (2 Kbit

8*1024 (2 Kbyte

1024*8*1024 (2 Mbyte

1024*1024*8*1024 (2 Gbyte

1024*1024*1024*8*1024 (2 Tbyte

5.2 Boolean Algebra

In decimal arithmetic we are familiar with the mathematical operations + - / and *. Each operatorrequires two input numbers and produces a third as the result.

In binary arithmetic we need to introduce a different set of basic operators. These are NOT, AND andOR.

The NOT operation only requires one input number and produces its result by changing the state ofeach bit in the number to its other value; i.e. it changes 1s to 0s and 0s to 1s.

The AND operation requires two input numbers and produces its result by comparing each number bitby bit. If either of the two corresponding bits are 0 then the resulting number has a 0 at that location.If both bits are 1s then the result is also a 1 at that location.

The OR operation requires two input numbers and produces its result by comparing each number bitby bit. If either (or both) of the two corresponding bits are 1 then the resulting number has a 1 at thatlocation. If both bits are 0s then the result is also a 0 at that location.

Additional Boolean operators include

SHIFT L(eft) which moves all bits one place to the left. A 0 is inserted in the right-most bit and theoriginal left-most bit is discarded.

SHIFT R(ight) which moves all bits one place to the right. A 0 is inserted in the left-most bit and theoriginal right-most bit is discarded.

XOR is exclusive OR. The XOR operation requires two input numbers and produces its result bycomparing each number bit by bit. If either one of the two corresponding bits are 1 then the resultingnumber has a 1 at that location. If both bits are 1s or 0s then the result is also a 0 at that location.

The following examples using 5-bit numbers illustrate how these operators work.

NOT(10100)=01011 AND(10101;01010)=00000 AND(11101;01010)=01000OR(10101;01010)=11111OR(11101;01010)=11111XOR(10101;01010)=11111XOR(11101;01010)=10111

SHIFT L(10101)=01010SHIFT R(10101)=01010




5.3 Gates

The Boolean operators are translated into so-called gates when realised as electronics circuits. The

following Figure 5-1: The 'gates' performing the basic Boolean operations are represented at the

single bit level: the NOT gate has one input and one output, the AND and OR gates both have two

inputs and one output.

Figure 5-1: The 'gates' performing the basic Boolean operations

The NOT gate is shown with a small circle on the output which signifies the NOT operation; without

the circle the gate would be a unity gate with the output echoing the input. This allows other types of

gate to be realised such as the NAND and NOR gates shown in Figure 5-2. Also shown is the XOR

gate.

Figure 5-2: Some addition logic gates

Some of the Boolean operations can be defined using more than two inputs. For example the AND

and OR functions could take any number of inputs whilst still having only one output, even though the

output may be duplicated for ease of use (Figure 5-3).

Figure 5-3: Gates with multiple inputs and outputs

Circuits can be designed with all sorts of different configurations of gates depending on the functional

behaviour required. A powerful design tool to help make sure the circuit does what is intended is the

truth table.

5.4 Truth Tables

The use of truth tables allows a complicated circuit to be traced through systematically to confirm its

function and also to aid in the initial design process.




A truth table identifies all the inputs and outputs and shows all the logical combinations of inputs and

the corresponding outputs. For more complicated circuits it can also be useful to include internal

nodes in the truth table.

Consider the gates, AND, OR, XOR and NOT. If we label the inputs A and B then the four truth tables

are shown in Table 5-2.

Table 5-2: Truth tables for the AND, OR, XOR and NOT gates

We can also construct truth tables for functions which we wish to perform. For example consider how

we would go about adding two binary numbers together. The two numbers would be strings of 0s and

1s and to add them together we would add each bit pair in turn taking care to follow any need to add

in a carry from the previous pair and to promote a carry to the next pair. This is illustrated in Table

5-3.

Table 5-3: Adding to bits together when summing to numbers

. . . . A . .

. . . . B . .

. . . . S . .

The truth table corresponding to this situation requires three inputs, A, B, and C in, and two outputs, S

and Cout, and so will have 5 columns. With three inputs each with two possible values the truth table

will have 8 rows. It is shown in Table 5-4.

The truth table can now be used to design a digital circuit which will perform this function. The design

process is a combination of inspection of the table to see which sub-structures can be performed by

known gates and then piecewise building around them to up to assemble the whole function. For

example in Table 5-4 there is a 4-row section, enclosed by a dashed line, which matches the XOR

table. Now we need to add additional logic circuitry around the XOR gate to achieve the second

output and add the additional conditions. There are multiple solutions to this and one is shown in

Figure 5-4.

Carr in C

Carry out Cout




Table 5-4: Truth table for general bit adding function

Cin A B S Cout

0 1 0 1 0

0 0 1 1 0

0 0 0 0 0

0 1 1 0 1

1 1 0 0 1

1 0 1 0 1

1 0 0 1 0

1 1 1 1 1

Figure 5-4: Logic circuit to perform bit addition with carry in and carry out

5.5 Basic Electronic Building Blocks

Electronic logic circuits work on voltage levels to define the high/low, 0/1, true false logic states. The

main workhorses are TTL or TTL-like components, often called ‘chips’. These contain active internal

circuits whose details we do not need to know in detail. They can be treated like ‘black-boxes’ with

logic level inputs and outputs. The actual voltage levels used to define 0 and 1 depends on the

explicit technology used to manufacture the circuit. The technology type is indicated in the part

numbers and, as the logic levels do vary, it is advised to always use the same family throughout any

particular design.

Figure 5-5: Typical logic voltage levels




All of these gates so far discussed are available as chips. In some cases the chips contain more than

one gate. For example the 7404 chip contains six NOT gates. It is normally referred to as an hex

‘inverter’. Each inverter is fully independent with its own input and output. The chip is an active chip

and requires a power supply to operate. Two pins on the chip are provided for this. Pin 7 is labelled

GND for ground and pin 14 is labelled Vcc and this is where the +5V power should be connected.Most chips have the same basic layout with the ground and V cc in the same corner locations.

However different chips may have different numbers of pins; 14 and 16 are most common.

Many chips provide both a normal output and an inverted output for convenience.

Figure 5-6: Pin configuration for the 7404 hex inverter gate.

General considerations when using logic chips are:-1. Use chips from the same family throughout

2. Keep your layout neat and tidy3. If you do not need to use all inputs on a multi-gate chip it is good practice to connect unusedgates to ground.

4. Outputs can be used to feed multiple inputs but check the data sheets to make sure you donot overdo it

5.6 Higher Level Functions

A bewildering range of higher level functions integrated onto single chips is now available. In additionTo the basic gates this includes

Registers and latches – for temporary storage of bit values

Counters

Shift registers – for sequentially transferring bitsMultiplexers – for selecting particular inputs

Clocks

Many circuits require the use of a clock to perform sequential operations. For example latches andregisters require a clock to make sure the information they need to store is accepted at the correcttime.

6 Operational AmplifiersYou may already have encountered operational amplifiers (op-amps) in the first year laboratory.

These consist of a rectangular piece of black plastic with dual-in-line pins for connection (see Figure

6-1). The microcircuit inside is complicated and typically contains over 25 transistors. Since an

understanding of one transistor is well beyond the scope of this course, trying to understand how so




many work in tandem would seem to be an impossible task. However the details need not concern us

and the approach here is to treat op-amps as ‘black boxes’ which are governed by a single equation.

The full extent of the usefulness of op-amps only becomes apparent when we apply the techniques of

feedback and show how they can perform mathematical operations. In fact the prospect of making

analogue computers was the driving force in the development of op-amps.

Figure 6-1: A 741 operational amplifier; left is a schematic functional diagram, right is the pin

assignment diagram

6.1 Differential Amplifiers

An op-amp is a differential amplifier. It has two inputs and one output. It amplifies the difference

between the inputs. Figure 6-1 shows the circuit symbol for an op-amp which we are going to treat as

a “black box” which obeys the following equation:

6-1

The two input voltages and are referred to as the inverting (-) and non-inverting (+) terminals

respectively. Note that the voltages may be AC or DC. The amplifier gain is large, typically 105.

Note that the gain is written in bold type since it is a complex quantity (i.e. the gain may be frequencydependent and may show a phase shift). Op-amps are voltage driven sources: they generate an

output voltage which is proportional to the input voltage(s) but is derived from the amplifier

supply voltages (pins 4 and 7 on the schematic of the packaged 741 module in Figure 6-1).

can never exceed . Op-amps are designed for relatively small voltages and a typical value for the

supply voltages is 15V. is often omitted in circuit diagrams. Input voltages are measured relative

to the common or earth line shown at the bottom of Figure 6-1 left hand side.

Since and then /

If 15V and | | 10 then 150μV. Hence

Op-amps therefore have a very limited working range. Thus a tiny difference in the values of and

will drive to or depending on the sign of the tiny difference (Figure 6-2). This condition

is in fact used to advantage in some applications including digital voltmeters, analogue to digital

converters and control systems. In these situations the op-amp behaves as a comparator (i.e.

compares with ). Final important properties of op-amps are their input and output impedance.

The input impedance is typically very large (M) while the output impedance is typically very small

(few ). The advantages of this are:

a voltage source connected to either of the input terminals will “see” a very large

impedance. If the source impedance of is relatively small the voltage divider rule ensures

that the true value of appears at the op-amp input




the amplifier output stage can be treated as a Thévenin source with an open circuit voltage

in series with a source resistance as discussed earlier. The amplifier output will

invariably be used to drive a display or loudspeaker circuit and provided is much smaller

than any input impedance of the display or loudspeaker circuit loading effects can be

neglected

The remarkable versatility of op-amps becomes apparent when feedback is applied. This is the action

of returning some, or all, of the output voltage to the inverting input (negative feedback) or the

non-inverting terminal (positive feedback).

Figure 6-2: Output limitations of op-amps

6.2 Negative feedback

Figure 6-3 shows an op-amp circuit where a fraction of the output voltage is directed back to

the inverting terminal: this is called negative feedback. Now and and

so that

Rearranging this gives the gain:

6-2

where is called the closed loop gain to indicate that feedback has been used. is always smaller

than

, the open loop gain (no feedback). If

is very large and not too small

1/ . This is an

extremely important result. It says that in a negative feedback system having a large open loop gain

the closed loop gain is independent of the open loop amplifier gain. How does negative feedback

work? Without feedback, if is a positive voltage then will be an amplified version of if there

is no voltage at the inverting terminal ( 0) then → . Returning part of to the inverting

terminal will try to produce an inverted (negative) voltage at the op-amp output preventing from

saturating.




Figure 6-3: An op-amp with negative feedback

6.3 Unity Gain buffer

Figure 6-4 shows a negative feedback circuit where all of the output voltage is returned to the

inverting input. Putting = 1 into equation 6-2 means that 1. It may seem odd to use an amplifier

which does not appear to amplify the input voltage. The circuit is called a unity gain buffer. The unity

gain is self-evident, the “buffer” part refers to the input and output impedances. In section 6.2 it was

stated that it is desirable for the op-amp to have large input impedance and small output impedance. Although the details will not be presented here another effect of the feedback is to increase

(decrease) the input (output) impedances compared with the open loop values. Section 2.11

described the requirements for impedance matching when joining circuits. The unity gain buffer

provides a method of achieving this.

In summary the unity gain buffer uses a negative feedback loop to:

Reduce the gain to 1 (0dB)

Significantly increase the bandwidth (see next section)

Produce a very high input impedance and low output impedance




Figure 6-4: An op-amp with unity negative feedback

6.4 Frequency dependent gain

Real components very rarely have ideal properties or exact values. For example manufacturing

tolerances mean the measured value of a resistor is rarely exactly the value indicated by the colour

code but will fall within the tolerance band (silver or gold). Integrated circuits are fabricated on

semiconductor substrates and are subject to variations arising from the many complicated processes

involved in their manufacture. Hence data sheets normally quote typical values and indicate the likely

variation from device to device. So the open-loop gain of an op-amp will state only a typical value,

say 105. In addition there may be variation in properties of internal components which would introduce

a frequency dependence, such as capacitances, including so called stray capacitances in integratedcircuits which act between unintentional conductors. Semiconductors also have relatively large

dielectric constants (for Si 13) which can enhance the stray capacitance. To avoid such stray

capacitance dominating the frequency dependence of the performance, and thus introducing too large

a variation from device to device, a capacitor is deliberately included in the op-amp output to

dominate all other capacitive effects. The net effect is to make the op-amp output stage resemble the

low pass circuit discussed at length in previous chapters. The result is shown in Figure 6-5.

Figure 6-5: Variation of open-loop gain with frequency




The gain has the familiar shape of a low pass filter but now with a large gain. Thus the op-amp has its

highest gain, , at low frequency, which then rolls-off with the usual -6 dB per octave (-20 dB per

decade). Just as we saw for the low pass filter there will also be a phase change between the input

and output voltage at higher frequencies and the open loop gain is given by:

0

0

1 f

f j

A A

6-3

Where is the corner frequency or 3dB point as found previously for the low pass filter; it is also

known as the bandwidth of the op-amp. In the case shown in Figure 6-5, ~ 5 Hz. Reiterating the

point that and are given only as typical values it would be difficult to choose an amplifier to have

a specific gain at a specific frequency. Fortunately this problem can be overcome by using feedback.

6.5 Gain-bandwidth product

Combining equations 6-2 and 6-3 the closed loop gain is:

)1(/(1

)1/(

)/1(1

)/1/(

1 00

00

00

00

A f f j

A A

f f j A

f f j A

A

AG

This has the same form as equation 6-3 but now has a modified closed loop gain of / 1 and

a modified corner frequency of 1 . We can therefore say that as a result of negative

feedback

The low frequency gain has been reduced by a factor of 1

The corner frequency has been increased by a factor of 1

Thus the product of the gain and the bandwidth is constant. If we take a specific case where the

feedback fraction 0.1 then the closed loop gain:

1/ 10

Figure 6-5 gives an open-loop gain of 10 and a bandwidth 5Hz and the gain–bandwidth

product is 5 10 Hz. With feedback the closed-loop gain is 10 and we now have a closed-loop

bandwidth given from the equation 10 5 10 which gives 5 10 Hz. Note that in

applying the gain-bandwidth product rule the linear gain and not dB values should be used. However,

we can still show the closed-loop behaviour on a Bode plot; a gain of 10 is equivalent to a gain of

20 dB and the closed-loop gain frequency dependence is plotted in Figure 6-5. The gain is

constant up to 5104 Hz and then follows the behaviour of the open loop gain reducing by 20dB per

decade. The bandwidth for a unity gain buffer having the same open loop values will be extended

even further since the gain is now reduced to 1.

Having an amplifier which operates over a much large frequency range is highly desirable. For

instance, an audio amplifier should have a nominally flat response over the range of human hearing

(20Hz – 20kHz).




6.6 Control of gain - the non-inverting amplifier

Figure 6-6: Op-amp configured as a non-inverting amplifier

Figure 6-6 shows how a fraction of the output voltage might be derived and returned to the inverting

terminal. The input voltage is connected to the non-inverting terminal so this configuration is known as

a non-inverting amplifier.

and act as a voltage divider so the potential at point A is: out A V R R

RV

21

2

Then the feedback fraction

21

2

R R

R

and ~1/ . For example, if 9kΩ and 1kΩ, then 0.1 and the closed loop gain 10. It

is also possible to have a variable gain by making a variable resistor.




6.7 The inverting amplifier

Figure 6-7: Op-amp configured as an inverting amplifier

Figure 6-7 shows another application of negative feedback and the one which is used most often with

op-amps. The feedback connection is made from the output through impedance

to the inverting

terminal. The input voltage, to be amplified, is also connected to the same terminal through another

resistor, . The non-inverting terminal is connected directly to the earth/ground rail. In section 6.2 it

was shown that a high open loop gain implies ~ and since 0 this means that the point S

must also be forced to zero volts by the action of the feedback. However S is not directly connected to

earth potential and, in this configuration it is called a virtual earth. The high input impedance of the op-

amp means that no current can flow into the inverting terminal and 0. Applying Kirchhoff’s current

rule gives . Substituting for and gives:

F F

out

i

in

in

i R

V

R

V i

00

Rearranging this equation gives us an expression for the closed loop gain of:

in

F

in

out

R

R

V

V

So once again the closed loop gain is dictated only by the relative values of the feedback impedances

and and the minus sign indicates that the output voltage is inverted relative to the input. The

gain-bandwidth product rule still holds for this (and the previous) configuration.




6.8 Differential feedback amplifier

It is sometimes necessary to amplify the difference between two voltages, and , in a controllable

way and Figure 6-8 shows an example of a difference amplifier with gain. Although an op-amp

without feedback is already a difference amplifier, the high gain and strong frequency dependence

(which can vary from one op-amp to another) are not desirable. Note that and, appear at boththe inverting and non-inverting inputs. The input to the non-inverting terminal is a voltage divider so:

1V R R

RV

niF

F

6-4

Figure 6-8: Op-amp configured as a differential amplifier

For we have need to recall that the input impedances of the op-amp are very large compared with

and and, since no current can flow into the inverting terminal . Equating these currents

gives

F

out

in R

V V

R

V V

2

which leads to:

F in

inout F

R R

RV RV V

2 6-5

The action of the feedback forces ~ and equating 6-4 and 6-5 gives (eventually)

)( 21 V V R

RV

in

F out

So the closed-loop gain is / and the output voltage is proportional to the difference between the

input voltages.

6.9 Other op-amp operations

Op-amps were developed to perform mathematical operations and are used extensively in control

circuitry, for example, servo control units in instrumentation (Scanning Tunnelling Microscopes). This

section covers some basic mathematical applications.




6.9.1 A summing amplifier – inverting adder

Figure 6-9 shows three voltage sources connected to the inverting input of an op-amp. Each input

produces a current

x

x

R

V (x =1,2,or 3) which sum at S, the virtual earth. If ∑ the three voltages

are added and amplified and: x

x

F

out

R

V

R

V

In fact, because the currents are added and a voltage given out the circuit is a current to voltage

converter , so that . The circuit can be used to add , and by having

or combine them in some predetermined proportion by varying the resistors. Such circuits are used in

broadcast studios where signals from the presenter’s microphone and a music signal are combined

or in a recording studio where the inputs from different instruments/vocals are mixed.

Figure 6-9: Op-amp configured as a summing amplifier

6.9.2 Integrators/differentiators

Figure 6-10: Op-amp configured as an integrator

Figure 6-10 shows a circuit which incorporates a capacitor in the feedback loop. S is again a virtual

earth since the non-inverting terminal is connected directly to the earth rail and so and all the

current from the source is directed through the capacitor. Now / and / where is the charge on the capacitor. Recalling that then / then:



dt V RC

V inout 1

so the output voltage is an integrated version of the input voltage. If is a constant (and negative)

voltage as shown in Figure 6-10 the integrator will generate a ramped voltage with a gradient that

depends on and . The integrator circuit is the basis of a ramp generator . A ramp voltage is used indigital voltmeters or analogue to digital converters.

Interchanging the resistor and capacitor will create a differentiator . Using a combination of integrators,

differentiators and adders it is possible to make an analogue computer which is capable of solving

differential equations.

6.10 Positive feedback

Positive feedback acts to reinforce any change at the input(s) to the amplifier and the circuit becomes

unstable. It might be thought that this would have no useful applications but this is far from the case.

Positive feedback is used to make oscillators (like the signal generators used in the laboratory) ormulti-vibrators which switch rapidly between positive and negative voltages generating a train of

square wave pulses for use in timing of digital circuits. However, these are somewhat beyond the

scope of this course.

electronics notes 2016

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