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September 25, 2007 Version 1.7

Aspects of Analog Electronics

- 1 -

Chemistry 838

Thomas V. Atkinson, Ph.D. Senior Academic Specialist Department of Chemistry Michigan State University East Lansing, MI 48824

Table of Contents Table of Contents............................................................................................................................ 1 Table of Figures .............................................................................................................................. 2 Table of Tables ............................................................................................................................... 5 1. Constant Signals ......................................................................................................................... 5

1.1. Conductor ............................................................................................................................. 5 1.2. Resistor ................................................................................................................................. 6 1.3. Voltage Source ..................................................................................................................... 8 1.4. Series Combinations of Resistors - Voltage Divider............................................................ 9 1.5. Parallel Combinations of Resistors – Current Splitter ....................................................... 14 1.6. Thevenin's Theorem ........................................................................................................... 15 1.7. Loading............................................................................................................................... 18

2. Time Varying Signals ............................................................................................................... 23 2.1. Capacitors - Ideal................................................................................................................ 23 2.2. Circuits Containing Capacitors (RC Response to a Step Function) ................................... 24

2.2.1. Charging Case .............................................................................................................. 26 2.2.2. Discharging Case ......................................................................................................... 27 2.2.3. RC Response to a Pulse Function ................................................................................ 28

2.3. Capacitors - Real ................................................................................................................ 29 2.4. RC Response to a Sine Wave ............................................................................................. 31

3. Power Supplies ......................................................................................................................... 36 3.1. Introduction ........................................................................................................................ 36 3.2. Two Simple but Flawed Approaches.................................................................................. 38

3.2.1. Simple Approaches 1 ................................................................................................... 40 3.2.2. Simple Approaches 2 ................................................................................................... 41

3.3. Linear Series Power Supply ............................................................................................... 41 3.3.1. Transformers ................................................................................................................ 42 3.3.2. Diodes........................................................................................................................... 44 3.3.3. Rectification ................................................................................................................. 49 3.3.4. Filter ............................................................................................................................. 53

Chemistry 838 Aspects of Analog Electronics Table of Figures

September 25, 2007 Version 1.7 - 2 -

3.3.5. Regulation .................................................................................................................... 55 3.4. Switching Power Supply .................................................................................................... 57 3.5. Shunt Regulator .................................................................................................................. 57

4. Operational Amplifiers ............................................................................................................. 61 4.1. Comparator - Ideal.............................................................................................................. 61 4.2. Comparator - Real .............................................................................................................. 62 4.3. Follower, Buffer ................................................................................................................. 64 4.4. Follower with Gain............................................................................................................. 67 4.5. Inverter ............................................................................................................................... 69 4.6. Current Follower ................................................................................................................ 75 4.7. Summing Amp.................................................................................................................... 78 4.8. Integrator ............................................................................................................................ 79 4.9. Differentiator ...................................................................................................................... 84 4.10. Difference (Instrumentation) Amp ................................................................................... 85 4.11. Potentiostat ....................................................................................................................... 86 4.12. Op Amp Terminology ...................................................................................................... 89 4.13. OpAmp Models ................................................................................................................ 90 4.14. OpAmp Realities .............................................................................................................. 90 4.15. OpAmp Applications........................................................................................................ 92 4.16. OpAmps: Feedback, Control Theory, and Stability ......................................................... 92

5. REVISION HISTORY........................................................................................................ 97

Table of Figures Figure 1 - Conductive Properties of Matter 6 Figure 2 – Resistor 7 Figure 3 - Characteristic Curve for the Resistor (Ideal) 7 Figure 4 - Characteristic Curve for a Resistor (Real) 7 Figure 5 - Voltage Source 8 Figure 6 - Voltage Source (Generalized Representation) 8 Figure 7 - Characteristic Curve (Ideal Voltage Source) 8 Figure 8 - Characteristic Curve (Real Voltage Source) 8 Figure 9 - Voltage Divider 9 Figure 10 - Voltage Divider (Alternative Representation) 9 Figure 11 - Generalized Voltage Divider 10 Figure 12 - Voltage Divider (Three Resistances) 11 Figure 13 - Voltage Divider (n Resistances) 13 Figure 14 - Arbitrary Network of R's and V's 15 Figure 15 - Thevenin Equivalent Circuit 15 Figure 16 – Simple Circuit (two R’s and One V) 16 Figure 17 – Voltage Divider – Short Circuit Case 17 Figure 18 - Alternate Form 17 Figure 19 - Thevenin Equivalent - Short Circuit Case 17 Figure 20 - Alternate Form 17 Figure 21 - Ideal Voltage Measuring Device 18 Figure 22 - Real Voltage Measuring Device 18 Figure 23 - Ideal Current Measuring Device 19



Figure 24 - Real Current Measuring Device 19 Figure 25 - Real Voltage Source 19 Figure 26 - Real Voltage Measurement 20 Figure 27 - Equivalent Circuit 20 Figure 28 - pH Measurement 21 Figure 29 - Ideal Capacitor 23 Figure 30 - RC Circuit 24 Figure 31 - Step Function 24 Figure 32 - RC Response to Step Function 27 Figure 33 - Step Function (Discharge) 27 Figure 34 - RC Discharge 27 Figure 35 – RC (=0.1) Response to Pulse (VC) 28 Figure 36 – RC (=0.1) Response to Pulse (VR) 28 Figure 37 – RC (=1.3) Response to Pulse (VC) 29 Figure 38 – RC (=1.3) Response to Pulse (VR) 29 Figure 39 - Real Capacitor 29 Figure 40 – Leakage 29 Figure 41 - Switching RC Charging On and Off 30 Figure 42 – Real vs. Ideal Capacitors 30 Figure 43 – Sine Wave 31 Figure 44 – Bode Plot of VC 34 Figure 45 – Phase of VC 34 Figure 46 – Bode Plot of VR 35 Figure 47 – Phase of VR 35 Figure 48 – Low Pass Filter 35 Figure 49 – High Pass Filter 35 Figure 50 – Higher Order Filters 36 Figure 51 – Ideal High Pass Filter 36 Figure 52 – Ideal Low Pass Filter 36 Figure 53 - Ideal Band Pass Filter 36 Figure 54 - Ideal Notch Filter 36 Figure 55 - Complete Power Supply 37 Figure 56 - Example Electronic Load 39 Figure 57 - Goal for Sample Power Supply 39 Figure 58 - Assumption 39 Figure 59 - Voltage Divider to Produce 5 Volts 40 Figure 60 - Loaded Voltage Divider 40 Figure 61 - Second Feeble Attempt 41 Figure 62 - Generalized Linear Power Supply 42 Figure 63 - Ideal Transformer 42 Figure 64 - Ideal Transformer with two Secondary Windings 42 Figure 65 - Center Tapped Transformer 44 Figure 66 - Real Transformer 44 Figure 67 - Diode Symbol 45 Figure 68 - Ideal Diode Behavior 45 Figure 69 - Ideal Diode Characteristic Curve 45



Figure 70 - Real Diode Characteristic Curve 46 Figure 71 - Solid State Diode 46 Figure 72 - Diode Sensitivity to Temperature and Light 46 Figure 73 – Si Diode Characteristic Curve 47 Figure 74 - Si Diode Characteristic Curve (Forward Region) 47 Figure 75 - Si Diode Characteristic Curve (Zener Region) 48 Figure 76 - Half Wave Rectifier 50 Figure 77 - Half Wave Rectifier (Positive Half Cycle) 50 Figure 78 - Half Wave Rectifier (Negative Half Cycle) 50 Figure 79 - Full Wave Rectifier 51 Figure 80 - Full Wave Rectifier (Positive half Cycle) 51 Figure 81 - Full Wave Rectifier (Negative half Cycle) 51 Figure 82 - Full Wave Bridge Rectifier 52 Figure 83 - Series Regulator 55 Figure 84 - Series Regulator - First Order Model 55 Figure 85 - Series Regulator – Equivalent form 56 Figure 86 - Series Regulator - Hydraulic Analog 57 Figure 87 - Switching Supply - Hydraulic Analog 57 Figure 88 - Series Regulator Efficiency 57 Figure 89 - Switcher Regulator Efficiency 57 Figure 90 - Zener Diode Regulator 59 Figure 91 - Zener Diode Regulator (Model) 59 Figure 92 - Shunt Regulator Design Criteria 59 Figure 93 – Comparator 61 Figure 94 - Transfer Function - Ideal Comparator 61 Figure 95 - Real OpAmp 62 Figure 96 - Operational Amplifier Characteristic Curve (i vs V) 63 Figure 97 - Operational Amplifier Characteristic Curve (iout) 63 Figure 98 - Comparator Application 64 Figure 99 - Follower Amp 64 Figure 100 - Follower - Transfer Function 66 Figure 101 – Follower Application – The Challenge 66 Figure 102 - Follower Application – The Solution 67 Figure 103 - Follower with Gain 67 Figure 104 - Follower with Gain (Alternate Schematic) 67 Figure 105 - Comparison of Transfer Functions 69 Figure 106 - Inverter Amp 69 Figure 107 - Inverter Transfer Functions 72 Figure 108 - Thermistor 73 Figure 109 - Inverter Application - Measurement of Temperature I 73 Figure 110 - Inverter Application - Measurement of Temperature II 74 Figure 111 - Symmetry of the two Configurations 75 Figure 112 - Current Follower 75 Figure 113 - Inputs for Current Follower and Integrator 75 Figure 114 - Current Follower Application 77 Figure 115 - Large Photon Flux 77

Chemistry 838 Aspects of Analog Electronics Table of Tables


Figure 116 - Small Photon Flux 77 Figure 117 - Summing Amp 78 Figure 118 – Integrator 79 Figure 119 - Integrator - Simple Sweep Generator 81 Figure 120 - Integrator Realities 81 Figure 121 - Integrator - Variable Inputs 82 Figure 122 - Integrator - Single Saw Tooth 82 Figure 123 – Saw Tooth Generator 83 Figure 124 – Saw Tooth Time Course 83 Figure 125 – Differentiator 84 Figure 126 - Difference Amp 85 Figure 127 – Potentiostat 86 Figure 128 - Potentiostat Typical 88 Figure 129- Potentiostat Typical (View 2) 88 Figure 130 - OpAmp Offset Voltage 91 Figure 131 - OpAmp Offset Voltage (Inverter Example) 91 Figure 132 - Follower 93 Figure 133 - Time Course for Follower 94 Figure 134 - Damping 95 Figure 135 - Simple Op Amp Configuration 95 Figure 136 - Time Course of the Simple Circuit 96

Table of Tables Table 1 RM of Example VMDs 22 Table 2 - Power Converters 38 Table 3 - Effects of Power on Digital Devices 39 Table 4 - Si Diode Characteristic Curve Data 49 Table 5 - Paths through the Bridge Rectifier 53 Table 6 - Filter Activities 54 Table 7 - Responses of the Series Regulator to Changes 56 Table 8 - Responses of the Shunt Regulator to Changes 60 Table 9 - Effect of Offset on the Follower 91 Table 10 - Effect of Offset on the Inverter 92 Table 11 - Time Course for Follower 93 Table 12 - Time Course for the Simple Circuit 96

1. Constant Signals The study of analog electronics begins with the examination of the properties of a number of different devices. Next combinations of the devices will be examined.

1.1. Conductor Figure 1 illustrates an arbitrary piece of material. Consider any two points (say a, b in Figure 1) on the surface of the material. One can talk about the voltage difference, ΔV, between the two points and the current, i, that flows between the two points. If ΔV is 0 regardless of the value of

Chemistry 838 Aspects of Analog Electronics Constant Signals

i, then the material is called a conductor. In fact, for such a material, the voltage difference between any and all points on the surface is 0 and the surface is called an “equipotential” surface.

Conductor_01.cdr 16-Sep-2004

ΔVia

b

Material Figure 1 - Conductive Properties of Matter

The conductor can carry any amount of current without any voltage drop. Such devices are used as “wires” to connect various parts of the circuits to be built. Of course, there is no such thing as a perfect conductor; any real material can not carry current with out there being an associated voltage drop across the device.

1.2. Resistor If ΔV is a function of i, then the material is called a resistor and the relationship of ΔV and i is given by the following equation, e.g. Ohm’s Law.

iRVVV ba =−= )(

GVRVV

RVi ba =

−==

)(

Figure 2 is the symbolic representation of the resistive material. The device has two connections and R is a constant. R is the resistance of the device. G = 1/R is the conductance. The units of R are ohms. The units of G are mhos. The conductor can be thought of as a resistor with resistance = 0.


Chemistry 838 Aspects of Analog Electronics Constant Signals Resistor_01.cdr 25-Sep-2003

Va

VVb

+ -

i

R

Figure 2 – Resistor

When defining the voltage across a device in this document, we will use the following conventions.

)( rightleftdevice VVV −= or )( belowabovedevice VVV −=

Another characteristic of a resistor is the fact that power, P, is consumed and dissipated as heat.

RVRiP )( 2

2 Δ==

Figure 3 is another way of presenting the description of the resistor. The line is linear and continues to ∞ and represents the behavior of an ideal resistor. Ideal behavior means that the behavior of the device will obey Ohm’s Law regardless of the value of any physical parameter other than ΔV, i, and R.

Resistor_02.cdr 16-Sep-2004

ΔV

i

slope = R

Figure 3 - Characteristic Curve for the

Resistor (Ideal)

Resistor_03.cdr 16-Sep-2004

ΔV

i

operatingregion

Figure 4 - Characteristic Curve for a

Resistor (Real)

Figure 4 represents the behavior of a real resistor. For any real resistor there is limit to how much current the resistor may carry before the dissipation of the heat attendant to the passage of that current can no longer be adequately dissipated. If this operational region is exceeded by too



much, the device will catastrophically destruct (The smoke comes out!). Furthermore, the performance of the device may also be a function of other physical parameters such as temperature. Thus, any real resistor will have to operate within a set of boundary conditions, including P to approximate ideal behavior.

1.3. Voltage Source The next device to be considered is the voltage source. A voltage source is a device with two connections that is capable of delivering a voltage ΔV between the two connections. Figure 5 is the typical representation of a voltage source. VS is typically constant when this representation is used. Figure 6 is a more generalized representation especially if VS is a function of time.

VoltageSource_01.cdr 16-Sep-2004

VS

+

-

iS

ΔVS

+

- Figure 5 - Voltage Source


VS

+

-

iS

ΔVS

+

- Figure 6 - Voltage Source (Generalized

Representation)

Figure 7 illustrates the behavior of an ideal voltage source. Note that the output of the device does not vary with the current that is delivered. The ideal voltage source will also output the same constant voltage regardless of any other physical parameter such as temperature, time, humidity, etc.


Figure 7 - Characteristic Curve (Ideal

Voltage Source)


Figure 8 - Characteristic Curve (Real Voltage

Source)

Figure 8 illustrates the behavior of any real voltage source. While not ideal, such a real source will be acceptable as long as the error in the output voltage, VError, is within the error tolerance for a given application. Notice that the behavior is not necessarily symmetric with respect to the sign of the current delivered. In fact most voltage sources are only capable of outputting voltages of one sign, positive or negative.



1.4. Series Combinations of Resistors - Voltage Divider Next is the exercise of combining the devices that we now know into various combinations. Such combinations of devices are called circuits. Figure 9 represents the combination of a voltage source with two resistors. The goal is to understand the behavior of this circuit. To that end, we will analyze the circuit by making several observations which will yield a family of simultaneous equations which we will then solve for a single equation that describes the behavior of the circuit.

Figure 10 is an equivalent representation of the circuit under consideration. In fact, this is the form we will most often use.

Divider_01.cdr 25-Sep-2003

VS

a

ΔV1

ΔVS

ΔV2

+

-+

-

+

-

+

- i1

iS i2

R1

R2

Figure 9 - Voltage Divider

Divider_02.cdr 25-Sep-2003VS

ΔV1

V1

ΔV2

V2+

-+

-

i1

iS

i2

R1

R2

ΔVS

+

-

Figure 10 - Voltage Divider (Alternative Representation)

The set of physical observations (Steps 1, 2, 3, and 4) yield a set of simultaneous algebraic equations which are then combined.

iiii S === 21 There are no sources or sinks of current in the path through the circuit other than the voltage source.

1

1111 iRRiV == By Ohm’s Law and 1 2

2222 iRRiV == By Ohm’s Law and 1 3

21 VVVS += By Kirchhoff’s Voltage Law 4

Algebra is all that remains.

)( 2121 RRiiRiRVS +=+= Plugging 2 and 3 into 4 5

)( 21 RRVi S

+= Rearranging 5 6



1

1

RVi = Rearranging 2 7

)( 211

1

RRV

RV S

+= Combining 6 and 7 8

)( 21

11 RR

RVV S += Rearranging 8. The final result or

the Voltage Divider Equation which describes the behavior of the circuit.

9

The above case is the special case where one side of the divider chain is at common. A more generalized form can be developed as follows.

Divider_03.cdr 25-Sep-2003Vc

Va

ΔV1

Vb

ΔV2

+

-+

-

i1

iS

i2

R1

R2

Figure 11 - Generalized Voltage Divider

Begin with a set of observations.

iiii S === 21 There are no sources or sinks of current in the path through the circuit other than the voltage source.

1

1111 iRRiVVV ab ==−=Δ By Ohm’s Law and 1 2

2222 iRRiVVV bc ==−=Δ By Ohm’s Law and 1 3

21 VVVV ac Δ+Δ=− By Kirchhoff’s Voltage Law 4

Solving the system of equations is what remains.

)( 2121 RRiiRiRVV ac +=+=− Plugging 2 and 3 into 4 5

)( 21 RRVVi ac

+−

= Rearranging 5 6

)( 211 RRVV

RVVi acab

+−

=−

= Rearranging 2 7



)()()(

21

1

RRRVVVV acab +

−=− Combining 6 and 7 8

)()(

21

1

RRRVVVV acab +

−+= Voltage Divider Equation 9

If Va = 0, then we get the same results as above.

)( 21

1

RRRVV cb +

=

The above considerations can be extended to the case of an arbitrary series combination of resistances. Again, a series of physical observations (Steps 1-6) yield a set of simultaneous equations. The rest is simple algebra.


ΔV1

V1

ΔV2

ΔV3

V2

V3

+

-

+

-

+

-

i1

iS

i2

i3

R1

R2

R3

ΔVS

+

-

Figure 12 - Voltage Divider (Three Resistances)

iiiii S ==== 321 There are no sources or sinks of current in the path through the circuit other than the voltage source.

1

1111 iRRiV ==Δ By Ohm’s Law and 1 2

11 VV Δ= Because the common point is on one side of R1

3



321 VVVVS Δ+Δ+Δ= By Kirchhoff’s Voltage Law 6

Algebra is all that remains.



)( 321321 RRRiiRiRiRVS ++=++= Plugging 2, 3, and 4 into 6 7

)( 321 RRRVi S

++= Rearranging 7 8

1

1

RVi = Rearranging 2 9

)( 3211

1

RRRV

RV S

++= Combining 7 and 8 10

)( 321

11 RRR

RVV S ++= The final result for R1. 11

The derivation can be continued to find an expression for V2.

212 VVV Δ+Δ= By Kirchhoff’s Voltage Law 12

21212 iRViRVV +=+Δ= Plugging 4 into 12 13

)( 3211

1

RRRV

RVi s

++== Combining 2, 3, and 11 14

2321321

12 )()(

RRRR

VRRR

RVV ss

+++

++=

Combining 11,13, and 14 15

)( 321

212 RRR

RRVV s +++

= Rearranging 15. The final answer for V2.

16

Any arbitrary number of resistances can be combined in the serial sense. Figure 13 illustrates the generalized serial combination of resistances with a voltage source, e. g. the generalized voltage divider equation.




ΔV1

V1

Vn-2

ΔV2

ΔVn-1

ΔVn

V2

Vn-1

Vn

+

-

+

-

+

-

+

-

i1

iS

i2

in-1

in

R1

R2

Rn-1

Rn

ΔVS

+

-

Figure 13 - Voltage Divider (n Resistances)

The behavior of the circuit in Figure 13 can be derived in a manner similar to those above. The final result is shown below.

∑

∑

=

== n

kk

i

jj

Si

R

RVV

1

1

That is, a particular divider voltage is given by the source voltage multiplied by a fraction with the numerator equal to the sum of resistances from the bottom of the chain up to and including the resistor of interest. The denominator of the fraction is equal to the sum of all of the resistances. As an example, a series combination of 4 resistances would yield the following voltages.

4321

11 RRRR

RVV S +++=

4321

212 RRRR

RRVV S ++++

=

4321

3213 RRRR

RRRVV S +++++

=



1.5. Parallel Combinations of Resistors – Current Splitter

Parallel_01.cdr 16-Sep-2004VS

V1 V2

iS

i1ΔV1

+

-

R1 ΔV2

+

-

i2 R2

SVVV == 21 All three are connected by a conductor.

1

111 RiV = By Ohm’s Law 2

222 RiV = By Ohm’s Law 3

21 iiiS += By Kirchhoff’s Current Law 4

Algebra is what remains.

1

221 RRii = 1, 2, and 3 5

2

112 RRii = 1, 2, and 3 6

2

111 RRiiiS += Plugging 6 into 4 7

2

211

2

11 )1(

RRRi

RRiiS

+=+= Rearranging 7 8

21

21

RRR

ii

S += One of two “Current Splitter”

equations that describe how the current is split between the two legs.

9

21

12

RRR

ii

S += The other “Current Splitter”

equation, which can be derived in a fashion analogous to 9.

10

The derivation can now be carried a few steps further to complete the description of the behavior of this circuit.



21

21

RRR

iRV

S

S

+=

Plugging 2 into 9 11

parallelSSS RiRRRRiV =

+=

21

21 Rearranging gives the final results for a parallel combination of two resistors.

12

Step 12 includes the recognition that the fraction 21

21

RRRR

+has the units

Ω+ΩΩΩ or ohms. Thus, the

combination of R1 and R2 behaves like a single resistance with the value Rparallel.

1.6. Thevenin's Theorem Thevenin’s Theorem states that any arbitrary network, e. g. Figure 14, of voltage sources and resistors that has two external points of connection, PNa and PNb, can be replaced with the circuit shown in Figure 15. If the proper values of VT and RT are chosen, the two circuits will be indistinguishable if your access is limited to the two sets of external connections, i.e. PNa and PNb, PTa and PTb.

a

Pna

Pnb

b

iN

iN

VN

Thevenin_00.cdr 20-Sep-2004

Figure 14 - Arbitrary Network of R's and

V's


VT VTO

iTORT

VRT

PTa

PTa

+ -

Figure 15 - Thevenin Equivalent Circuit

As an exercise, we will find the Thevenin’s Equivalent for the simple circuit shown in Figure 16. We will examine two cases in order to derive the two parameters VT and RT.




VD

VDOi1

iDOi2

R1

R2 PDa

PDb Figure 16 – Simple Circuit (two R’s and One V)

Open Circuit Case: (iDO=iTO=0) For the first case, there is no external current path, i.e. connections outside of the box, between the external points, PTa and PTb, or PDa and PDb. To begin, we look at the circuit of Figure 16.

21

1

RRRVV DDO +

= Since iDO = 0, the circuit in Figure 16 behaves as a voltage divider yielding this result for the open circuit case.

1

Now look at the Thevenin circuit.

TTO VV = Since iTO = 0, Ohm’s law dictates that the voltage drop across RT will be zero and the voltage on both sides of RT will be the same.

2

21

1

RRRVVVV DDOTOT +

=== Combining 1 and 2, we obtain the value for VT.

3

Short Circuit Case: (VDO=VTO=0) For the second case, a conductor connects the external points, PTa and PTb and a conductor connects PDa and PDb. To begin, we look at the voltage divider circuit of Figure 16 which becomes Figure 17 with the placement of the short across the outputs. But the short results in VDO = 0 and Figure 18 describes the behavior of the voltage divider.




VD

VDOi1

iDOi2

R1

R2 PDa

PDb

Figure 17 – Voltage Divider – Short Circuit Case


VD

iDO

i2 R2

Figure 18 - Alternate Form

01 =i , DOii =2 Since VDO = 0, Figure 17 becomes Figure 18.

4

22 R

Vii DDO == By Ohm’s Law 5

Now we move to the Thevenin circuit.


VT VTO

iTORT PTa

PTa

Figure 19 - Thevenin Equivalent - Short Circuit Case

Thevenin_03.cdr 8-Oct-2003

VT

iTO RT

Figure 20 - Alternate Form

T

TTO R

Vi = By Ohm’s Law 6

DOTO ii = This must be true, since the two circuits are to behave the same.

7



2RVi

RVi D

DOT

TTO === Combining 5, 6, and 7 8

D

TT V

VRR 2= Solving for RT. 9

D

D

T VRR

RVRR 21

1

2+

=

Combining 9 with 3 10

21

21

RRRRRT +

= Simplifying 10 yields the answer 11

Notice that for this case the Thevenin equivalent resistance is the parallel combination of the two resistors in the other circuit. This is a non-intuitive result.

1.7. Loading Measurement of voltage is very important to electronic instrumentation. This section will examine a few of the realities of making voltage measurements. As always, the goal is to determine the value of the parameter of the system being studied without perturbing that system, i.e. changing it as a consequence of the measurement. Changing a system as a result of measurement is termed “loading.” The measurement system should not provide a load on the system being studied.

Figure 21 represents an ideal voltage measurement device, which displays a value for eM. For the ideal voltage measurement device i+ = i- = 0. Figure 22 is a first order model of a real Voltage Measurement device.

Display:show value

of eM

iM+

iM-

VMD_00.cdr 29-Sep-2003

Figure 21 - Ideal Voltage Measuring Device

Idea

l VM

D+

-

RM

iM-

iM

iM+

RealVMD_01.cdr 29-Sep-2003

Figure 22 - Real Voltage Measuring Device

Figure 23 represents an ideal current measuring device. In the ideal case, eM = 0. Figure 24 represents a real current measuring device. In the real device RM and eM will not be zero, but hopefully small.



Display:show value

of iM

iMCMD_00.cdr 29-Sep-2003

Figure 23 - Ideal Current Measuring Device

IdealCurrent

MeasuringDevice

iMCMD_01.cdr 29-Sep-2003

RM

Figure 24 - Real Current Measuring Device

Figure 25 is the first order model of a real voltage source. The object is to deliver VS to the output connections, e.g. when trying to measure the value of VS, but only eS and iS are accessible.

RealVoltageSource_01.cdr 29-Sep-2003

+ -

Figure 25 - Real Voltage Source

A real measurement consists of connecting a real voltage measurement device to the real source. Figure 26 illustrates that measurement. Figure 27 is an equivalent representation of the real measurement.



Idea

l VM

D

Real Voltage Measuring Device

(VMD)

Real Voltage Source

+

-

RM

iM-

iM

iM+

Loading_01.cdr 2-Oct-2003

+ -

Figure 26 - Real Voltage Measurement

Loading_02.cdr 29-Sep-2003

VS

VM

iM

iS

RM

RS

Figure 27 - Equivalent Circuit

The relationship of VM and VS is given by the following.

MS

MSM RR

RVV+

=

This can be rearranged as follows.

1

1

+=

M

SSM

RRVV

Examine the following limits.

As then ∞→MR 0→M

S

RR and SM VV →

As then 0→SR 0→M

S

RR and SM VV →

As then ∞→SR ∞→M

S

RR and 0→MV

As then 0→MR ∞→M

S

RR and 0→MV

The deduction to be made from the above is that RM should be as large as possible and RS should be as small as possible to make the best measurement. In real life, there is nothing to be done with RS, as it is a characteristic of the system being studied. RM is a characteristic of the VMD




op across

Thus,

At this point, an example is in order. Figure 28 illustrates the measurement of pH using a glass n

that is to be used. To make RM larger, one must rebuild the measurement instrument, or more realistically, choose another that is more appropriate for the measurement to be made.

Another view of this is that the current iS should be as small as possible. The voltage drRS is the error in the measurement.

VVVVV +=+= errorSRSSM

SSerror RiV =

electrode. This system can be modeled with the real voltage source of Figure 25. VS is a functioof pH (typically 59 millivolts for a neutral solution and RS = 100 Mohms). To measure pH, one measures VS and then uses the function relating pH and VS to determine the pH.

pHElectrode_01.cdr 1-Oct-2003

Figure 28 - pH Measurement

As part of this example of a real r real voltage measuring l

measurement, we will look at foudevices that exist in our lab, candidates for use in this measurement. Table 1 contains typicavalues of RM for a number of VMDs available in CEM 838.


Table 1 RM of Example VMDs

VMD Typical RM (ohms)

DMM 10M

Oscilloscope 1M

VOM 10000

LF411 (FET input Op Amp)

1012

These will give the following results when used to measure eS and, hopefully, VS.

VDM Displayed Value of SM

MSM RR

RVe+

=

VOM SSSSM VVVVe 00009999.010001

11010001

101010

104

4

84

4

==×

=+

=

Oscilloscope SSSSM VVVVe 0099.0101

110101

101010

106

6

86

6

==×

=+

=

DMM SSSM Vx

VVe 091.0101.1

101010

108

7

87

7

==+

=

LF411 SSSSM VVVVe 99990.00001.11

100001.110

101010

12

12

812

12

==×

=+

=

Clearly, LF411 is by far the best choice to make the measurement. The minimization of loading is especially important in the case of the pH measurement. If the current through the cell is not zero, the equation relating pH and VS no longer holds. Electrochemical reactions begin to occur at the electrode surfaces and the system being measured changes.

One might ask why not make the measurement and plug the values for VM, RS, and RM into the

equation MS

MSM RR

RVV+

= and solve for VS? While this is algebraically appropriate, this

approach is not practical for the following reasons.

1. Typically we don’t know RM and RS and they are usually very hard to measure.


Chemistry 838 Aspects of Analog Electronics Time Varying Signals

2. RM and RS may vary with time, temperature, geometry of the electrodes, composition of the solution…

3. In some cases, in particular the pH measurement, the system being measured might change if iS is not zero. Indeed the relationship relating pH and epH, is predicated on the constraint that iS = 0. If this is not true, electrochemistry takes place and reactions go on at the electrodes and change the solution.

However, these concerns are minimized as the loading is decreased, allowing the measurement to be made without knowing the values of RM and RS, knowing how they are changing, or having the system being measured be perturbed.

2. Time Varying Signals 2.1. Capacitors - Ideal A capacitor consists of two metal plates separated by space. Each plate has a conductor attached that is available for external connections. Figure 29 is the schematic drawing for such a device.

Capacitor.cdr 9-SEP-2003

Figure 29 - Ideal Capacitor

We will assume that the charges on the two plates are equal and of opposite sign.

qqq == −+ Equation 1

The following describes the behavior of the capacitor, where C is a constant called capacitance and is related to the geometry of the two plates.

( ) ( )CtqtVC = Equation 2

Since current is the movement of charge, the following two relations hold.



( ) ( )dttdqti = Equation 3

( ) ∫= idttq Equation 4

Combining the three expressions above leads to the following.

( ) ( )Ci

dttdq

CdttdVC ==

1 Equation 5

( ) ∫= idtC

tVC1 Equation 6

Equations 2, 5, and 6 are different expressions that describe the behavior of the capacitor.

2.2. Circuits Containing Capacitors (RC Response to a Step Function) The next step is to investigate the behavior of circuits containing a capacitor and other components. Figure 30 shows a simple combination of a R, a C, and a voltage source. The object is now to understand the behavior of this circuit. As it turns out, the behavior of the circuit will depend on the nature of the excitation signal, VRC(t). To begin, consider the case of VRC(t) being a step function such as shown in Figure 31.

RC.cdr 10-SEP-2003

R

V (t)RC

V (t)Ri(t)

V (t)C

Figure 30 - RC Circuit

StepFunction.cdr 10-SEP-2003

V (t)RC

Figure 31 - Step Function

At the beginning, assume that the capacitor is discharged, i.e. q(0) = 0 and hence 0)0( =CV . The objective is to understand the behavior of , , and . )(tVC )(tVR )(ti

( ) ( ) ( ) ( ) ( )RtV

RtV

RtVtVti CRCCRC −=

−= Equation 7



RtitVR )()( = (by Ohm’s Law) Equation 8

)()()( tVtVtV CRRC += (by Kirchhoff’s Voltage Law) Equation 9

)()()( tVtVtV CRCR −= (by Kirchhoff’s Voltage Law) Equation 10

Before time = a, the capacitor is discharged, i.e. q = 0. The voltage drop across R will be 0, since both VC and VRC are 0. Thus, no current will be flowing and the state of the capacitor will not change. At time = a, VR will instantaneously be E. and i(a) will be equal to E/R. However, the flow of current will deposit charge on the plates of the capacitor leading to an increase in VC which will decrease the driving voltage VR which will decrease i. The object is to derive exact expressions for the variables of the circuit.

( ) ( ) ( )RtV

RtV

RVti CRCR −== Equation 11

( )( ) ( )

( ) ( )RC

tVtVCR

tVtV

Ci

dttdV CRC

CRC

C −=

−

== Equation 12

Using the fact that VRC(t) is constant after time a yields the differential equation below.

( ) ( )RC

tVEdttdV CC −

= Equation 13

Make the following definition

( )tVE C−=ξ Equation 14

Differentiating the above yields.

( ) ( )dttdV

dttd C−=

ξ Equation 15

( )RCdt

td ξξ−= Equation 16

( )RCdttd

−=ξξ Equation 17



Integrating both sides yields

tt

RCt

00ln −=ξ Equation 18

RCtt −=− )0(ln)(ln ξξ Equation 19

eRCtt −

=)0()(

ξξ Equation 20

e)ξ(ξ(t) RCt

0−

= Equation 21

eCVEtVV RCt

CRC))0(()( −=−

−

Equation 22

RCt

CC eVEEtV−

−−= ))0(()( Equation 23

2.2.1. Charging Case

The charging case occurs when the boundary conditions are VC(0) = 0 and VRC=E.

RCt

C EeEtV−

−=)( Equation 24

)1()( RCt

C eEtV−

−= Equation 25



RC Response to Step Funcltion

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

Time

VC, I VC (RC = 0.5)

VC (RC = 1.0)VC (RC = 2.0)I (RC=1.0)

Figure 32 - RC Response to Step Function

2.2.2. Discharging Case

StepFunction_1.cdr 14-SEP-2003

V (t)RC

Figure 33 - Step Function (Discharge)

RC Discharge

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7

Time

Vc

Vc, Ic (RC=1.0)Vc, Ic (RC=0.5)Vc, Ic (RC=2.0)

Figure 34 - RC Discharge

The discharging case occurs when the boundary conditions are VC(0) = E and VRC(t)=0. In this case and before time a, the capacitor has been charged to E and the driving force VRC(t) - VC(t) = E – E = 0 and i = 0. At the instant (t = a) that the step occurs, the driving voltage is 0 – E and current starts to flow and remove charge from the capacitor, i.e. the capacitor is being discharged. But as the charge is removed, the driving voltage decreases, and hence, the discharging current decreases. The exact behavior is found by plugging the boundary conditions into the general solution, i.e. Equation 23.

RCt

CC eVEEtV−

−−= ))0(()( Equation 26



RCt

C eEtV−

−−= )0(0)( Equation 27

RCt

C EetV−

=)( Equation 28

2.2.3. RC Response to a Pulse Function StepFunction_3.cdr 14-SEP-2003

V (t)RC

RC Response to PulseRC=0.1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

Time

VC

Figure 35 – RC (=0.1) Response to Pulse (VC)


-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

Time

VR

Figure 36 – RC (=0.1) Response to Pulse (VR)




-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

Time

VC

Figure 37 – RC (=1.3) Response to Pulse (VC)


-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7

Time

VR

Figure 38 – RC (=1.3) Response to Pulse (VR)

2.3. Capacitors - Real In case of the ideal capacitor (See Figure 29), the two plates are separated by a perfect insulator, i.e. no current passes between the two plates. Any flow of charge occurs via the external connections. A real capacitor (see Figure 39) has the two plates separated by a real insulator. Such an insulator, being not perfect, allows current to flow between the two plates. This leakage can be represented by a resistance, RInsulator and a current, ileakage as illustrated in Figure 40.

Capacitor01.cdr 9-SEP-2003

Figure 39 - Real Capacitor

Capacitor02.cdr 9-SEP-2003

Rinsulator

Figure 40 – Leakage

The better the insulator, the larger RInsulator will be and, hence, the smaller the current, ileakage, will be for any given VC.



RC1.cdr 15-SEP-2003

R

V (t)RC

V (t)Ri(t)

V (t)C

Figure 41 - Switching RC Charging On and Off

Revisiting the step function response of the RC circuit, the non-ideal behavior of a real capacitor can be explored with the circuit shown in Figure 41. If the same conditions are applied as before, e.g. VC(0) = 0, E= 1.0, RC=1.0 plus the switch is closed in the beginning and opened at time = 1, then the behavior of Figure 42 will be seen. In the ideal case, once the switch is opened, there is no driving force present to cause a current to flow and VC will be constant and equal to the value at the time the switch was opened. In the case of the real capacitor, the leakage current, ileakage, will discharge the capacitor with a time constant equal to RInsulatorC. In the illustration RInsulatorC is 6 times RC. As better quality capacitors are used, the value for RInsulator will increase and the decay become slower and slower.

Real vs Ideal Capacitors - Leakage

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1 2 3 4 5 6 7 8

Time

Vc

Ideal CaseReal Case

Figure 42 – Real vs. Ideal Capacitors



2.4. RC Response to a Sine Wave

π/2 (90 )°

0

3 /2 (270 )π °

π (180°)

π/4 (45 )°3 /4 (120 )π °

5 /4 (225 )π ° 7 /4 (315 )π °

ω

Periods0 0.5

1.0

-1.0

0Vp Vpp

1.0 1.5 2.0 3.0 3.5

Sine.cdr 15-Sep-2003

Figure 43 – Sine Wave

Now we will examine the case where the input function VRC is a sine wave such as described in Figure 43. Again the goal is to understand the behavior of the circuit. To begin, the following observations are as in the step function case.

( ) ( ) ( )tVtVtV CRRC += (Kirchhoff’s Voltage Law) Equation 29

or, upon rearrangement,

( ) ( ) ( )tVtVtV CRCR −= Equation 30

( ) RtitVR )(= (Ohm’s Law) Equation 31

Combining the last two equations

( ) ( ) ( )RtV

RtVti CRC −= Equation 32

Recalling the relationships between current and charge and the basic relationships governing the capacitor the following are true.

( ) ( )CtqtVC = Equation 33

( ) ( )dttdqti = Equation 34

( ) ∫= idttq Equation 35



( ) ( )Ci

dttdq

CdttdVC ==

1 Equation 36

Applying the sine wave as the excitation function of the circuit

tVtV RCpRC ωsin)( = Equation 37

Combining the above

( ) ( ) tVtVtV RCpCR ωsin=+ Equation 38

( ) tVCtqRti RCp ωsin)( =+ Equation 39

( ) tVCtq

dttdqR RCp ωsin)(

=+ Equation 40

We arrive at the differential equation that governs the behavior of the circuit.

( ) tRV

RCtq

dttdq RCp ωsin)(

=+ Equation 41

Solving the above differential equation is beyond the scope of this discussion but yields a solution that contains an initial transient part which decreases exponentially with time and leads into a steady state part. The steady state solutions include expressions for VC(t), VR(t), and i(t) which will describe the behavior of the system at after the initial transient has died down.

)sin()( CCpC tVtV φω += Equation 42

)sin()( RRpR tVtV φω += Equation 43

)sin()( RRp tRV

ti φω += Equation 44

Notice that all of the resultant sine waves have the same frequency as the excitation signal but vary in amplitude and phase. The relations for the various parameters for the capacitor are the following. The steady state solutions yield the following relationships. XC is defined and called the reactance of the capacitor.



fCCXC πω 2

11== Equation 45

The relationships for VC are the following.

( ) ( )22 RX

XVV

C

C

RCp

Cp

+= Equation 46

CC X

R−=φtan Equation 47

2)(tan)(tan 11 πφ −=

−= −−

RX

XR C

CC Equation 48

The relationship for current is the following.

( )( ) ( )

( i

C

RCp tRX

Vti φω +

+−= sin

22) Equation 49

The relationships for VR are.

( ) ( )22 RX

RVV

CRCp

Rp

+= Equation 50

°−== 90CRi φφφ Equation 51

2)(tan)(tan 11 πφ +== −−

C

CR X

RRX Equation 52

Now, look at a special case where ω0 is defined to be the frequency at which VC = VR

RC1

0 =ω Equation 53

Plug this into Equations 46 and 50 and simplify,

( ) ( ) 21

11

1)(220 =

+== ωω

p

Cp

VV

Equation 54



( ) ( ) 21

11

1)(220 =

+== ωω

p

Rp

VV

Equation 55

A qualitative understanding of the behavior of this circuit can be seen by taking the limit of the relationship between the amplitudes of the input and output sine waves, the frequency of the input sine wave, and the values of R and C.

( ) ( ) 222

1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=+

=

C

C

C

RCp

Cp

XRRX

XVV

Taking the limit of this result:

As ω → ∞ then 01⎯→⎯=

CXC ω

and ∞⎯→⎯⎟⎟⎠

⎞⎜⎜⎝

⎛2

CXR and 0⎯→⎯

RCp

Cp

VV

As ω → 0 then ∞⎯→⎯=C

XC ω1 and 0

2

⎯→⎯⎟⎟⎠

⎞⎜⎜⎝

⎛

CXR and 1⎯→⎯

RCp

Cp

VV

Thus, the amplitude of VC tends to 0 as the frequency increases. A more quantitative view is given by plotting the functions for VC as seen in Figure 44 and Figure 45.

Bode Plot Vc

-80.00

-70.00

-60.00

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

10.00

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

log(w/w0)

20*lo

g(Vc

p/Vr

cp) =

VpC

/Vrc

p ,d

B

Figure 44 – Bode Plot of VC

Vc Phase Angle

-100.00

-90.00

-80.00

-70.00

-60.00

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

10.00

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

log (w/w0)

Phi V

c (d

egre

es)

Figure 45 – Phase of VC

Similarly, the behavior of VR can be seen by taking various limits of the relationship governing the behavior of VR.

( ) ( )

1

1222

+⎟⎠⎞

⎜⎝⎛

=+

=

RXRX

RVV

CCRCp

Rp



Taking the limit of the above:

As ω → ∞ then 01⎯→⎯=

CXC ω

and 02

⎯→⎯⎟⎠⎞

⎜⎝⎛RXC and 1⎯→⎯

RCp

Rp

VV

As ω → 0 then ∞⎯→⎯=C

XC ω1 and ∞⎯→⎯⎟

⎠⎞

⎜⎝⎛

2

RXC and 0⎯→⎯

RCp

Rp

VV

Again, a more quantitative view can be obtained by plotting the relationships for VR.

Vr Bode Plot

-80.00

-70.00

-60.00

-50.00

-40.00

-30.00

-20.00

-10.00

0.00

10.00

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0

log (w/w0)

20*lo

g(Vr

p/Vr

cp) =

Vrp

/Vrc

p, d

B

Figure 46 – Bode Plot of VR

Vr Phase Angle

-10.00

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

log (w/w0)

Phi V

r (de

gree

s)

Figure 47 – Phase of VR

A common practice is to consider VRC as an input signal and VR and VC as output signals. Figure 44 can then be thought of as the transfer function of the circuit in Figure 48. Notice that the circuit tends to pass input sine waves with a frequency less than ω0 with little attenuation. In contrast, the amplitude of the input sine waves are increasingly attenuated as the frequency increases beyond ω0. For this reason, this circuit is called a Low Pass Filter.

Filter_10a.cdr 19-SEP-2004

R

Vini Vout

Figure 48 – Low Pass Filter

Filter_13.cdr 20-SEP-2004

RVin i Vout

Figure 49 – High Pass Filter

Turning to VR, Figure 46 is the transfer function for the circuit found in Figure 49. In this case, Frequencies above ω0 are passed through the circuit with little attenuation. The lower frequencies are attenuated. Thus, this circuit is called a High Pass Filter.

These two filters are first order filters and are the simplest of an unbounded set of filters resulting from more complicated combinations of resistors, capacitors, and inductors. As the number of components increase, the order of the filter and the steepness of the fall-off increase. Figure 50 illustrates the Bode plot for higher order Low Pass Filters. The unattainable limit is the ideal filters show in Figure 51 and Figure 52.


Chemistry 838 Aspects of Analog Electronics Power Supplies

Bode Plot Vc

0.00

10.00

-80.00

-70.00

-60.00

-50.00

-40.00

-30.00

-20.00

-10.00

-5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

log(w/w0)

20*lo

g(V

cp/V

rcp)

=V

pC/V

rcp

,dB

Figure 50 – Higher Order Filters


Filter_1.cdr 15-SEP-2003 Filter_2.cdr 15-SEP-2003

Log(

V/V

)O

UTp

INp

Log(

V)

OU

TpIN

p/V

Figure 51 – Ideal High Pass Filter Figure 52 – Ideal Low Pass Filter

Filters can be combined to make more complicated filters. Figure 53 and Figure 54 are ideal Band Pass and Notch filters.


Log(

V


Log(

V/V

)IN

p

/V)

INp

OU

Tp

OU

Tp

lo high

Figure 53 - Ideal Band Pass Filter

lo high

Figure 54 - Ideal Notch Filter

3. P

veloped many “devices,” i.e. instrument systems, which can be cts of our environment. These devices are based on physical principles:

ower Supplies 3.1. Introduction Over the centuries, man has deused to measure aspemechanical, hydraulic, chemical, electrical, etc. Over the last century, the vast majority of instrument systems have come to include electronic devices. Such electrical and electronic



get

w the device to be powered. The vast majority of modern instrument

ved

as

devices require power to operate. This section will concentrate on how the delivery of the electrical power is achieved.

Another way of saying this: “All laws of thermodynamics are always obeyed.” “You don’tsomething for nothing.”

Figure 55 generalizes the problem. A primary power source generates the power that is somehodelivered to the “load” orsystems derive the necessary power from the ubiquitous electrical power provided by your local electrical power company and delivered to your premise via wires. Of course, if you are involin remote sensing applications, e.g. the planetary probes, other power systems will prevail.

The electrical power industry can be said to begin in 1882 when Thomas Alva Edison opened the first power plant on Pearl Street in New York City. This system distributed DC power and winitially used to power street lights. During the 1880’s and 1890’s the power industry struggled to decide if electrical power should be distributed in DC form or AC form. Edison, Lord Kelvin,and other notables were on the side of DC distribution. Westinghouse, Tesla, and others were onthe side of AC distribution. The intensity of this conflict is reflected in the fact that when the DC proponents were loosing, they began to widely publishing statements alluding to the inherent dangers of AC power as “illustrated” by the “electric chair” which had recently been invented as a means of performing executions. However, as history will attest, the AC forces prevailed. Except for a very few pockets of DC power distribution, the world distributes power in the AC form.

PowerSupply_01.cdr 17-Sep-2004

PrimaryPowerSource

DistributionSystem

Complete Power Supply

Steam (gas, oil, coal, nuclear) driven generatorhydroelectric generator

batteryfuel cell

solar cellwindmill

tide generator

ACheatersmotorslights

DCheatersmotorslights

electronicscomputers

What is usually called a Power Supply

"Converter""Power Supply"

or"Conditioner"

Device(Load)

Figure 55 - Complete Power Supply

Thus, electrical power is com n the form of AC, i.e. sinusoidal electrical signals with a frequency of 50 or 60 Hz and amplitudes of 110-120 VRMS,

monly available through out the world i



e

al ll have to be a “converter” employed to match the power available to that needed. There

f if a

t

Type

208-240 VRMS, 440 VRMS, etc. The actual forms available depend on where you are in thworld.

The devices to be powered as alluded to in Figure 55 require various forms of power. In generthere wiare 4 possibilities. The role of the converter is to convert the available form of power into the form that is appropriate for the load and while consuming as little power as possile in the process. The efficiency of the conversion is important for two reasons. First, there is the cost othe power lost in the process. For most instrument systems this cost is not great. However,load consumes a lot of power, the converter will also consume a large amount and the actual coscan be significant. The other motivation for efficiency is that the power consumed is converted into heat. This heat must be dissipated. Otherwise the system overheats and may even catastrophically destruct.

Table 2 - Power Converters

Examples

AC→AC Transformers

DC→DC Auto ignition systems, switcher power supplies

DC→AC inverters

AC→DC What one usually means with the term “power supply”

.2. Two Simple but Flawed Approaches following two examples of converting AC .

These two examples are very naïve and inappropriate except to illustrate the possibility of erroneous approaches to solving a real problem.

Assum un as illustra y large and expensive

3To emphasize all of these concepts, consider the power to what is needed for an electronic system

WARNING!

e that you have a large computer that requires 100 amps of 5 volt DC power to rted in Figure 56. This is a very realistic problem as there are man

computing systems in this category. The power being supplied to such digital devices must becarefully controlled as described in Table 3.



5 VDC

100 amps

Computer

Figure 56 - Example Electronic Load

Table 3 - Effects of Power on Digital Devices

Input Power* Effect

> 5.5 volts Device will not work and will eventually catastrophically destruct (The smoke will come out.)

4.5 to 5.5 Proper operation

0 to 4.5 Either:

1.) Device does not work

2.) Device appears to work but gives erroneous results.

<0 Device will not work and will eventually catastrophically destruct (The smoke will come out.)

*These specifications are nominal. The exact values will vary with particular devices.


115 VAC 100 ampsPowerSupply

5 volts

CPU

Figure 57 - Goal for Sample Power Supply

Figure 58 illustrates a simplifying assumption that will be made for these two examples. The assumption is only partially far fetched. The assumption is that the AC power signal provided by the power company can be converted to a constant voltage of 163 volts which corresponds to the maximum amplitude of the 60 Hz 115 VAC sine wave. This follows from the traditional practice of using volts RMS (Root Mean Square) for the units of the voltage delivered by the power grid.

peakpeak

RMS VV

V ∗== 707.02

for a sine wave.


V = 163 voltsp 163 volts0

assume thiscan be done

Figure 58 - Assumption



3.2.1. Simple Approaches 1

We know that a voltage of 5 volts can be generated with the voltage divider shown in Figure 59. However, this will not work.


V = 163 voltsS

iS

R =158K/1632

R =5K/1631

ΔV = 158 volts2

+

-

Δ V = 5 volts1

V1+

-

Figure 59 - Voltage Divider to Produce 5 Volts


V = 163 VDCS

VCPU

iS

iCPU

R =158K/1632

R =5K/1631 CPU ΔV1

V1+

-

ΔV2

+

-

Figure 60 - Loaded Voltage Divider

5163

516315855163

163158

1635

1635

16321

11 ==

+=

+=

+=

KK

K

RRRVV S

Ω== 05.0100

5ampsvoltsRCPU

Ω==+

=+

=+

= 049919.07248.30

5337.105.06748.30

163250

05.01635

05.0*1635

1

1

K

K

RRRRR

CPU

CPUparallel

voltsKRRR

VVparallel

parallelS 008394.0

38.969049919.0163

33.969049919.0049919.0163

163158049919.0

049919.01632

1 ==+

=+

=+

=

As you can see, the voltage divider is incapable of delivering the desired power to the load. This should not be surprising, since the derivation of the voltage divider equation is predicated on the constraint that i1 = i2, i.e. no current is pulled off from the junction between the two resistors.



3.2.2. Simple Approaches 2 PowerSupply_06.cdr 18-Sep-2004

V = 163 VDCS

i = 100 ampsS

R =(163-5)/100

2

=(158)/100 =1.58

R =0.051 CPU ΔV1 = 5 volts

+

-

ΔV2 = 158 volts

+

-

Figure 61 - Second Feeble Attempt

Figure 61 illustrates the second attempt to achieve the desired results. This circuit would, indeed, deliver the 100 amps at 5 volts. However, there are three serious problems with this approach. First of all, the circuit is not able to adapt to changes in voltage of the power from the power grid. Second, the circuit is not able to adapt to changes in the load presented by the computer. This can be a problem with many computer systems as sub-systems are kept on-line when needed and are turned off when not needed. The final problem is seen by examining what is happening with the 1.58 Ω resistor. The power dissipated by this resistor is calculated below.

wattsampsvoltsViP 158001001582 =∗=Δ=

As you can see an immense amount of power is dissipated. This is of concern because of the cost of this wasted energy and because of the attendant amount of heat that must be removed from the system to prevent destruction of the devices or the environment. Furthermore, finding a resistor that can handle such power can be a challenge. The resistors used in the CEM 838 Lab are typically ¼ watt. Thus, you would need about 64000 of these resistors in parallel to handle such power.

As you will see as this section unfolds, the power supply is constructed to counteract these problems and deliver power to the load at a constant voltage.

3.3. Linear Series Power Supply Figure 62 illustrates the generalized linear power supply, which takes the AC power delivered by the electric grid and converts the power into a constant DC supply that is, to a large degree, impervious to variations in the power grid and variations in the load. The remainder of this section will investigate the various sub-systems shown in Figure 62.

Figure 62 illustrates a linear or series power supply. The term series is derived from the fact that the regulator and the load are in series. The depicted system is completely generalized. Any given implementation may not have some of the sub-systems. Furthermore, discussion of the line side and load side protections is beyond the scope of this document.



PowerSupply_01a.cdr 19-Sep-2004

LineSide

ProtectionTrans-former

FuseCircuit Breaker

Ground Fault InterrupterSurge ProtectorBattery Backup

Over Voltage Disconnect

RFI Filters

Over Current DisconnectOver Temperature DisconnectEMI Schielding

Rectifier Filter RegulatorLoad

(Computer,instrument, etc.)

LoadSide

Protection

0 0 0 0 0

Figure 62 - Generalized Linear Power Supply

3.3.1. Transformers

VSVp np

ns

ip

is

core

Transformer_01.cdr 16-Sep-2004

Figure 63 - Ideal Transformer

VS2

VS1

Vp np

ns2

ns1

ip

is2

is1core

Transformer_02.cdr 16-Sep-2004

Figure 64 - Ideal Transformer with two

Secondary Windings

The first sub-system to be considered is the transformer, a device that transforms AC power from one form to another. This device will be used in the power supply to get the maximum voltage of the sine wave power closer to the desired voltage to be delivered to the load. In the simplistic case, the amplitude of the sine wave should be equal to the desired constant DC voltage desired for the load. In reality, the various sub-systems require some head room, i.e. the input amplitude will have to be greater than the output amplitude. Thus, the real role of the transformer is to transform the input power to that with the minimum amplitude needed for the remaining sub-systems to operate and still deliver the desired voltage at the end of the power supply. Again, the idea is to minimize voltage drop over the sub-systems to minimize power dissipation in any component other than the load. Figure 63 illustrates the symbol used for an ideal transformer.

A transformer consists of two coils of wires wound around a core of magnetic material. The input winding is called the primary winding or primary for short. The other winding is the output of the device and is called the secondary winding. np is the number of turns (coils) of wire in the



primary coil. ns is the number of turns (coils) of wire in the secondary coil. The connections to the primary and secondary have traditionally been called “taps.”

The behavior of the device is given by the following relationships. All signals here are sine waves of the form


) sin( φω += tVV a

where V is a function of time, Va is the peak amplitude and is constant, ω is the frequency in radians, and φ is the phase angle. For the ideal transformer, the following are true.

)sin()sin( tVtVV PpPPpP ωφω =+=

)sin()sin( tVtVV SpSSpS ωφω =+=

P

S

Pp

Sp

nn

VV

=

SSPP ViVi =

Thus, the input and output are both sine waves of the same frequency and phase. The ratio of the amplitudes of these two sine waves is the ratio of the number of the coils in each winding. This ratio is often called the “turns ratio” of the transformer. The last relationship immediately above is an indication that there is no power consumed by the transformer.

If then PS nn > 1>Pp

Sp

VV

and . This is called a “Step up” transformer. PpSp VV >

If then PS nn < 1<Pp

Sp

VV

and PpSp VV < . This is called a “Step down” transformer.

If then PS nn = 1=Pp

Sp

VV

and PpSp VV = . This is called an “Isolation” transformer. There

can be separate commons on the primary and secondary sides of the circuit.

A transformer may have more than one secondary winding. Figure 64 illustrates a transformer with two secondary windings. The following describes the behavior.

)sin()sin( tVtVV PpPPpP ωφω =+=

)sin()sin( 1111 tVtVV pSSpSS ωφω =+=

)sin()sin( 2222 tVtVV pSSpSS ωφω =+=

P

S

Pp

pS

nn

VV 11 =

P

S

Pp

pS

nn

VV 22 =


Figure 65 illustrates the special case when ns1 = ns2 and the two nearest ends of the secondary windings are tied together. Figure 65 also shows the different ways the outputs can be considered by changing the definition of the common point.

pSpS VV 21 =

pSpSpSpSSp VVVVV 2121 22 ==+=

pSapS VV 11 −=

pSapS VV 22 =

VS2 VS2a

Vp np

ipis

Transformer_02a.cdr 16-Sep-2004

is

VS1 VS1a

ns

ns VS

+

+-

-

+

+

-- 0

0

0

VS2

VS1

+

+-

-

VS

+

-

0

0

VS2a

VS1a

+

+

--

Figure 65 - Center Tapped Transformer

Finally, we take a quick look at the real transformer shown in the second order model of Figure 66. Perhaps the most pronounced non-ideality of the circuit is the resistances, RP and RS, of the windings and RLeakage which introduce i2R heating. CLeakage introduces phase shifts.

nP

nS

RS

RP

RLeakage

CLeakage

ip

isTransformer_05.cdr 21-Sep-2004

VP VS

Figure 66 - Real Transformer

3.3.2. Diodes

The next subsection of the power supply converts the AC signal to a time varying DC signal, i.e. one with a single sign. The diode is a device that will enable this transformation. Figure 67 is the symbol used to represent the diode.



Vd

+ -

id

Diode_10.cdr 14-Sep-2006

Figure 67 - Diode Symbol

3.3.2.1. Ideal Diode

Figure 68 and Figure 69 illustrate the behavior of the ideal diode. Notice that the device acts as either a perfect “open” or a perfect “short” depending on the sign of the voltage across the device.

Vd

V >0 (forward biased)d

V <0 (reverse biased)d

+ -Diode_11.cdr 14-Sep-2006

id

id x

Figure 68 - Ideal Diode Behavior

Vd

Diode_11a.cdr 21-Sep-2004

id

forward bias regionreverse bias region

Figure 69 - Ideal Diode Characteristic

Curve

3.3.2.2. Real Diode

Figure 70 shows the behavior of a real diode. There are three regions of behavior: forward biased, reverse biased, and breakdown. Figure 73, Figure 74, Figure 75, and Table 4 contain the iD,VD data that characterizes a real Silicon Zener diode used in the CEM 838 Lab.

Real diodes are used for a number of applications. The actual devices are optimized for one of these applications.

• Signal – used to switch signals and thus is optimized for fast changes between the forward and reverse biased regions.

• Power – used to rectify AC signals

• Light sensitive – When Vd is held at a constant value in the reverse biased region, id is a function of the intensity of light impinging on the junction of the diode. Figure 72 shows



the qualitative relationship between light intensity and id. This allows the device to be used in the measurement of light intensity.

• Light emitting – the diode can be constructed so that the device will emit light of a fairly narrow band of wavelengths when operated in the forward biased region. The intensity of the light is a function of id. Used for indicators and light sources.

• Temperature sensing – When Vd is held at a constant value in the reverse biased region, id is a function of the temperature of the junction of the diode. Figure 72 shows the qualitative relationship between temperature and id. This allows the device to be used in the measurement of temperature.

• Zener – typically operated in the breakdown region in order to regulate a voltage.

Vd0.6 (Si)


id

forward bias regionreverse bias region

amplitude of current in negative region is exaggerated

acce

ptab

le o

pera

ting

regi

on

Zener, avalanche, or breakdown region

Figure 70 - Real Diode Characteristic Curve

Vd

id

np


+ -

Figure 71 - Solid State Diode

temperature or light intensity of diode junction


id

Vd is held constant inthe reverse bias region

Figure 72 - Diode Sensitivity to Temperature and Light



Silicon Diode Characteristic Curve

-15

-10

-5

0

5

10

15

-6 -5 -4 -3 -2 -1 0 1 2

vd (volts)

id (M

A)

Figure 73 – Si Diode Characteristic Curve


-1

1

3

5

7

9

11

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

vd (volts)

id (M

A)

Figure 74 - Si Diode Characteristic Curve (Forward Region)




-14

-12

-10

-8

-6

-4

-2

0

-5.5 -5.3 -5.1 -4.9 -4.7 -4.5 -4.3 -4.1

vd (volts)

id (m

a)

Figure 75 - Si Diode Characteristic Curve (Zener Region)




Table 4 - Si Diode Characteristic Curve Data

vd (V) id (ma) vd (V) id (ma) vd (V) id (ma)

-5.1600 -12.9997 -4.6100 -0.0184 0.3000 0.0000

-5.1500 -11.5726 -4.5600 -0.0157 0.4010 0.0005

-5.1100 -5.0456 -4.5000 -0.0134 0.4960 0.0065

-5.1000 -3.4787 -4.1500 -0.0050 0.5010 0.0076

-5.0900 -2.1115 -4.0400 -0.0037 0.5390 0.0182

-5.0800 -1.2931 -3.9900 -0.0034 0.5610 0.0516

-5.0700 -0.8939 -3.8000 -0.0020 0.5810 0.0942

-5.0600 -0.3151 -3.6000 -0.0013 0.6010 0.1919

-5.0500 -0.2032 -3.4000 -0.0008 0.6070 0.2340

-5.0300 -0.1364 -3.2300 -0.0006 0.6240 0.4116

-5.0200 -0.1195 -3.0000 -0.0004 0.6440 0.8178

-4.9800 -0.0914 -3.0000 -0.0004 0.6490 0.9923

-4.9700 -0.0832 -2.5000 -0.0002 0.6690 1.9903

-4.9500 -0.0738 -2.0000 -0.0001 0.6800 2.9783

-4.9200 -0.0641 -2.0000 -0.0001 0.6920 4.5552

-4.9000 -0.0575 -1.5000 -0.0001 0.7000 6.4115

-4.8600 -0.0468 -1.0000 -0.0001 0.7030 6.9304

-4.8000 -0.0371 -0.5000 0.0000 0.7100 8.9264

-4.7800 -0.0337 0.0000 0.0000 0.7180 11.7209

-4.7000 -0.0250 0.2000 0.0000

3.3.3. Rectification

Thus far, the input AC power has been converted to a sine wave with amplitude that is closer to the amplitude of the desired DC voltage to be delivered to the load. This section will examine techniques for converting this AC power to DC. This process is called rectification.

3.3.3.1. Half Wave Rectifier

The first method of rectification is illustrated in Figure 76. In this section, the primary winding and core of the transformer will not be drawn, and the presence of these two parts of the circuit will be assumed.


RL

VLVS

iL

Rectifier_01.cdr 18-Sep-2004

0 0

Figure 76 - Half Wave Rectifier

To derive the behavior of this circuit, consider the behavior during the two half cycles of the input power. During the positive half cycle, the circuit can be represented as in Figure 77 and the diode will be forward biased and conduct. Thus, the diode is replaced with a short. Consequently, VL = VS.

RL

V =L VSVS

iL

Transformer_03a.cdr 18-Sep-2004

0 0

Figure 77 - Half Wave Rectifier (Positive Half Cycle)

RL

V =0LVS

iL

Rectifier_01b.cdr 18-Sep-2004

x

0 0

Figure 78 - Half Wave Rectifier (Negative Half Cycle)

Figure 78 shows the behavior during each negative half cycle of the input power where the diode will be an open. This circuit is called a half wave rectifier since power is only delivered half of the time. The output is now DC, i.e. the sign of the voltage and current are always positive. However, the signal is not constant. If the diode is reversed, the output of the circuit will be a half wave with the voltages and currents negative, i.e. the inverse of the output seen above.

3.3.3.2. Full Wave Rectifier

Next, we investigate the circuit shown in Figure 79 which is constructed with two diodes and a center tapped transformer.



RL

VL

ns2

d2

d1

ns1 iL


VS

0

0

0

VS2

Figure 79 - Full Wave Rectifier

Again, the solution results from considering the two half cycles of the input power. As Figure 80 and Figure 81 show, the direction of current flow through the load is always the same. Thus, the output is DC, i.e. all of one sign, but time varying. Now, there is power delivered to the load during both half cycles of the input power, hence the name full wave rectifier.

RL

ns2

d2

d1

ns1 iL

Rectifier_02a.cdr 5-Sep-2006

VS1

VS2

V =L VS1

0

0

0

Figure 80 - Full Wave Rectifier (Positive

half Cycle)

RL

ns2

d2

d1

ns1 iL

Rectifier_02b.cdr 5-Sep-2006

VS

VS2

V =L VS2

0

0 0

Figure 81 - Full Wave Rectifier (Negative

half Cycle)



3.3.3.3. Full Wave Bridge Rectifier

VS

VL

i (negative half cycle)L

i (positive half cycle)L

RL

1

23

4


+

-

Figure 82 - Full Wave Bridge Rectifier

Figure 82 shows a circuit constructed with four diodes and a simple transformer. To arrive at the behavior of this circuit, identify the paths through the circuits as shown in Table 5. There are only four unique paths through the circuit that do not involve doubling back over segments.

Each path may be examined in turn for each half cycle of the input power. The two paths on the right have back to back diodes. One of the two will always be reverse biased and, thus, open. These two paths will never conduct. The other two paths will only conduct in one of the two half cycles of the input power. These two conducting paths through the circuit are illustrated in Figure 82. Notice that the direction of the current through the load is always in the same direction. The final result is that the load sees a full wave rectified DC signal. The same result as with the circuit of Figure 79.



Table 5 - Paths through the Bridge Rectifier Positive half-cycle + + + +

Polarity Negative half-cycle - - - -

Path

+

- +

-

Positive half-cycle - - - -

Polarity Negative half-cycle + + + + Positive half-cycle N Y N N

Conducts? Negative half-cycle Y N N N

3.3.4. Filter

The next task is to convert the time varying DC signal to a constant DC signal, essentially by averaging over time. Table 6 shows how the simple first order RC filter reacts to various inputs. Notice that Examples 1, 2, 3, and 4 are the simple RC circuit that has been discussed before and assumes that there is no leakage through the capacitor. Thus, Examples 1 and 4 have only charging currents and once the capacitor is charged to the maximum value, no further change is observed. Examples 5 and 6 have a diode added to the input. This is essentially the rectifier of the previous section. The diode limits the direction of any current flow from the input to be zero or a charging current. Examples 7 and 8 have a load added and, hence, there is a path that is constantly discharging the capacitor. The time varying component of the output in Examples 7 and 8 is called “ripple.” The load in example 8 is greater (RL is smaller) than that of example 7.



Table 6 - Filter Activities

# Vin Circuit Vout

1



R

Vini Vout

Filter_10b.cdr 19-SEP-2004

2



R

Vini Vout


3 0


R

Vini Vout

0

4

Filter_16.cdr 22-Sep-2004

0a


R

Vini Vout

Filter_16a.cdr 22-Sep-2004

0a

5


0a


R

Vini Vout


0a

6


0a


R

Vini Vout


0a



7


0a


R

RLVini Vout

Filter_16b.cdr 22-Sep-2004

0

VRipple

8


0a


R

RLVini Vout

Filter_16c.cdr 22-Sep-2004

0

VRipple

The input signals in examples 1 and 2 in Table 6 are single pulses and are not periodic. The input signals in examples 3 to 8 in Table 6 are periodic. Examples 3 to 6 in Table 6 do not have a “load,” i.e. there is no current path across the outputs. Examples 7 and 8 in Table 6 do have loads, RL, on the output with the load in Example 8 being larger, i.e. RL is smaller and the current through the RL is larger, and the ripple is worse.

Thus, the filter smoothes out the time varying part of the DC output of the rectifier. Better filters can be built to do a better job and lower the amount of ripple for a given load. Higher order filters can be built with more components as one way to achieve better filtering. Or, a circuit that actively smoothes the variation can be added to the power supply as discussed in the following section.

3.3.5. Regulation This sub-system actively senses changes in the input power and the load and makes changes to keep the voltage, VL, delivered to the load constant.

Regulator_01.cdr 14-SEP-2006

VRaw

Vload

iLoadRLoad

PassElement

Control

Figure 83 - Series Regulator

Regulator_02.cdr 14-SEP-2006

VRaw

iLoadRLoad

RPass

Control

ΔVPass

Vload

Figure 84 - Series Regulator - First Order

Model

PassLoadRawPassRawLoad RiVVVV −=Δ−=



Pass

LoadRawLoad RR

RVVLoad

+=

Regulator_03.cdr 20-Sep-2004

VRaw

VLoad

ΔVPass

+

-+

-

iLoad

RLoad

RPass

Figure 85 - Series Regulator – Equivalent form

Table 7 contains the changes performed by the series regulator to maintain a constant VL in the presence of changes in VRaw and RLoad. In the table only one variable is allowed to change at a time. In reality, both dimensions will see changes at the same time. The regulators are nimble enough to affect the complex combined responses.

Table 7 - Responses of the Series Regulator to Changes

Constraints Changes Corrective Action

VRaw is constant RLoad decreases, iLoad increases decrease RPass

VRaw is constant RLoad increases, iLoad decreases increase RPass

RLoad is constant VRaw decreases increase RPass

RLoad is constant VRaw increases decrease RPass

Figure 86 illustrates a hydraulic analog of the series regulator being discussed here. In this case, the level of the fluid is maintained at a constant level, i.e. Δy is constant, by increasing or decreasing the amount of fluid being introduced by the faucet. The greater the volume of fluid in the reservoir, i.e. the larger the Δx, the smaller the change in Δy and the easier it is to keep the level constant.




ΔxΔy

Fluid

Figure 86 - Series Regulator - Hydraulic Analog

Regulator_07.cdr

ΔxΔy

Fluid

Figure 87 - Switching Supply - Hydraulic Analog

3.4. Switching Power Supply On a simple level, a switching supply is very similar to the series power supply discussed above. The switching power supply uses a switching regulator which is very similar to that of Figure 84 except in the strategy used to control the Pass Element. The Pass Element of the series regulator is continually varied over the range of resistances, as in Figure 88, to control iPass. In this case i2R power dissipation occurs continually. In the switching regulator, the Pass Element is turned all the way on, i.e. RPass is the minimum value, for short intervals of time. Otherwise the Pass Element is turned all the way off, i.e. RPass is the maximum value. Thus, i2R is minimized. Figure 87 illustrates the hydraulic analog of the switching regulator and filter. This technique requires that enough “aliquots” of charge can be delivered to the filter element of the supply to maintain the “level” of charge of the filter element sufficient to keep the voltage delivered to the load at the desired and constant value.


i=V

/RP

ass

Load

Pas

s

Time Figure 88 - Series Regulator Efficiency


i=V

/RPa

ssLo

adP

ass

Time0

Figure 89 - Switcher Regulator Efficiency

A real switching supply is much more complicated as the supply is optimized for maximum efficiency and minimum size for a given amount of power to be delivered.

3.5. Shunt Regulator Figure 90 illustrates a circuit that is often used to regulate a raw DC voltage, VRaw, and produce a new voltage, VL, which will be essentially independent of changes in RL and VRaw. The steep slope of the Zener region of the characteristic curve of the diode makes this possible. Forcing the



circuit to operate only in this portion of the characteristic yields the desired result. Observe the actual characteristics of a diode given in Table 4 and Figure 75. Specifically, the following two points on the curve demonstrate the small ΔVZ (0.18 volts) associated with a fairly large ΔiZ (12.90083 milliamps).

(-4.98 volts, -0.0914 milliamps)

(-5.16 volts, -12.9997 milliamps)

This section will first explore how the parameters of the circuit are chosen to yield such a result. Figure 92 emphasizes the Zener region of the diode characteristic and defines a number of operating points of interest. Notice that the subscripts “min” and “max” imply relative values of the absolute values of the parameters, not the algebraic relationships of signed numbers. iZ-Limit is the negative current limit for the diode, larger (more negative) currents will result in catastrophic damage to the diode.

The task is to choose values for VZ-Nominal, RRaw, and VRaw so that VL will behave as desired. The choice of VZ-Nominal is easy, VZ-Nominal = VL-Nominal, the desired voltage for the load.

To begin, notice that the following relationships are always true.

• LZ Raw iii +=

• Load

LLoad R

Vi =

• Raw

LRawRaw R

VVi )( −=

One operating point to be allowed will be for the case of the load not being present, i.e. the following.

• ∞=−minL , 0min =−L , and thus R iRaw

LRawNoloadRawZ R

VVii )( NominalNominalmax

−−−−

−==



RL

ZenerRegulator_01.cdr 24-Sep-2002

RRawiRaw

iZ iL

VL

VRaw

Figure 90 - Zener Diode Regulator

RLRZ


RRawiRaw

iZ iL

VL

VRaw

Figure 91 - Zener Diode Regulator (Model)

VZ-nominal

iZ-min

iZ-max

iZ-Limit

VL-min

VZ

iZ


VL-max

Figure 92 - Shunt Regulator Design Criteria

Next a set of decisions have to be made as to what the operating region of the circuit will be. Two extremes are to be dealt with.

• Maximum VRaw at the same time as minimum iL. This results in the maximum iZ.

• Maximum iL at the same time as minimum VRaw. This results in the minimum iZ.

Don’t forget that VL will be essentially constant over the operating range of the circuit.

• Decide on a value for the maximum iL = iL-max to be encountered (allowed).

• Decide what is the maximum VRaw = VRaw-max to be encountered.

• Decide what is the minimum VRaw = VRaw-min to be encountered.

• Decide what is the maximum acceptable variation in VL = (VL-max - VL-min) will be.



• Raw

ZRawRaw R

VVi )( nominalminmin

−−−

−=

• Raw

ZRawRaw R

VVi )( nominalmaxmax

−−−

−=

• maxminmin −−− −= LRawZ iii

• LimitZRawLRawZ iiiii −−−−− ≤=−= maxminmaxmax

Thus, the operating point of the circuit will move up and down the Zener portion of characteristic curve as VRaw and RL change. Specifically, the diode will stay on the characteristic curve between the points (VZ-min, iZ-min) and (VZ-max, iZ-max).

Of course, if RL goes below the operating range, e.g. it becomes a short or VRaw goes outside the design range, then the desired behavior is no longer achieved. In such cases, one or more of four things will happen: VL is wrong, the Zener diode burns up, the load stops working, the load burns up.

The Zener diode is essentially a variable resistor as depicted in Figure 91. The Zener automatically adjusts the resistance so the device stays on the characteristic curve. In this case the resistance varies to keep VL at the desired value. Table 8 shows how the variable resistance changes to accommodate changes in VRaw and RL. In real life however, both VRaw and RL may change at the same time and in opposite directions.

• Z

ZZ iVR =

Table 8 - Responses of the Shunt Regulator to Changes

Constraints Changes Corrective Action

VRaw is constant RL decreases, iL increases increase RZ

VRaw is constant RL increases, iL decreases decrease RZ

RL is constant VRaw decreases increase RZ

RL is constant VRaw increases decrease RZ

The Zener diode shunt regulator is a simple, inexpensive regulator that is often used when small amounts of power are needed at voltages different than those provided by the main power supplies of a system. For instance, if a reference voltage of 7.0 volts were needed in a system that only had +15 and -15 volt supplies, the 15 volt DC supply voltage could be used as VRaw for a shunt regulator circuit built with a 7.0 volt Zener.

One disadvantage to this circuit is the power dissipation that occurs in RRaw and the diode.


Chemistry 838 Aspects of Analog Electronics Operational Amplifiers

4. Operational Amplifiers Operational amplifiers are a category of devices that have wide ranging applications in the world of instrumentation. In addition, the operational amplifier provides the tools necessary to build the bridge between the analog and the digital worlds.

This section will introduce the operational amplifier by showing the derivations of the fundamental equations, i. e. transfer functions, which describe the behavior of these devices. These derivations should give the reader a feeling for the nature of the devices as well as providing more practice in the analysis of electronic circuits. Applications of the devices will also be illustrated.

4.1. Comparator - Ideal A comparator answers the question “Is A > B?” Figure 93 illustrates an electronic comparator. The output has two states indicating whether e+ > e- or not.

OpAmps_01.cdr 20-Sep-2003

e+

e-

i-io

i+ eo

Figure 93 – Comparator

OpAmpsTF_01.cdr 27-Sep-2003

(e - e ) + -

(e > e ) + -(e < e ) + -

e o

VOL

VOH

Figure 94 - Transfer Function - Ideal Comparator

The ideal electronic comparator is defined by the following descriptions of the device’s behavior:

If (e+ > e-), then eo = VOH



If (e+ < e-), then eo = VOL

If (e+ = e-), then eo is indeterminate.

i+ = i- = 0

io = whatever needed Figure 94 is the graphical form of these definitions.

4.2. Comparator - Real


e+

e-

i-io

i+iPS-

iPS+

VPS-

VPS+

eo

Figure 95 - Real OpAmp

The first order model of an operational amplifier, which we will see can be used as a real electronic comparator, is defined by the following descriptions of the device’s behavior. A is a constant and is called the Gain or Open Loop Gain (more about this later). In general, Gain is the ratio of the output quantity to the input quantity of a device or function.

If (e+ - e-) > VOH/A, then eo = VOH

If (e+ - e-) < VOL/A, then eo = VOL

If VOL /A < (e+ - e-) < VOH/A, then eo = A(e+ - e-)

i+ = i- = 0

|io| < iLimit

Figure 96 and Figure 97 are graphical statements of these conditions.




(e - e ) + -

e o

VOL

VOH

VOL/A

(V - /AOH V )OL

VOH/A

comparatorregion

operationalregion

comparatorregion

Figure 96 - Operational Amplifier Characteristic Curve (i vs V)

|i |outiLimit0

1OpAmps_05.cdr 24-Sep-2003

e (actual)out

e (theoretical)out

Figure 97 - Operational Amplifier Characteristic Curve (iout)

VOH is typically approximately equal to VPS+. VOL is typically approximately equal to VPS-. Furthermore, VPS+ is often 12 to 15 volts and VPS- is often -12 to -15 volts. Typical values of A are 10000 to 100000. This leads to the following range of typical values for the width of the operational region.

millivoltsvoltsAAA

VV OLOH 0.3003.010000

3030)15(15====

−−=

−

millivoltsvoltsAAA

VV OLOH 3.00003.0100000

3030)15(15====

−−=

−

Thus, the operational region is very narrow with respect to the typical signal amplitudes to be used and the device behaves as a comparator except when the two inputs are very close in value.

Figure 98 gives an example of how the comparator may be used to shape signals. Notice that the edges of the output signal corresponds to the points when the signal, SIG, makes a transition through the threshold value ref. This device can be used as a threshold detector.



et

t

t

eo

VOH

VOL

eo

VOH

VOL

ref

ref

Sig

Sig

eo

ref

Sigeo


Figure 98 - Comparator Application

4.3. Follower, Buffer

Follower_01.cdr 20-Sep-2003

ein

i-io

i+eo

Figure 99 - Follower Amp

To derive the behavior of this circuit, we will first make a few assumptions.



)( −+ −= eeAeout , Limitout ii < OpAmp is operational. 1

0== −+ ii assumption 2 Next we will make a few observations.

inee =+ By definition. 3

outee =− By connection 4

All that remains is algebra, i.e. solve the above system of 4 equations.

)( outinout eeAe −= Plugging 3 and 4 into 1 5

outinout AeAee −= Expanding 5 6

outoutin AeeAe += Rearranging 6 7

)1( AeAe outin += Collecting terms 8

)1( AAee inout +

= Dividing both sides by )1( A+ yields the exact answer.

9

However, we can go one more step, by assuming A is very large.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+=

)11(

1

A

ee inout

Dividing the numerator and denominator of the fraction by A

10

inout ee = Limit as ∞→A or 01→

A 11

Thus, the circuit outputs a signal that is exactly the same as the input. i.e. the output “follows” the input. Why bother? We bother because of Assumption 2. If this is true, the circuit is a perfect VMD, i.e. senses the voltage of a source without providing a load. Thus, the circuit is also called a “buffer” amp.

Boundary Conditions:

The derived behavior is predicated on the two basic assumptions stated in Steps 1 and 2. Thus, Step 1 is a set of boundary conditions on the output of the operational amplifier that are required for the derived behavior to be realized.

VOL < (e+ - e-) < VOH



|io| < iLimit

These general boundary conditions will be realized in this circuit if the following boundary conditions are imposed on the input of the circuit.

VOL < ein < VOH

Figure 100 is the graphical form of the behavior of the Follower circuit. Such information is called the transfer function since it relates of the output of the device as a function of the input(s).


e in

e out

VOL

VOH

VOL

(V -V )OH OL

VOH

Figure 100 - Follower - Transfer Function

An example of how this circuit is typically used is shown below when the follower is used to decrease loading in the voltage measurement. Figure 101 illustrates a real voltage measurement. As shown before, iM needs to be minimized to minimize the error in the measurement.

Idea

l VM

D


(VMD)

Real Voltage Source

+

-

RM

iM-

iM

iM+


Figure 101 – Follower Application – The Challenge



Idea

l V

MD


(VMD)

Real Voltage Source

+

-

RM

iM-

iM

iM+


Figure 102 - Follower Application – The Solution

In the ideal case i+ = 0, Thus iS = 0 and eM = eout = es = VS and the ideal measurement is achieved. Of course, iM will not be zero, but the operational amplifier will be able to provide this current without disturbing the Voltage Source as long as the required |iM| is less than iLimit.

4.4. Follower with Gain


ein

i-i1 i2

io iLoad

i+eout

R1 R2

Figure 103 - Follower with Gain


ein

i-

i1

i2

io iLoad

i+eout

R1

R2

Figure 104 - Follower with Gain (Alternate Schematic)

Figure 103 is the next operational amplifier circuit to be considered. Figure 104 is an alternate way to draw the schematic drawing of the circuit. To derive the behavior of this circuit, we will first make a few assumptions.

)( −+ −= eeAeout , Limitout ii < Op Amp is operational. 1



0== −+ ii assumption 2

Next we will make a few observations.

inee =+ By definition. 3

( )21

1

RRRee out +

=− By connection, the voltage divider equation, and assumption 2.

4

( )21

1

RRR+

=β For convenience, make this definition.

5

( ) outout eRR

Ree β=+

=−21

1 Substituting 5 into 4 6

All that remains is algebra, i.e. solve the system of 4 equations.

)( outinout eeAe β−= Plugging 3 and 6 into 1 7

outinout eAAee β−= Expanding 5 8

outoutin eAeAe β+= Rearranging 6 9

)1( βAeAe outin += Collecting terms 10

)1( βAAee inout +

= Dividing both sides by )1( βA+ yields the exact answer.

11

However, we can go one more step, by assuming that A is very large.

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+=

)1(

1

βA

ee inout

Dividing the numerator and denominator of the fraction by A

12

inout eeβ1

= Limit as ∞→A 13

1

21

RRRee inout

+= The final result 14





VOL < (e+ - e-) < VOH

|io| < iLimit

These general boundary conditions will be realized in this circuit if the following boundary conditions are imposed on the input of the circuit. In addition, the output current will also have to be bounded appropriately.

βVOL < ein < βVOH


e in

e out

VOL

VOH

VOL

(V -V )OH OL

VOH

OpAmpe = A(e - e )o + -

Figure 105 - Comparison of Transfer Functions

4.5. Inverter

Inverter_01.cdr 20-Sep-2003

ein

i-i1 i2io

i+eout

R1 R2

iLoad

( )

Figure 106 - Inverter Amp



To derive the behavior of this circuit, we will first make a few assumptions.

)( −+ −= eeAeout , Limitout ii < OpAmp is operational. 1


0=+e By connection. 3

−−= Aeeout or Aee out−=− Combining 1 and 3 4

( )1

1 Reei in −−

= By Ohm’s Law 5

( )2

2 Reei out−

= − By Ohm’s Law 6

−+= iii 21 Kirchhoff’s Current Law. 7 All that remains is algebra, i.e. solve the above system of equations.

12 ii = Using Assumption 2 8

( ) ( )12 Ree

Ree inout −− −

=− Combining 5, 6, and 8 9

2211 ReReReRe inout −− −=− Cross multiplying 10

2211 RAeReReR

Ae out

inoutout +=−− Substituting 4 into 10 11

2211 ReRAeRAeRe outinoutout +=−− Multiplying both sides by A 12

2211 RAeReRAeRe inoutoutout =−−− Collecting terms 13

)( 211

2

RARRARee inout ++

−= The exact transfer function. 14

However, we can go further by assuming that A is very large.

)( 21

1

2

ARR

AR

Ree inout

++−= Dividing the numerator and

denominator of the fraction by A 15



inout eRRe

1

2−= Taking the limit ∞→A or

01→

Ayields an approximate

answer that we will typically use.

16



VOL < (e+ - e-) < VOH

|io| < iLimit


-(R1/R2)VOL > ein > -(R1/R2)VOH

Alternative Derivation

An alternative and shorter derivation is possible by making use of the fact that the Operational Region is very narrow, only a few millivolts wide. Again we begin with the two assumptions.

−+ ≅ ee , Limitout ii < Op Amp is operational. Now we use the shortcut definition of operational behavior, i.e. that the two input signals will be within a few millivolts of each other.

1


0== −+ ee By connection and Step 1. 3

( ) ( )11

1 Re

Reei inin =

−= − By Ohm’s Law 4

( ) ( )22

2 Re

Reei outout −=


−+= iii 12 Kirchhoff’s Current Law. 6 All that remains is algebra, i.e. solve the above system of equations.



12 ii = Using Assumption 2 and Equation 6

7

( ) ( )12 Re

Re inout =− Combining 5, 6, and 8 8

inout eRRe

1

2−= The final answer. 9

A Few Conclusions Notice that the current coming from the source is the following.

( )1

1 Rei in=

Thus, this circuit is not a buffer. Care must be used in applying the circuit to real measurement activities. Still, this is a very valuable circuit and is the basis of many related circuits.

This circuit is an inverting circuit, i.e. the sign of the output is the opposite of the sign of the input. The remaining circuits to be studied are all inverting circuits. The comparator, follower, and follower with gain are the only non-inverting circuits.


e in

e out

VOL

VOH

VOL 2VOL

(V -V )OH OL

-VOH-2VOH

OpAmpe = A(e - e )o + -

Figure 107 - Inverter Transfer Functions

Applications: An application of the inverter is the measurement of resistance. If three of the four parameters (ein, eout, R1, R2) of the circuit are known, the fourth can be determined. For instance, putting an unknown resistance in the circuit for R1or R2, the output voltage can be measured and the unknown resistance determined as seen in Equation 56 and Equation 57.



Equation 56

inout

eeRR 2

1 −=

Equation 57

in

out

eeRR 12 −=

Themistor_01.cdr 25-Sep-2003

10 15 20 25 30 35 40 45

Res

ista

nce

(Ohm

s)

Temperature (Celcius)

Themistor Response Curve

0.5

1.0

1.5

2.0

2.5

3.0

0

Figure 108 - Thermistor

A particular example of this type of application is the measurement of temperature using a thermistor. Figure 109 and Figure 110 show two configurations in which a thermistor has been placed in the inverter circuit. The output voltage is measured and the resultant value is plugged into Equation 58(for Figure 110) or Equation 59 (for Figure 109) yielding the value of RT. The temperature is then looked up in Figure 108.


ein

i-i1i2

ioi+

eout

R1

RT

T

iLoad

( )

Figure 109 - Inverter Application - Measurement of Temperature I




ein

i-i2

ioi+

eout

i1

R2

RT

iLoad

( )T

Figure 110 - Inverter Application - Measurement of Temperature II

Equation 58

in

outT e

eRR 1−=

Equation 59

inout

T eeRR 2−=

One problem with this measurement originates in the fact that the thermistor is a resistor. To make the measurement, a current passes through the thermistor, resulting in i2R heating of the system being observed. If the measured system is a large vessel of well stirred liquid, this is probably not a problem. However, if the mass of the system is small relative to the size of the thermistor, the heat generated by the measurement can significantly alter the temperature of the measured system. This is undesirable. This effect is usually called self heating. This effect can be minimized by minimizing the current passing through the thermistor with appropriate choices of ein and the other resistance. However, the current must be large enough so that an accurate measurement of eout can be made. Another approach would be to minimize the resistance of the thermistor (often this is not a choice). The third choice is to only turn on the driving voltage, ein, long enough to make the measurement of eout.

Figure 111 shows the similar appearance of the follower with gain and the inverter. The difference in the appearance of the two circuits is where the input signal is tied to the circuit and where the common is tied on the input side of the circuit.



ein

eout

R1 R2

InvFWG_01.cdr 24-Sep-2003

ein

eout

R1 R2

e =out ein R1

R +1 R2

e = -out ein R1

R2

Figure 111 - Symmetry of the two Configurations

4.6. Current Follower

CurrentFollower_01.cdr 20-Sep-2003

i-

i1

ifio

i+eout

Rf

iLoad

( )

Figure 112 - Current Follower

Inverter_02.cdr 5-Oct-2004

C

ein

iin

Rin ( )

iin ( )

iin

qin

( )

Figure 113 - Inputs for Current Follower

and Integrator

If the input to the circuit is considered to be the current i1 rather than the e1 above, the circuit becomes the ideal load for a current source, since the source is being connected directly to common (albeit virtual). The derivation of the transfer function for this circuit is very easily derived.



−+ ≅ ee , Limitout ii < OpAmp is operational. Now we use the shortcut definition of operational behavior, i.e. that the two input signals will be within a few millivolts of each other.

1


0== −+ ee By connection and 1. 3

( ) ( )f

out

f

outf R

eReei −=


−+= iii f1 Kirchhoff’s Current Law. 5 All that remains is algebra, i.e. solve the above system of equations.

1ii f = Using 2 and 5 6

( ) 12

iReout =− Combining 4 and 6 7

1iRe fout −= The final answer. 8



VOL < eout < VOH

|io| < iLimit


-RfVOL > iin > -RfVOH

Applications:

Figure 114 illustrates a spectrophotometric detector system, a common application of the current follower. The Photo Multiplier Tube, PMT, is a transducer that converts a beam of photons into iphoto, a photo current or stream of electrons that flow out of the tube into the rest of the detector



circuit. Figure 115 and Figure 116 show the response of the PMT to high and low intensity beams of light. The circuit converts iphoto into a voltage that can be measured or recorded.

The low photon flux case shown in Figure 116 is rather difficult to observe. The pulses are very narrow and, thus, require very fast electronics. In addition, the PMT must be cooled to very low temperatures to reduce noise.

CurrentFollower_PMT.cdr 5-Oct-2004

iphoto

e = -i Rout photo f

Rf

VPS-

( )

PMT

λ

Figure 114 - Current Follower Application

PMT_01.cdr 6-Oct-2004

Photon Flux Figure 115 - Large Photon Flux

PMT_02.cdr 6-Oct-2004

Time Figure 116 - Small Photon Flux



4.7. Summing Amp

SummingAmp_01.cdr 20-Sep-2003

e1

i-i1 if

ioi+

eout

R1

e2

i2

R2

Rf

iLoad

( )SP

Figure 117 - Summing Amp

As always, we desire to understand the behavior of this circuit. In all of the remaining examples, we will use the short hand definition of operational behavior to derive the behavior of the circuit. First, we will list our assumptions.


1

0=−=+ ii assumption 2


0==+ −ee By connection and 1. 3

( ) ( )11

11 R

eReei in=

−= − By Ohm’s Law and 3 4

( ) ( )2

2

2

22 R

eReei =


( ) ( )f

out

f

outf R

eReei −=


21 iiiii ff +==+ − Kirchhoff’s Current Law and 2. 6 All that remains is algebra, i.e. solve the above system of equations.



2

2

1

1

Re

Re

Re

f

out +=− Combining 5, 6, and 8 7

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

2

2

1

1

Re

ReRe fout

The final answer. 8

Notice that the output is a function of the sum of the two inputs, hence the name of this opamp configuration. In addition, notice that the output is the negative of the sum. Thus, this is also an inverting circuit.

This result can be generalized for an arbitrary number of input ei and Ri segments.

∑=

−=n

i i

ifout R

eRe1



VOL < eout < VOH

|io| < iLimit

The set of boundary conditions on the inputs is more difficult to specify for this circuit since there are now more than one input, and hence dimension, that must be bounded. The following is one form of the boundary conditions. The issue is that the allowed values for one input vary with the values of all of the other inputs at that point in time. One must consider all of the inputs together.

-RfVOL > Σ(ein/Rin) > -RfVOH

4.8. Integrator

Integrator_01.cdr 20-Sep-2003

ein

i-iin ifio

i+eout

R CVC

iLoad

( )SP

Figure 118 – Integrator



Again, we begin with the two assumptions and the shortcut definition of operational behavior.


1



( ) ( )Re

Reei inin

in =−


outoutC eeeV −=−= − By definition, 3 5

outfCC edtiC

VC

tqqV −=∫+=+

=1)0()()0( Basic capacitor definition. q(0) is

the charge on the capacitor at time 0. q(t) is the charge added during charging.

6

ffin iiii =+= − Kirchhoff’s Current Law and 2. 7 All that remains is algebra, i.e. solve the above system of equations.

)1)0(( ∫+−= dtiC

Ve inCout Using 6 and 7 8

∫+−= )1)0(( dtRe

CVe inCout

Using 4 9

)1)0(( ∫+−= dteRC

Ve inCout Bringing the constant R outside of the integral.

10

∫−= dteRC

e inout1 If VC(0) = 0, i.e. the capacitor is

discharged at time 0. 11

Thus, the output is a function of the integral of the input. A very common application of this circuit occurs when is constant. ine



))0(( ∫+−= dtRCe

Ve inCout

Bringing the constant ein outside of the integral.

12

)()0( 0ttRCe

Ve inCout −−−=

Evaluating the integral 13



VOL < eout < VOH

|io| < iLimit For this case, the boundary conditions on the input are also more difficult to specify since time now enters into the behavior of the circuit. In fact, if the circuit runs indefinitely, the output will always go to either the positive or the negative limit and stay there. This is because the integral is unbounded. Thus, the circuit will only be operational for some period of time, dictated by the RC of the circuit.

Applications: The Integrator circuit has a number of practical applications. One category of these applications is the variations of the sweep generator illustrated in this section. In this category of applications the input signal, ein, to the Integrator circuit is constant with time or constant with time over segments of time. Figure 119 illustrates two examples of the time course for the circuit as predicted in the equation of Step 13 above. Notice that eout is a linear function of time with a slope with the opposite sign from that of ein. This is in contrast to the non-linear exponential charging of the simple RC circuit. Also notice that eout becomes pegged to the output limit (The limits are often called the “rails.”) of the operational amplifier.

OpAmpTimeCourse_01.cdr 30-Sep-2004

Time

e out

VOL

0

VOH

Figure 119 - Integrator - Simple Sweep

Generator


Time

e out

VOL

0

VOH

Figure 120 - Integrator Realities



Figure 120 illustrates what happens when the input signal, ein, is turned off at time ta. If the components were all ideal, eout would be held at the value at the time ta. With real components, the capacitor will be discharged by leakage and other non-ideal attributes of the components.

Figure 121 illustrates the behavior of the integrator when ein is switched to a second value of the opposite sign at time ta during the observation. Notice that the circuit eventually ends up pegged to one of the rails.


Time

e out

VOL

0

VOH

Figure 121 - Integrator - Variable Inputs


Time

e out

e in

e 1

e 2

0

0

Time Figure 122 - Integrator - Single Saw Tooth

Figure 122 illustrates a case of this where the two inputs are carefully chosen to leave the capacitor discharged at the end of the sequence.

Figure 123 is a more complicated circuit that will yield a continuous saw tooth function. The Switch Control is a circuit which will cause the switch S to be placed into Position Px when a rising edge occurs at the Go to Px input. Or, the switch will be placed in Position Py when a rising edge occurs at the Go to Py input. Once the switch is in a given position, no change occurs until a rising edge is seen at the opposite input to Switch Control. Figure 124 shows the resultant behavior when the capacitor is discharged initially and switch S is initially in position Px. In this example, e1 is negative and e2 is positive. Notice that the first cycle is atypical.



SawToothGen_01.cdr 1-Oct-2004

e1

eout

eout2

eout3ebR1 C

Se2

Py

Px

R2

OA1

ea

OA2

OA3SP

SwitchControl

Go To Px

Go To Py

Figure 123 – Saw Tooth Generator

OpAmpTimeCourse_05.cdr 4-Oct-2004

Time

e out

e a

e b

0

Hi

Lo

Hi

Lo

eout3

e out2

Figure 124 – Saw Tooth Time Course

Thus, the sweep rates are controlled by e1 and e2 and the sweep range is controlled by ea and eb. These are but a few examples of function generators with linear sweeps that can be implemented with the Integrator and a set of constant input signals.



4.9. Differentiator

Integrator_01.cdr 20-Sep-2003

ein

i-iin ifio

i+eout

RCSP

iLoad

( )VC

Figure 125 – Differentiator

Again, we begin with the two assumptions.


1


0=+e By connection. 3

0== +− ee By 1. 4

ininC eeeV =−= − By definition and the fact that ein =0.

5

( ) ( )Re

Reei outout

f −=−


dtdeC

dtdVC

dtdqi inC

in === By the nature of the capacitor 7

−+= iii fin By Kirchhoff’s Current Law. 8 All that remains is algebra, i.e. solve the above system of equations.

inf ii = Using 2 and 8 9

dtdeC

Re inout =− Combining 6, 7, and 9 10



dtdeRCe in

out −= The final answer. 11

Thus, this circuit differentiates the input. The circuit is an inverter circuit.

4.10. Difference (Instrumentation) Amp

Difference_01.cdr 20-Sep-2003

e2

e1

i-i1

i3 i4

i2io iLoad

i+eout

R1

R3 R4

R2

Figure 126 - Difference Amp

The shorter derivation that is based on the fact that the Operational Region is very narrow, only a few millivolts wide, is used. Again we begin with the two assumptions.

−+ ≅ ee , Limitout ii < Assumtion: OpAmp is operational. Now we use the shortcut definition of operational behavior, i.e. that the two input signals will be within a few millivolts of each other.

1

0== −+ ii Assumption 2

13 RR = Definition 3

24 RR = Definition 4 Next we will make a few observations.

21

22 RR

Ree+

=+ Step 2 allows the use of the Voltage Divider Equation.

5

21

111 )(

RRReeee out +

−+=− By the generalized Voltage Divider Equation. Again, Step 2 allows the use of the Voltage Divider Equation.

6



T


he rest is algebra.

21

11211 )()(RR

ReeRRee out

+−++

=− Expanding 6. 7

21

1112111

RRReReReRee out

+−++

=− Rearranging 8

21

121

RRReRee out

++

=− Simplifying 9

21

22

21

121

RRRe

RRReReee out

+=

++

== +− Combining 1, 5, and 9 10

22121 ReReRe out =+ Simplifying 11

21221 ReReReout −= Rearranging 12

)( 121

2 eeRReout −= Solving for eout 13

.11. Potentiostat 4Potentiostat_01.cdr 20-Sep-2003

ein

ea

i-

i-

i1

i2

io iLoad

i+eout

R1

R2

Figure 127 – Potentiostat

An alternative and shorter derivation is possible by making use of the fact that the Operational Region is very narrow, only a few millivolts wide. Again we begin with the two assumptions.



1



21

1

RRRee outa +

= By Voltage Divider Equation 4

ain eee −= − The voltage drop across a device is the difference between the voltages on either side.

5

All that remains is algebra, i.e. solve the above system of equations.

ina ee −= Plugging 3 into 5. Notice that ea does not depend on R1 or R2 only ein.

6

inaout eRRRe

RRRe

1

21

1

21 +−=

+= Using 2, 4 and 6 7

A more useful version of the potentiostat is shown below. This form provides two input signals, ex and ey, that can be used to program the behavior of ea



ea

ea

i-if

i1

i2

io iLoad

i+eout

i+

R1

R2

ey

iy Ry

Rf

ex

ix Rx

Figure 128 - Potentiostat Typical

Potentiostat_03a.cdr 21-Sep-2005

ea

ea

i-if

i1

i2

io iLoad

i+eout

R1

R2

ey

iy Ry Rf

ex

ix Rx

Figure 129- Potentiostat Typical (View 2)

Again we begin with the two assumptions.

−≅+ ee OpAmp is operational. Now we use the shortcut definition of operational behavior, i.e. that the two input signals will be within a few millivolts of each other.

1



0=−=+ ii assumption 2


0==+ −ee By connection and 1. 3

21

1

RRRee outa +

= By Voltage Divider Equation. Valid because of the follower.

4

−+=+ iiii fyx By Kirchhoff’s Current Law. 5

x

xx R

ei =

3 and Ohm’s Law. 6

y

yy R

ei =


f

ff R

ei −=


All that remains is algebra, i.e. solve the above system of equations.

f

a

y

y

x

x

Re

Re

Re

−=+ 2 and 5 9

( )y

fy

x

fxa R

Re

RR

ee +−= The final answer. Rearranging 9. Notice that ea is not dependent on R1 and R2

10

4.12. Op Amp Terminology We have seen a number of circuits that perform “mathematical operations” on analog signals. These are summarized below. Many others exist. You should now be able to understand why the term “Operational” is used with these devices and their associated circuits.



Circuit Mathematical Operation

Follower Y = 1*X

Follower with gain Y = A*X

Inverter Y = -A*X

Summing Amp Y = -(A1X1 + A2X2)

Integrator ∫−= XdtAY

Differentiator dtdXAY −=

Difference Y = A(X1 – X2)

Multiplier* Y = X*Z

Divider* Y = X/Z

Logarithmic* Y = logX

Exponentiation* Y = eX

Many others*

* Not discussed in this document.

4.13. OpAmp Models OpAmpModel_01.cdr 20-Sep-2003

e+

ioutif

eout

Diff

eren

ce

e-

(e-e

)+

-

VPS-

VPS+

RLoad iLoad

iPS-

iPS+

Con

trol

4.14. OpAmp Realities

• Limit to output current • Non-linear Characteristic curve as you approach the output voltage limits (rails) • Input resistances (currents) • Input offsets



• Time response • Common Mode Rejection

eout

ein

OpAmpOffset_01.cdr 6-Oct-2004

Voff

Figure 130 - OpAmp Offset Voltage

eout

e1

R2

Voff

R1 OpAmpOffset_02.cdr 6-Oct-2004

Figure 131 - OpAmp Offset Voltage

(Inverter Example)

Table 9 - Effect of Offset on the Follower

ein eout

0.00 0.03

0.50 0.53

1.00 1.03

2.00 2.03

3.00 3.03

-0.50 -0.47

-1.00 -0.97

-2.00 -1.97




Table 10 - Effect of Offset on the Inverter

(R2\R1)= 1 (R2\R1)= 10ein

eout eout

0.00 -0.03 -0.30

0.50 -0.47 -4.70

1.00 -0.97 -9.70

2.00 -1.97 VOL

3.00 -2.97 VOL

-0.50 0.53 5.30

-1.00 -1.03 10.30

-2.00 -2.03 VOH

4.15. OpAmp Applications A few categories of applications are listed here.

• Comparator (trigger, signal shaping) • Resistance Measurement (Themistor - measurement of Temperature) • Current measurements (PMT - photo current measurement) • Signal generator • Signal processors (integration, differentiation, ...) • Digital to Analog Converters • Analog to Digital Converters • Sample/Hold • Analog Multiplexer

4.16. OpAmps: Feedback, Control Theory, and Stability

Negative Feedback We will now perform a simplistic thought experiment that will illuminate the source of stability of the circuits which we have been discussing in this document. Take the simple follower, Figure 132, realized by simply connecting the output of the OpAmp to the negative (inverting) input of the device. We have ignored the dynamic behavior of these devices up to this point in time. In this thought exercise, we will take into account the natural time response of the device. Simply stated, the op amp cannot instantaneously respond to changes in the inputs to the device.



ein

eout

Figure 132 - Follower

Assume that the gain A = 100000. We will use the more accurate form of the transfer function for the follower.

ininininout ee

A

eAAee 99999.0

00001.11

111

1==

+=

+=

The following table has the results at several arbitrary times along the time course of operation.

Table 11 - Time Course for Follower

ein = e+ eout = e-

(Actual)

e+ - e- A(e+ - e-) eout = e- = A(e+ - e-)

(Predicted)

Comments

1 0.99999 0.00001 1 0.99999 Before the step. We assume the Op Amp is in a stable state.

2 0.99999 1.00001 100001 VOH At the instant of the step, the inertia of the system precludes an instantaneous change of the output. Driving force causes the output to begin moving in a positive direction.

2 1.00000 1.0 100000 VOH Output continues to move in a positive direction.

2 1.99 +0.01 1000 VOH Output continues to move in a positive direction.

2 1.9999 +0.0001 10 10 Change of output slows down.

2 1.99997 +0.00003 3 2.99997 Nearing the goal, rate of change of the output is decreasing.

2 1.99998 +0.00002 2 2 At the correct answer, but the inertia keeps the output moving in a positive direction.

2 1.99999 +0.00001 1 1 Output overshoots. A(e+-e-) indicates that output should move in the opposite direction. Driving force is now in the negative direction, but inertia keeps the output moving in a positive direction.



2 2 0 0 0 Overshoot continues. Driving force is even greater in the opposite direction.

2 2.0001 -0.0001 -10 -10 Driving force is even greater in the opposite direction. Output eventually turns around.

2 2.00001 -0.00001 -1 -1 Moving back towards goal.

2 1.9999 +0.0001 10 10 Over shoots again. Driving force is now back towards 2.0 volts.

2 1.9998 +0.0002 20 VOH Still moving in a negative direction, but driving force dictates a change in direction. Output will eventually turn around.

Changes in the output will eventually damp down and settle to the desired value. This is because the driving force is always pointed toward the correct answer. Figure 133 illustrates what might be seen.

Feedbook_01.cdr 25-Sep-2003

(e - e ) + -

(e = e ) – out

e in

t

t

t

1

1

1

0

2

2

Figure 133 - Time Course for Follower

The actual path followed for these signals will depend on the dynamic characteristics of the system. While the details are beyond the scope of this document, Figure 134 illustrates the range of behavior possible. The top case of Figure 134 represents a system that is strongly damped. The middle case of Figure 134 illustrates a less well damped system, e. g. the output does not smoothly settle to the target value. This damped oscillation is typically termed “ringing.” Finally,



the bottom case of Figure 134 illustrates a case with little or no damping and the system oscillates and never settles to the target value.


(e = e ) – out

t

1

2

(e = e ) – out

t

1

2

(e = e ) – out

t

1

2

Figure 134 - Damping

Positive Feedback

We will now turn to a second simplistic thought experiment. The circuit illustrated in Figure 135 can certainly be built. All that is needed is to connect the output to the positive (non-inverting) input. We will now look at the time course of how this circuit would respond to a simple step function illustrated in Figure 136.


ein

eout

Figure 135 - Simple Op Amp Configuration

At the beginning we will assume (as it turns out this is a very poor assumption) that the circuit will be operational and e+ will equal e- and the output will be 1 volt and will stay at that value as long as there are no changes in the inputs. After the step, we might naively assume that the



output should be 2.0. Again, we will exaggerate the natural inertia of the device as we explore the time course of the circuit responding to the step function on the input.


e in

t 1

2

(e = e ) – out

t

1

0

2

(e - e ) + -

t

-1

-2

0

Figure 136 - Time Course of the Simple Circuit

Table 12 - Time Course for the Simple Circuit

ein = e- eout = e+

(Actual) e+ - e- A(e+ - e-) eout = e+ =

A(e+ - e-)

(Predicted)

Comments

1 1.00001 0.00001 1 1 Before the step.

2 1 -1 -100000 VOL At the instant of the step, the inertia of the system precludes an instantaneous change of the output. Notice that the value of the output predicted by the transfer function is a very large negative value. Output will begin to move toward the negative limit.

2 0.9999 -1.0001 -100010 VOL Output will continue to move in the negative direction.

2 0 -2 -200000 VOL Output will continue to move in the negative direction.

2 -1 -3 -300000 VOL Output will continue to move in the negative direction until it reaches VOL.

Thus, the output never reaches a stable state and never reaches the “correct” value. The driving force is always away from the “correct” answer. The conclusions from these two examples and is


Chemistry 838 Aspects of Analog Electronics REVISION HISTORY

- 97 - September 25, 2007 Version 1.7

borne out by control theory is that negative feedback typically leads to stability while positive feedback typically leads to instability, e.g. oscillations.

5. REVISION HISTORY Revision History for Aspects of Analog Electronics

Version Date Authors Description

1.0 8-Oct-2003 T V Atkinson This document is the transcription of my lecture notes as distilled over 3 decades of CEM 838. This is the first edition of the material in this form. Shared with CEM 838 FS03.

1.1 23-Sep-2004 T V Atkinson Added Voltage Sources, Current Splitter, Power Supplies details. Shared with CEM 838 FS04.

1.2 5-Oct-2004 T V Atkinson Added a few more Integrator examples.

1.3 20-Sep-2005 T V Atkinson Corrected a number of errors and added a few clarifications. Fixed the Instrumentation Amplifier and Potentiostat sections.

1.4 29-Aug-2006 T V Atkinson Minor edits for correctness and clarity.

1.5 19-Sep-2006 T V Atkinson Minor edits for correctness and clarity. Major corrections to the long derivation of the inverter OA. Converted the derivation of the integrator to include q(0).

1.6 26-Sep-2006 T V Atkinson Expanded the Sawtooth Generator section.

1.7 25-Sep-2007 T V Atkinson Corrected the arithmetic in the simple Approaches section

aspects of analog electronics - chemistry.msu.edu · chemistry 838 aspects of analog electronics...

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