fundamentals of electricity and electrical circuits

FUNDAMENTALS O F E L E C T R I C I T Y AND ELE C T R I CAL C I R C U I T S

J. ZEILER NIELSEN

Materials and components employed in electrical and electronic circuits may conveniently be divided in three groups according to their physical properties :-

Conductors, Semi-conductors and Insulators.

In the conductor group we find copper wires used primarily for intercon- nections of circuit components or wound up to form solenoids or coils. By proper choice of material and dimensions, other conductors may in a limited space offer considerable resistance to transportation of electricity and are accordingly called resistors.

Modern semi-conductors comprise such devices as diodes for rectifying and transistors for amplifying purposes. In this group we also find the Hall genera- tors.

Insulators are employed for the separation of circuit sections which carry current as for instance in a multi-core cable. When used for separation of the metal foils in a capacitor the insulator is called a dielectric.

R E S I S T A N C E

The resistance R of a given conductor is expressed by the formula:-

'1 R = Q - (ohms)

A where e is a constant related to the material,

1 is the length and A is the cross-sectional area of the conductor.

This resistance is not necessarily constant for a given resistor, but may change with such factors as temperature, light and pressure. These phenomena are utilised in devices like thermistors or NTC resistors, photo-resistors and strain gauges, respectively.

From the Department of Electronics, the Technical Highschool, Aarhus, Denmark.

24 J. ZEILER NIELSEN

OHM’S LAW

In an electrical circuit comprising in its simplest form an electromotive force E-for instance a battery-and a resistor R (fig. l ) , the relation between the current I and the voltage E applied can be expressed in a very simple way by Ohm’s law:-

E 1 R . I

It must be emphasised that we have here neglected the resistance of the connecting leads as well as of the ammeter A indicating the current. From the following section it will be seen, however, that we can easily calculate the influence of these additional resistors.

I

Fig. 1 .-Simple electrical circuit.

R E S I S T A N C E S I N C O M B I N A T I O N , VOLTAGE A N D C U R R E N T D I V I D E R S

In more complicated circuits we may, of course, have more than one electromotive force (EMF) as well as several resistors, and the question arises how they are mutually connected. Generally we distinguish between series and parallel connections. In figure 2 we have a series connection of three resistors R,, R, and R, supplied by the EMF E. In this circuit the current I is com-

1:

Fig. 2.-Series connection of resistors.

FUNDAMENTALS OF ELECTRICITY 25

I: -- --

In figure 3 we have a similar parallel connection of three resistors R,, R, and R,. In this circuit it is not the current, but the voltage E which is common to all three resistors. The individual currents again follow Ohm’s law, SO

we get:-

E and I, = -

E E I, = --, I, = -

Rl R2 R3

According to the well-known Kirchhoff’s first law the current I supplied by the EMF equals the sum of the three individual currents:-

or more conveniently


indicating that the reciprocal value of the total resistance is equal to the sum of the reciprocals of the resistance values. We often replace the reciprocal value of a resistance by its ability to conduct the current, by its conductivity. Consequently, the last equation may read as follows :-

Gtotal = GI + G, + G3,

where G is the symbol for conductivity. In addition, the above equations indicate that we here get a division of

the initial current inversely proportionate to the resistance of the individual branches.

T H e V E N IN’ S P R I N C I P L E (E Q U I VAL E N T C I R C U I TS)

Very often we have to handle circuits more complicated than the pure series or parallel circuits we have just dealt with. A circuit like the one we have in the left-hand section of figure 4 at first sight looks fairly complicated, but after all it is possible to decompose it by means of the principles mentioned above. For instance, the parallel combination of R, and R3 may be substituted by a new resistance R, and similarly R,, R, and R, by R,. The total resistance of the circuit will then be equal to the sum R, + R, + R,. Actually this may become a streneous procedure, but still worse it might be if the process had to be repeated for, e.g., ten different values of say R,.

Fortunately, this may be avoided by application of Thtvenin’s principle illustrated in the right-hand section of figure 4. This new circuit proves to be-as far as voltages and currents related to R, are concerned-entirely

Fig. 4.-Combined series and parallel connection of resistors (left). Equivalent constant- voltage circuit (right).


Fig. 5.-Equivalent constant-voltage and constant-current circuits for figure 4

1:s

(left).

equivalent to the original circuit. But it must be emphasised that the EMF Eoc is not equal to E, and that Ri is generally different from any of the original resistors.

Eoc-the so-called open-circuit voltage-is the voltage which can be measured between the terminals a-b of the original circuit with a non-loading voltmeter when R,, and the instruments are disconnected.

Similarly Ri-the internal resistance-is the resistance which can be measured with an ohm-meter between terminals a-b when the original EMF is “short-circuited,” that is E must be removed and the remaining terminals of the circuit then interconnected.

It remains to be stated that it is possible and very often worth while to calculate Eoc and Ri.

MAYER and NORTON have pointed out that the equivalent circuit may be rearranged as illustrated in figure 5 (right). Here Ri is again an internal resistance equal to the above-mentioned, but instead of the EMF & we find a so-

called “constant-current generator” supplying a current Ik = -. This cur-

rent, in accordance with the current-division principle mentioned above, di- vides between Ri and whatever is to the right of the terminals a-b.

Corresponding to the constant-current generator notation we often see the Thtvenin circuit mentioned as a “constant-voltage generator.” One type of generator may during the calculations conveniently be converted into the other type. Moreover, it must be mentioned that either type is equally convenient for solving problems concerning direct and alternating currents.

Eoc

Ri

E L E C T R O M O T I V E F O R C E

In our previous figures we have used a battery signature to indicate the presence of an electromotive force in the circuit. EMF’S may be based on chemical processes as in dry cells and various types of accumulators, but they may also depend on thermo- or photo-electrical phenomena as in the thermo-


couple and the modern sun battery, respectively. Similarly, the EMF generated by a dynamo is based on an electro-magnetic action.

All these sources of EMF generate so-called direct voltages which are- within certain limits-constant with respect to time. Other sources are made in such a way that their voltages deliberately change in a controlled manner; they are said to deliver alternating voltages.

It is characteristic of both types of voltage sources that their voltages under load conditions are lower than under no-load conditions. This is due to the fact that in any generator the EMF must be considered inseparably connected to an internal resistance as illustrated by the diagrams of the above-mentioned equivalent circuits.

D I R E C T A N D A L T E R N A T I N G C U R R E N T S

The characteristics of constant (direct) and alternating EMF'S and the currents they cause in circuits consisting of resistors may be summarised as follows :-

A direct voltage (DV) and a direct current (DC) do not change with time (see fig. 6), and the latter will in a given circuit flow in one direction only.

Contrary to this an alternating voltage (AV) will incessantly change its instantaneous value and periodically reverse its polarity, and similarly an alternating current (AC) will periodically reverse its direction of flow as

Fig. 6.-Time diagram of a direct voltage or current.

Fig. 7.-Time diagram of an alternating voltage or current.


illustrated in figure 7, where positive values refer to one polarity or direction of flow and negative values to the opposite polarity or direction of flow.

Figure 7 also illustrates that the phenomenon is repeating itself at certain intervals of time indicated by T on the time axis. T denotes the period or cycle and is generally measured in seconds. The number of periods or cycles per second is called the frequency of the alternating phenomenon and is measured in Hertz (Hz) or cycles per second (c/s).

By means of a mathematical analysis it may be proved that any function repeating itself periodically may be considered to consist of a summation of functions of a very simple type: the sine function. Consequently, the majority of electrical treatments on alternating phenomena are concentrated on sinusoidal curves.

Fig. 8.-Time diagram of a sinusoidal voltage or current.

A single period (or cycle) of a sinusoidal voltage or current versus time curve is illustrated in figure 8 along with its construction by means of a radius rotating counter-clockwise with a constant angular velocity.

Mathematically, a sinusoidal voltage (or current) may be expressed by the equation :-

e = Em,, sin ( o t + pl), where

e is the instantaneous value of the voltage, Em,, is the maximum value or amplitude of the voltage,

2n (r) = - = 2 n f is the so-called angular velocity,

T t is the time, T is the time of one cycle, f is the frequency, cot + pl is called the phase of the function, and 9 is the initial phase corresponding to t = 0.

In figure 8, pl is evidently zero, while in figure 9, in which two sinusoidal phenomena, two voltages, two currents or a voltage and a current are outlined, the dotted curve has an initial phase different from zero and is said to be


leuding the full curve which is similarly said to be lagging. The phase of one curve is said to be shifted as compared with the other one. p may be measured in units of time, but is more conveniently referred to as being a fraction of a full period.

Apparently, a voltage of this type is uniquely defined by its maximum value Emax and by either the angular velocity o or the frequency f and the initial phase p. But for some practical purposes it is convenient to be familiar with two more characteristic values frequently used, viz. the average value and the RMS value, respectively.

Phase shift Fig. 9.-Phase shift between two sinusoidal phenomena.

The average value of a full period of a sinusoidal phenomenon is evidently equal to zero, and consequently the average value is defined for half a period. For a sine curve it is easily proved that the average value EaV is equal to

Figure 10 indicates that the area below the half sine curve is equivalent to the area of the rectangle with the height EaV.

The value most often used to characterise an alternating voltage or current is the root-menn-square value (RMS value). I t is based on a comparison of the power dissipated in a resistor by an alternating current and by a direct current. When the two powers are equivalent, the alternating current is said to

Fig. 10.-Average value of a sinusoidal voltage.


have an RMS value equal to the number of amps of the direct current. For a sine current it is easily proved that

2

S Q U A R E WAVES A N D PULSES

In addition to the sinusoidal voltages and currents mentioned above modern electronics employ a couple of other waveforms: square waves and pulses which are illustrated in figures 11 and 12. They are both characterised by an abrupt shift of the instantaneous value from, say, a negative to a positive value which is maintained for a while. Then a similar shift in the opposite direction followed by a constant negative value is observed. This total period is then repeating itself.

For square waves, intervals of opposite polarity occupy equal times and positive and negative amplitudes are equal. This is not the case for pulses, but a closer inspection of figures 11 and 12 reveals that for both types of curves positive and negative areas are equal.

In figure 13, it is illustrated how an alternating phenomenon may be super- imposed on a constant (or direct) one. The latter is here conveniently considered to be a dislocated zero line for the sine curve.

Fig. 1 1.-Time diagram of a square-wave voltage or current.

Fig. lP.-Time diagram of a pulsed voltage or current.


E. I

Fig. 13.-Time diagram of a combined direct and alternating voltage or current.

CAPACITANCE A N D I N D U C T A N C E

In addition to resistance we meet in electrical and electronic circuits devices with properties referred to as capacitance and inductance. They are characterised by reacting differently to direct and alternating currents.

In the circuit illustrated in figure 14 the capacitance is concentrated in the capacitor C, which principally consists of two metal conductors such as wires or sheets of metal foil mutually separated by an insulator. With the switch S open, no voltage difference initially exists between the plates of the capacitor, but immediately after the switch is closed, a deflection of the ammeter indicates that a transportation of electricity takes place. This finally results in a concentration of charges of opposite polarity on the two capacitor plates followed by a voltage difference equal to the EMF of the battery. The transportation-and the ammeter deflection-cease when this state is obtained, but the voltage difference remains even if the switch is reopened. The charge stored in the capacitor, which will theoretically remain in it for ever, proves to be proportional first of all to the voltage applied, but also to the area of the plates and inversely proportionate to their mutual distance. Moreover, it depends on the type of insulating material employed. The ability

1 7- Fig. 14.-Electrical circuit with capacitance.


of the capacitor to store electrical charges is expressed by its capacity C in the formula :-

&A

d C = --, where

C is the capacity measured in farads, A is the area of the plates, d is the distance between the plates and E is a constant related to the insulating material, the dielectric constant.

Fig. 15.-Magnetic field caused by a current I flowing in an electrical conductor.

The charge may be removed from the capacitor by connecting a resistor between its terminals. The capacitor then for a while acts as an EMF and delivers a discharge current-in opposite direction to the original charge current-which in turn decreases the EMF. Finally, the EMF will reach zero and the discharge current stops.

We notice that during charge and discharge intervals a current exists, but it does not remain constant as the direct currents in the circuits we have studied up till now.

If we connect a capacitor to a generator giving an alternating EMF, the capacitor will be exposed alternately to charge and to discharge (the discharge current is here delivered to the generator instead of a resistor). This means that in an AC circuit with capacitor we must have a current shifting in value and direction continuously, i. e. an alternating current.

This may be recapitulated in the following way: A constant EMF is unable to maintain a current through a capacitor, whereas an alternating EMF will be able to pass an alternating current through a capacitor. Consequently, a capacitor may be utilised to separate direct and alternating currents in a common circuit.

I t is a well-known fact that whenever a conductor is carrying current, it will be surrounded by a magnetic field as illustrated in figure 15. If the conductor is wound up as a solenoid or coil, the magnetic fields of the separate


windings will add so that we get an electromagnet. To a stationary direct current such a coil will act as an equivalent length of straight wire, but whenever we try to increase or decrease the current, a similar change in the magnetic field will take place. This again means that the coil conductor is situated in a changing magnetic field. According to the well-known law of induction this results in an EMF being induced in a conductor whenever it is changing its current. This EMF, will always be induced in such a direction that it tends to cancel the action which created it. It will never success entirely in this obstruction, but at any rate it will delay the action of the change considerably.

This phenomenon is generally called self-inductance. The voltage induced, e, proves to be proportional to the rate of change of the current, generally

di

dt expressed by the derivative: -, and to a constant L related to the coil.

The corresponding formula :-

comprises a minus indicating the opposing action; e is said to be a counter-EMF. L depends on the geometry of the coil and is generally proportional to the

number of turns to the second power. Moreover, L may be increased considerably by winding the coil on a core of magnetic material. L is called the coeffi- cient of self-inductance and is measured in the unit called henry.

If a self-inductance is connected to a generator giving an alternating EMF, the counter-EMF succeeds in reducing the instantaneous value of the current to a value much smaller than that decided by the instantaneous value of the voltage and the so-called ohmic resistance of the copper wire.

The circuit signature of a self-inductance is illustrated in figure 16.

6 6 Fig. 16.-Circuit signature for a self-inductance.

IMPEDANCE

If an AC generator is connected to an ohmic resistor, the instantaneous value of the resulting alternating current will be given in simple accordance with Ohm’s law. Consequently, the corresponding voltage and current curves will follow each other as illustrated in figure 17, i.e. no phase shift is observed.


If a purely inductive component (i.e. an imaginary coil without ohmic resistance) is connected to an AC generator, the instantaneous value of the current cannot be derived directly from Ohm's law. It may be proved that the voltage e from the generator is related to the current i as expressed by the equation:-

di e = L -

dt

If e is following a sine curve, we find that i follows a similar cosine curve as illustrated in figure 18. We evidently have the current lagging the voltage by a quarter of a period, or with reference to the circular diagram of figure 8, where one period occupies 360 degrees, we may express the phase shift as 90 degrees.

4 ' . I

Fig. 17.-Time diagram ofvoltage and current in an AC circuit with purely ohmic resistance.

Fig. 18.-Time diagram of voltage and current in an AC circuit with purely inductive resistance.

Fig. 19.-Reactance XL of a self-inductance L as a function of frequency f.


The equation above may also be expressed in terms of RMS values. It then states that E,,, = UL I,,, = 2 d L I,,,. The value XL = WL = 2 d L is called the reactance of the coil and is apparently proportional to the frequency of the generator voltage as illustrated in figure 19.

In circuits containing a purely capacitive component (i. e. capacitor with a perfect insulation between the plates) we must again desist from a direct application of Ohm’s law. The equation between the instantaneous values of applied voltage e and corresponding current i here reads :-

e = - i dt. C ‘s

Fig.20.-Time diagram of voltage and current in an AC circuit with purely capacitive resistance.

For e following a sine curve we again find i following a cosine curve, but now of opposite sign as illustrated in figure 20, The phase shift is again 90 degrees, but now the current is leading the voltage, which is referred to as negative phase shift.

In terms of RMS values the formula is changed to:-

1 1 1 ~-

WC cot 2nfc Erms = - I,,,. The value Xc = - = -

is similarly called the reactance of the capacitor and is apparently inversely proportional to the frequency as illustrated in figure 21, where the curve is drawn below the axis to correspond to the negative phase shift.

In practical coils and capacitors the pure reactances mentioned above will always be accompanied by ohmic resistances representing the inevitable losses. These resistors may be considered connected either in series or in parallel to the appropriate reactance and influence the actual ‘‘resistance to AC” as well as the phase shift which for this basic type will always be less than 90 degrees.

This “resistance to AC” of a component is generally termed its impedance. The impedance-as we have already seen-may be purely ohmic, purely reactive or a combination of both, and corresponding to this it may have no,


90 degrees, or less than 90 degrees of phase shift, respectively. Generally, impedances are measured in ohms, but additional information of the phase shift and the frequency will be necessary to describe the phenomenon com- pletely.

Fig. 2 1 .-Reactance Xc of a capacitance C as a function of frequency f.

Impedances-like pure resistances-may be connected in series or in parallel. If an inductive and a capacitive impedance are combined, their reactances

, i.e. where 1

will oppose each other and at a frequency fo where 2nf0L = ~

2nfoC

1 fo = - - =, they will simply cancel each other. This frequency fo is called

2nVLC

the resonance frequency of the combination, which is generally called a tuned circuit. Tuned circuits are used for selection of one frequency, the one corresponding to the resonance frequency, among a group of available frequencies as is known from radio and television sets.

T R A N S F O R M E R S

Whenever a self-inductance-as coil 1 in figure 22-is connected to a generator giving an alternating EMF, the resulting current and the magnetic field accompanying it will alternate. If we place another (secondary) coil in the vicinity of the first (primary) one, it will be exposed to this alternating magnetic field, and an alternating EMF will be induced in it. Such a set of coils is called a transformer.

The voltage induced in the secondary coil depends on its number of turns and on the rate of change of the fraction of the magnetic field reaching it. This fraction can be greatly increased by winding both coils on a common iron core. In this way the coupling between the coils may become nearly


Coil I

I .< ,ad

Coil 2

Fig. 22.--Simple transformer.

perfect, and the ratio of the secondary voltage induced to the generator voltage will then be equal to the corresponding number of turns of the windings. Thus, the secondary voltage may be stepped up or down to a deliberate level simply by choosing an appropriate number of secondary turns as compared with the number of turns of the primary winding. The secondary of the transformer may be equipped with as many independent windings as is found suitable for the present purpose.

A load may be connected to any of the secondary windings, and the voltage induced will act as a generator sending an alternating current through the load. The power consumed by the load will be delivered by the generator connected to the primary winding through the interaction of the magnetic field. In this way an insulating separation of the primary and the secondary circuits is obtained.

POWER

The power W consumed by a resistor forming part of a DC circuit can easily be calculated by multiplying the voltage E applied to it by the current I flowing in it:-

W = E . I (watts)

By means of Ohm’s law this formula may be written as follows:-

R If a resistor is fed by a sinusoidal AC, the instantaneous power w can

similarly be expressed as :-

w = e . i = Em,, sin wt . I,,, sin wt


This value may be shown to oscillate around an average value W :-

Emax . Imax - w = Erma . Irms 2

so that the DC and AC formulae as might be expected from the definition of the RMS values are very much alike.

If an impedance is substituted for the resistor, a phase shift must be expected, and a correction factor, cos v, is added to the power formula:-

W = Erma . Irma * cos v where v is a measure of the phase shift between the voltage and the current curve as illustrated in figure 23.

Fig. 23.-Phase shift between voltage and current in an AC circuit.

If the impedance is a pure resistor, the formula becomes equal to the simple formula stated above because v then is equal to zero, making cos v = 1.

If the impedance is a pure reactance, cp becomes 90 degrees and cos v = 0, so that no power is consumed corresponding to what is mentioned above.

E F F I C I E N C Y

In the circuit illustrated in figure 24, “Gen.” is a generator, and RG denotes its inevitable internal resistance. I t must be emphasised that the actual generator terminals correspond to the two points marked A and B, while the junction between the generator and RG is inaccessible. In practice, this means that whenever we connect a load resistor RL to such a generator, we must face the fact that we have no means of avoiding that part of the total power delivered by the generator is consumed by the internal resistance RG instead of being fully utilised by the load resistance RL.


The so-called efficiency 7 of the generator is defined as the fraction:-

WRL - WHL 7=---- ~~

Wcen WRG + WRL

where WGen, WR, and W R ~ denote the power delivered by the generator and consumed by the internal and the load resistances, respectively. The efficiency 7 is generally measured as a percentage and is always less than 100~o.

\

Fig. 24.-Circuit illustrating matching of a load resistance RL to the internal resistance Rcen of the generator.

In power supplies, such as public power plants, the efficiency is generally very high compared to what can be obtained in electronic circuits, where the total power available from a given generator is very often so small that we want to transfer the maximum part of it to the load resistance in question without considering such a minor factor as efficiency.

By fairly simple mathematical calculations it may be shown that the maximum amount of power will be transferred to the load resistance when this has a value equal to the generator resistance. I t is easily seen that this means that the power is divided evenly between generator and load resistance giving an efficiency of 50 yo. In such cases, the load is said to match the generator resistance.

In practical circuits it may be very difficult or even impossible to make the load resistance equal to a given generator resistance. But for AC circuits the transformer here comes in with another property. As well as being able to step voltages up or down it proves to be able to step impedances up or down, i.e. a generator connected via its own internal resistance to the primary terminals of a transformer acts as if it were loaded by an impedance equal to the one really connected to the secondary terminals of the transformer multiplied by a factor equal to the squared ratio of number of primary turns to number of secondary turns. By choosing this ratio in a proper way we may match pearly any load to any generator.

fundamentals of electricity and electrical circuits

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