Alternating Current Circuits

If it were not for fear of being plagiaristic, this chapter might well be entitled Alice in Wonderland, with sincere apologies to Lewis Carroll. This thought stems from the puzzled expressions observed appearing on the faces of countless students when they first encounter this subject matter. This puzzle-ment follows from the fact that most students know quite a bit about steady state dc circuits and their mindset is to try to push the hearsay knowledge they have of ac circuits into this same framework. This does not work well at all and is similar to working on an automobile with a set of English wrenches when all of the nuts and bolts are metric. A few wrenches will fit and then only approximately. Any real craftsman knows that in order to do jobs properly, one must have the appropriate tools. If your knowl-edge of this subject matter is only cursory, it might be well to put aside what you may already know and begin acquiring a new set of tools or concepts as they are introduced in the following. The goal is to arrive at a thorough basic understanding rather than try to memorize a set of mysterious rules. When you thor-oughly understand something, you are able to write the rulebook yourself.

The first circuit to be considered appears as Fig. 8-1. The British physicist who first analyzed this circuit is one of the boyhood heroes of many budding physics students. His name was William Thomson, later Lord Kelvin. Thomson was Professor of Physics at Glasgow University from 1846 to 1899. The analysis of this circuit, in about 1850, was perhaps the least of his achievements but was a crucial step toward making possible modern communications.

The circuit of Fig. 8-1 is a series connection of the three idealized passive circuit elements. It consists of a pure resistor, a pure capacitor, and a pure inductor all connected in series (departure from ideal behavior exhibited by real resistors, capacitors, and inductors will be discussed after development of the necessary tools). In a series circuit, the electrical current is taken as a reference as it is common to

each of the circuit elements. The arrow in the diagram indicates the sense of the current when it is considered to be a positive current. This circuit, as is true of most circuits, exhibits two types of behavior.

These are referred to as the transient solution and the steady state solution. For the moment, only the steady state solution will be considered. As the name implies, the steady state solution describes the behavior of the circuit after it has been connected for a reasonable time. What constitutes a reasonable time can only be answered after a study of the tran-sient solution which will be treated in Chapter 22 Signal Processing on Laplace transforms.

Alternating currents are currents which either periodically or aperiodically reverse sense, i.e., are sometimes positive and sometimes negative. Peri-odic currents alternate between positive and nega-tive and back again after the elapse of a definite interval of time or period, T. Even periodic alter-nating currents exist in many waveforms. The wave-form is the wave-form of the current as viewed on an oscilloscope or plotted on a piece of graph paper.

Some common waveforms are sine, cosine, sawtooth, triangle, square, etc. Of these, the sine and cosine are essentially the same in that they differ only in the starting point reference on the oscillo-scope sweep. Additionally, the sine and cosine, unlike the others, contain only a single frequency component, f. This single frequency, f, is the recip-rocal of the period T. The other waveforms, even though they each have a definite period, are made up of a fundamental frequency component, which is the reciprocal of the period, as well as harmonic frequency components which are multiples of the fundamental frequency. This understanding is the result of the work of the French mathematician and physicist Jean Fourier (1768-1830). Fourier’s work will be studied in more detail later.

The analysis of the circuit of Fig. 8-1 is begun by assuming the circuit current is given by

(8-1)

In this statement the dependent variable is the electrical current which is denoted by the symbol i.

The current is called the dependent variable because its value at any instant depends on the value which is assigned to the independent variable which is the time, t. The value of i is linked to the value of t through the functional properties of the cosine. The cosine function, of course, is tabulated in terms of an angle such as θ. In this instance the angle θ is given by

(8-2) Figure 8-1. Series LCR circuit.

L C

R i = I_mcos(2πft)

θ = 2πft

114 Chapter 8 The angle θ is called the phase angle of the current. It should be noted that the value of the phase angle is directly proportional to the value of t.

Starting at t = 0, the time which must elapse before θ takes on the value 2π is called the period, T. That is,

(8-3) or

(8-4)

This last equation says that the frequency is the reciprocal of the period. Finally, one defines a quan-tity called the angular frequency or radian frequency as follows:

(8-5) The current can now be expressed as

(8-6) The maximum value that the cosine function attains is one and hence the maximum value that the current can take on is I_m. I_m is called the amplitude of the current. The current at any instant as well as the current amplitude is measured in amperes, A. An ampere is a coulomb of charge per second. Elapsed time and the period are measured in seconds, the frequency is measured in reciprocal seconds or hertz (Hz), and the angular frequency is measured in radians per second (rad/s). Fig. 8-2 is a graph of the assumed current over two periods where t is expressed in units of the period T.

The question to be answered is “What voltage must be applied across the input terminals such that

the current in the circuit will be the assumed current?” This question is answered by the applica-tion of Kirchhoff’s laws I and II. Law I is based on the conservation of electric charge and for a series circuit requires that the current everywhere be the same. Law II is based on the conservation of energy and requires that the voltage applied across the input terminals be equal to the sum of the voltages across the individual circuit elements in the series connected circuit. Therefore it is necessary only to determine the voltage which must exist across each circuit element and then add them up.

Unlike Lord Kelvin, who pursued the analysis by means of the differential and integral calculus, use will be made of the tools provided by Steinmetz when working with phasors. In what follows, the current will be represented by a phasor where it is understood that only the real part of the phasor represents the actual current.

(8-7) One can easily determine the voltage that must exist across the resistance. By the definition of resis-tance, the voltage across a resistance is the product of the resistance and the current. Therefore,

(8-8) That was easy enough! The phasor representation of the voltage across the resistance is the same as the current when scaled by a factor equal to the resis-tance. In particular, it should be noted that the phase angle of this voltage phasor is the same as that of the current. Therefore the voltage across a resistance is in phase with the current.

When one considers the voltage across the induc-tance, however, the going gets a little tougher. The definition of self-inductance requires that the product of the inductance with the slope of the current curve gives the voltage across an inductance at any instant, that is,

(8-9)

The last equation introduces the mathematical operator d/dt. This operator is called the derivative with respect to time. In this instance, the operation is to be applied to the function representing the current and tells one to generate a new function whose value at any instant is the same as the slope of the current curve at that instant. In the language of phasors, this is accomplished simply by multiplying the current phasor by jω. Hence,

Figure 8-2. The circuit current over an interval of two periods.

Instantaneous current in units of the current amplitude

i = I_me ^jωt

v_R = Ri = RI_me ^jωt

v_L Ldi dt

---=

Interfacing Electrical and Acoustic Systems 115

(8-10)

At this point it is worth noticing that whereas the phase angle of the circuit current is ωt, the phase angle of the voltage across the inductance is ωt + (π⁄2). The phase angle of the voltage across the inductance is greater than the phase angle of the current by an amount of π⁄2. This is the reason for the expression; “The voltage across an inductance leads the current.”

Now for the capacitor voltage, one again goes back to fundamentals. The capacitance is defined to be the transferred charge divided by the resulting potential difference or voltage. The charge and the current are related by

(8-11)

Therefore, the voltage across the capacitor is given by (8-12)

In the language of phasors, integration with respect to time is accomplished simply by division by jω. Therefore,

(8-13)

Note that the phase angle of the voltage across the capacitance is ωt − (π ⁄ 2) and is less than the phase angle of the current by an amount of π⁄2. This is the origin of the statement; “The voltage across a capacitor lags the current.”

The voltage that must be applied across the input terminals of this circuit in order to produce the

assumed current is then .

Fig. 8-3 depicts the phasor sum of the individual voltages. The figure is drawn for the instant of time such that t = 0. In the construction of the figure, it is arbitrarily assumed that the voltage across the inductance is greater than the voltage across the capacitance.

An examination of Fig. 8-3 will indicate that the phasor sum of the individual voltages yields

(8-14)

8.2 Impedance

All of the foregoing analysis can be made much more compact by defining a new term called the circuit impedance, Z. The circuit impedance, Z, is defined to be the ratio of the complex applied voltage to the complex current which results from the application of that voltage.

(8-15)

Upon applying this definition, the impedance of the series LCR is found to be

(8-16)

The last expression is the complex circuit imped-ance written in rectangular form. When written in the complex exponential form it appears as

(8-17) with the angle ϕ being given by

(8-18)

Figure 8-3. Phasor sum of resistor, capacitor, and inductor voltages.

116 Chapter 8 In general, the definition of the impedance implies a knowledge of two quantities. One is called the magnitude of the impedance, which is the ratio of the applied voltage amplitude to the amplitude of the resulting current. The other is called the angle of the impedance, which is the phase angle of the applied voltage minus the phase angle of the current that results from the application of that voltage. The magnitude of the impedance is denoted by |Z|. For the case at hand,

(8-19)

where,

V_m is the amplitude of the applied voltage, I_m is the amplitude of the circuit current.

At this point, it is well to re-examine the circuit in a slightly different way. The circuit consists of a pure inductor, a pure capacitor, and a pure resistor connected in series. Each of these circuit elements has its own impedance. These respective imped-ances are

(8-20) (8-21) (8-22) The impedance of the inductance is purely imagi-nary. Such an impedance is termed a reactance. In this instance it is a positive reactance as it falls on the positive imaginary axis. This reactance is denoted by X_L with

(8-23) The impedance of the capacitance is also purely imaginary indicating that it is also a reactance. In this case, however, it is a negative reactance as it falls on the negative imaginary axis. This reactance is denoted by X_C with

(8-24)

Finally, the impedance of the resistance is purely real with

(8-25)

The total impedance of the circuit is thus

(8-26)

when written in rectangular form, or

(8-27)

when written in exponential form. Notice that the magnitude of the total impedance is the square root of the sum of the resistance squared and the net reactance squared while the angle of the impedance is the angle whose tangent is the ratio of the net reactance to the resistance.

Impedances may be summed as was done above or they may be summed graphically by drawing a diagram in the complex plane. This diagram is similar to a phasor diagram as they are both drawn in the complex plane. Unlike the phasor diagram, the impedance diagram does not rotate with the passage of time. An impedance diagram is fixed, not changing with time as long as the circuit compo-nents remain unchanged. Fig. 8-4 is an impedance diagram for the present circuit.

Now that the concept of impedance has been introduced and, hopefully, is understood, circuit analysis is greatly simplified. The procedure to be employed is outlined in the following steps assuming that the current is a known quantity.

1. Calculate the total circuit impedance and express it in exponential form.

2. Express current as a phasor.

3. Multiply current phasor by the total impedance to obtain the voltage phasor.

4. Take the real part of the voltage phasor to obtain the actual voltage as a function of time.

Z V_m

Figure 8-4. Impedance diagram for the series LCR circuit.

Interfacing Electrical and Acoustic Systems 117 When the applied voltage is a known quantity,

the procedure is listed in the following steps:

1. Calculate the total circuit impedance and express it in exponential form.

2. Express the applied voltage as a phasor.

3. Divide the voltage phasor by the circuit imped-ance to obtain the current phasor.

4. Take the real part of the current phasor to obtain the actual current as a function of time.

All of this is best illustrated through a numerical example.

A series LCR circuit supports a current

The series inductance is 0.04henry (H), the series capacitance is 0.0002farad (F), and the series resis-tance is 20ohms (Ω). Find the voltage (V) which must be applied to produce the given current (I).

The impedance of the resistor is 20Ω The impedance of the inductance is

The impedance of the capacitor is

The total impedance is

The current phasor is

The voltage phasor is

The actual voltage is thus

Alternately let the voltage applied to the same circuit be

What is the circuit current in this case?

The total impedance is

The voltage phasor is

The current phasor is

The actual current is thus