Network Theorems

Much reference has been made to voltage and current sources as well as passive circuit elements in all of the previous work. Voltage sources and current sources are also circuit elements, which are further classified as being active circuit elements.

The distinction between an active circuit element and a passive one is that the active circuit element contains or controls a source of electrical energy. It is true that capacitors and inductors can temporarily store energy but they do not inherently contain inde-pendent sources of energy and hence are classified as passive elements.

Active elements or networks must contain devices or agencies which have the ability, in the thermodynamic sense, to reversibly convert some other form of energy into electrical energy. The amount of energy so converted in the device or agency per unit of charge passing through the device or agency has historically been called the electromo-tive force or emf. The appearance of the word force in the historical terminology is a misnomer as energy rather than force is the applicable concept.

An emf is measured in volts and a volt is a joule of energy per coulomb of electric charge. In the case of an emf, the energy referred to is the amount of some other form of energy converted into electrical energy per unit of electrical charge. Potential differ-ence, previously referred to as simply voltage, is also measured in volts. Potential difference, however, represents the change in electrical poten-tial energy per unit of electrical charge experienced in moving between two points. The letter E will be used here to represent a sinusoidal emf. The letter v

will continue to be used to represent a time depen-dent potential difference.

In order to sustain a current in a closed conducting circuit, the circuit must contain a source of electrical energy provided by an emf. The only exception to this statement occurs in a closed super-conducting coil. Even in the supersuper-conducting coil case, an emf is necessary in order to first establish a direct current in the coil. Sources of emf appear in many forms. A mechanically driven alternating current generator is a source of emf in which a portion of the mechanical energy of rotation of the generator shaft is converted into electrical energy.

Chemical electric cells of both the primary and secondary categories are sources of a dc emf. In this instance an exoergic chemical reaction in the cell makes electrical energy available. A solar panel is a source of a dc emf. Here a portion of the luminous energy from the sun impinging on the cell is converted into electrical energy. One could easily list many other examples. A common property of all of these sources of emf is that they must possess a conducting path between their terminals and along this path will occur losses. For example, the coils in a mechanically driven alternating current generator will possess both resistance and inductance and perhaps significant distributed capacitance. This means that any source of emf will also have some impedance internal to its structure. This situation is depicted in its simplest form in Fig. 8-27.

The elements depicted on the left in Fig. 8-27 represent a physically realizable source of emf. The element on the right in Fig. 8-27 represents the impedance of some circuit which is connected across the terminals of the source of emf. The source of emf is represented by two elements. A circle containing one period of a sine curve, this being the representation of an ideal source of alternating emf (one without losses) and a rectangle representing the internal impedance of the conducting path through the source of emf. The internal impedance of the Figure 8-26. Impedance angle after correction.

Corrected Impedance Angle

10² 10³ 10⁴ Frequency–Hz

0.8 0.6 0.4 0.2 0

−0.2

−0.4

−0.6

−0.8

−1.010¹

Impedance angle

Figure 8-27. A source of alternating emf and its exter-nally applied load.

E Z₀

Z_x

Interfacing Electrical and Acoustic Systems 133 source of emf is Z₀ and the impedance of the

external circuit is Z_x. The arrow indicates the posi-tive sense of the current in the circuit. The sustained current which will exist in a simple conducting loop is the algebraic sum of the emfs (if there are more than one) divided by the total series impedance of the loop including all source internal impedances. In this instance the current is given by

(8-94)

The first circuit theorem to be discussed is the Maximum Power Transfer Theorem. This theorem deals with the proposition that given a particular source having a fixed emf and a fixed internal impedance, what value of load impedance must be connected to the terminals of the emf such that the power dissipated in the external load will be a maximum? In answering this question, first rewrite the equation for the current in the circuit of Fig. 8-27 by explicitly displaying the complex nature of both the internal and external impedances.

(8-95)

Earlier it was learned that the power dissipated in a load is directly proportional to the square of the current multiplied by the real part of the load imped-ance. The next step is to examine the expression for the current and inquire what adjustment can be made to the load impedance that will not change its real part but will make the denominator in the current expression smaller and consequently, the current larger. Upon recalling that reactances can be both positive and negative, the denominator can be made smaller by having the reactance of the load be just the negative of the reactance of the source thus yielding a net reactance of zero. The current now becomes

(8-96)

The average power dissipated in the external load is then

(8-97)

where,

E_m is the amplitude of the sinusoidal emf.

The next and final step is to find the value of R_x which will make the power expression a maximum.

Readers familiar with the differential calculus would differentiate the power expression with respect to R_x and set the derivative equal to zero thus obtaining a simple algebraic expression which can easily be solved for the magic value of R_x satisfying the problem. Those readers not familiar with calculus can still arrive at the answer after the expenditure of some additional effort. It is only necessary to let R_x be a variable expressed in fractions and multiples of R₀ and graph the function

(8-98)

A carefully drawn graph will have a maximum occurring when R_x equals R₀.

The conclusion is that the power in the external load is a maximum when the load impedance is the complex conjugate of the source impedance. As a reminder, complex conjugate means equal real parts with imaginary parts equal but opposite in sign.

It is important to note, however, that when the power dissipated in the load has been maximized, the power transfer efficiency is only 50% as an equal amount of power is being dissipated within the source itself. Additionally, under these conditions, the voltage applied to the load is only one-half of the open circuit voltage of the source. This type of match is often desirable in communications systems where a premium is placed on signal power. It is not employed in commercial power systems where a premium is placed on voltage regulation and overall efficiency. Note also, this theorem can not be applied to audio power amplifiers because the emf associated with the power amplifier output is not a constant as required by the theorem, but rather is a function of the load impedance to which it is connected. Power amplifiers are purposely designed to have low output impedances (high damping factors) and are designed to work into load impedances equal to or greater than a specified minimum which is significantly larger than the output impedance.

T h e s e c o n d t h e o r e m t o b e d i s c u s s e d i s Thévenin’s Theorem. This theorem deals with the proposition that a linear network of any number of emfs and any combination of impedances which communicates with the external world through only two terminals is equivalent to a single emf E₀ in series with a single impedance Z₀. E₀ is equal to the voltage present between the actual network’s termi-nals when it is disconnected from the external world and Z₀ is the impedance measured between the actual network’s terminals when it is disconnected from the external world and all internal emfs are inactive. The value of this theorem and an

elabora-i E

134 Chapter 8 tion of the terminology used therein can best be explored through an example problem.

Fig. 8-28A displays a circuit problem where the objective is to determine the current which exists in the physical inductor located in the central branch.

The circuit contains two sinusoidal generators each having a time dependence of cos(ωt) and each having internal resistance as indicated. The emfs associated with the generators are the rms values.

The inductor is connected between the points labeled a and b. One could solve this problem by assuming mesh currents similar to the technique employed in the section dealing with the impedance bridge. One would then have to write two simulta-neous equations, solve them, and then use knowl-edge of the mesh currents to determine the net current in the central branch. The Thévenin approach is simpler in this instance. First, discon-nect the inductor between a and b and find the potential difference or open circuit voltage which exists between a and b under these conditions. This circuit appears in Fig. 8-28B. In Fig. 8-28B the net emf is 10V − 5V = 5V and the total series imped-ance is 2Ω + 3Ω = 5Ω . The current in the loop is then 5V divided by 5Ω or 1A. From the generator on the left, V_ab is then 10V less the drop across its internal resistance which is 1A times 2Ω giving 10 V − 2V = 8V. Now one determines the imped-ance which exists between a and b when the emfs of the generators are set equal to zero but with the generators still in place. In this instance, one has 2Ω paralleled by 3Ω giving a value of 1.2Ω. The Thévenin equivalent of the outer loop is then a generator having an emf of 8V and an internal impedance which is a pure resistance of 1.2Ω.

Finally, in Fig. 8-28C, this source is connected to the load inductor and it is a simple matter to determine the current. The impedance of this series loop is

The current in this loop which is the current in the inductor has an rms value of 8V divided by 5Ω or 1.6 A and the current lags behind the voltage of the generator by an angle of 0.927radian.

In summary, Thévenin’s theorem says that any linear two terminal active network can be replaced by an ideal source of emf in series with an imped-ance. The value of the emf is equal to the open circuit voltage of the network between the two terminals and the impedance is the measured imped-ance between the terminals when the sources internal to the network are inactive.

A related network theorem is called Norton’s Theorem. Norton’s Theorem states that any linear two terminal active network can be replaced by an ideal current source paralleled by an admittance. The value of the current source is the short circuit current which could exist between the two terminals of the actual network and the admittance is the admittance measured between the two terminals of the network when the sources internal to the network are inactive.

The Norton equivalent which could replace the Thévenin equivalent of the sample problem can be derived from the Thévenin equivalent and is shown in Fig. 8-29. The Norton values are determined as follows. When a is shorted to b in the Thévenin equivalent, the current is 8V⁄1.2Ω = 6²⁄³A. The admittance between a and b with the voltage set equal to zero in the Thévenin equivalent is the recip-rocal of 1.2Ω or ⁵⁄⁶siemen (S).

In order for each and every one of the foregoing network theorems to be valid, it is necessary that the circuit or system be linear. Being linear means that the laws of physics governing the system appear in the form of linear differential equations. One of the properties of such equations is that if there exist 1.2 1.8 j4+ +

Figure 8-28. Application of Thévenin’s theorem.

a

Interfacing Electrical and Acoustic Systems 135

several solutions to a particular linear differential equation, then a sum of these solutions is also a solu-tion to the equasolu-tion. This leads to what is termed the principle of superposition or the Superposition Theorem. In a network containing several sources, either voltage or current, and several impedances, one applies superposition by calculating the effects of each source acting individually with all other sources set equal to zero. This is done for each source in turn. When this is done, it is important to note that even though a source may be inactive, its impedance still remains in the circuit. The final solu-tion when all sources are active simultaneously is just the sum of the solutions obtained when the sources are acting one at a time. As an example, superposition will be applied to the circuit of Fig. 8-28B in order to determine the potential differ-ence between a and b. Consider that the 10V source is active with the 5V source set equal to zero. In this instance, the 3Ω and 2Ω resistors form a voltage divider across the 10V source and V_ab is then

Now consider that the 5V source is active with the 10V source set equal to zero. The situation here has the 2Ω and the 3Ω resistors forming a voltage divider across the 5V source yielding a value of V_ab of

When both sources act simultaneously, V_ab is the sum of these two individual solutions yielding the value of 6V + 2V or 8V. There are many more network theorems, many of which are more specialized. The ones studied here are the ones most often invoked in routine circuit analysis.