Electrical Circuits 3(2+1)

LESSON 11. Reciprocity Theorem

The circuit in the figure is a concrete example of reciprocity. The reader should solve the circuit, and determine the values of the current I in the two cases, which will be equal (0.35294 A). If E is reversed, then the direction of I is reversed, so the direction does not matter so long as both E and I are reversed at the same time.

Fig. 11.1

A non-bilateral element, such as a rectifying diode, destroys reciprocity, as the circuit at the left shows. Here it is obvious that I = 0 on the right, while I ≠ 0 on the left. Even if the polarity of E is reversed so that the diode is forward-biased on the right, the currents are not reciprocal. On the left, I will be 0.21294 A, while on the right it will be 0.32824 A, as the reader can show by solving the circuits.

Fig. 11.2

A nonlinear element also destroys reciprocity. The circuit at the right includes a resistor whose voltage drop is proportional to the square of the current, V = 10i². In the circuit at the left, I = 0.3798 A, while on the right I = 0.3397 A. The inclusion of controlled sources or active elements may also destroy reciprocity.

Fig. 11.3

11.1. Two-Port Networks

Consideration of reciprocity leads naturally to two-port networks. These are networks with four terminals considered in two pairs as ports at which connections are made. The emf E in the reciprocity theorem is considered to be connected to one port, say port 1, while the current is at port 2, assumed to be short-circuited. The ports result from breaking into two of the branches of the network. One terminal of each port is denoted by (+) to specify the polarity of the voltage applied at the port, and currents are positive when they enter the (+) terminal. The fundamental variables are V₁, I₁, V₂ and I₂. Any two of these variables are functions of the remaining two. For certain networks, some of the four choices are not admissible. In most cases, the variables appearing in the models are variations from DC bias conditions, not the DC variables themselves.

Fig. 11.4

 A single resistor forms two two-ports, depending on whether it is in series or shunt. For the series resistor, it is normal to take the dependent variables as I₁ and I₂, and the independent variables V₁ and V₂. The coefficients are called the admittance parameters, since admittance is the ratio of current to voltage. If the resistor is connected in shunt, the natural independent variables are I₁ and I₂, while V₁ and V₂ are the dependent variables. The coefficients in this case are the impedance parameters, since impedance is the ratio of voltage to current. In both cases, we see that the off-diagonal or transfer coefficients are equal.

Consider the admittance model, where the coefficients are not necessarily equal to the simple ones for a single resistor. If an emf E is applied at the input, E = V₁, and the output is shorted, V₂ = 0, we see that I₂ = y₂₁E. If we apply the same voltage E to the output, and short the input, we have I₁ = y₁₂E. Therefore, the currents are equal, and the network is reciprocal. Reciprocity is the result of the equality of the transfer admittances. A similar result can be obtained for the impedance model, where reciprocity holds if z₁₂ = z₂₁, as the reader can easily show. For our simple example of a single resistor, these conditions hold.

We note that there is no impedance model for the series resistor, and no admittance model for the shunt resistor. In general, it is possible to transform from impedance to admittance and vice versa, but the determinant of the coefficients is in the denominator of the transformation formula, and for a single resistor this determinant is zero. If Δ is the determinant of the admittance coefficients, then z₁₁ = y₂₂/Δ, z₂₂ = y₁₁/Δ, z₁₂ = y₂₁/Δ and z₂₁ = y₁₂/Δ. Similar formulas hold for the inverse transformation. A full list of transformations is given in the first Reference. The conclusion is that if y₁₂ = y₂₁, then z₂₁ = z₁₂ as well. This, of course, is a general result. The particular way of modeling a network does not affect the property of reciprocity.

A popular model, especially for active elements, is the hybrid model in which the independent variables are V₂ and I₁. The equations for the series resistor are V₁ = RI₁+ V₂ and I₂ = -I₁. The hybrid parameters for this case are h₁₁ = R, h₂₂ = 0, h₁₂ = -h₂₁ = 1. Note that the off-diagonal elements are of opposite sign in this case, but of the same magnitude. This is the mark of reciprocity in the hybrid model. As an exercise, the reader may show that for the shunt resistor h₁₁ = 0, h₂₂ = 1/R, and the transfer coefficients are the same as those for the series resistor.

There is a fourth model in which the output variables I₁ and V₁ are expressed in terms of the input variables V₂ and I₂: V₁ = AV₂ - BI₂, and I₁ = CV₂ - DI₂. Note the change in sign of I₂, which is conventional so it will be in the same direction as I₁ for an identity transformation A = D = 1, B = C = 0. If networks are cascaded, these matrices simply multiply. In this model, reciprocity is expressed by AD - CB = 1. This can be proved by expressing A, B, C and D in terms of the impedance parameters. A = z₁₁/z₂₁, B = z₁₁z₂₂/z₂₁ - z₁₂, C = 1/z₂₁ and D = z₂₂/z₂₁. The result is AD - CB = z₁₂/z₂₁ = 1 if the network is reciprocal. It is also clear that any cascade of reciprocal networks is also reciprocal, since the determinant of the product of matrices is the product of the determinants.

Fig. 11.5

Any model can be expressed in terms of controlled sources and impedances. This is shown at the right for they hybrid model. The h₁₁ is represented by an impedance, h₂₂ by an admittance. h₁₂ and h₂₁ are the reverse and forward transfer ratios, which are pure numbers. A current source appears in the output, while a voltage source appears in the input. This is the model often used to represent a transistor, in which h₂₁ is usually denoted h_fe or β, the base to collector current gain. The input is between base and emitter, the output between collector and emitter. Since h₁₂ is much smaller than h₂₁, the transistor is not reciprocal. A good approximate model results if h₁₁ = h₂₁ (25Ω/I_c), h₁₂ = h₂₂ = 0. I_c is the DC collector current. A typical value for h₂₁ is 100. For accurate work, the hybrid coefficients can be expressed as functions of the bias conditions.

A vacuum tube can be represented by an admittance model, in which the plate current I_p is a joint function of the grid voltage V_g and the plate voltage V_p. The input port is connected to grid and cathode, the output port to plate and cathode. As long as the grid is biased negatively with respect to the cathode, I_g = 0, and two of the admittance parameters are zero. The remaining equation can be written I_p = g_mV_g + V_p/r_p, where y₁₂ = g_m is the transconductance and 1/y₂₂ = r_p is the plate resistance. When I_p is held constant, -g_mr_pV_g = V_p, which defines the amplification factor μ = g_mr_p. The same model can be used for an FET. We note that none of these models are reciprocal, which is worth remembering. These models are extremely useful in circuit design.

11.2. Proof of the Reciprocity Theorem

We wish to show that in a network of linear, bilinear elements, that is, in one constructed of ordinary impedances, that if when a voltage V is inserted in one loop the current I in another loop due to the insertion of this voltage is the same as the current at the first position due to the insertion of a voltage V in the second loop, or that the network is reciprocal. We shall do this by explicitly calculating the currents in the two cases, and observing that they are equal.

Consider the network as made up of N independent loops, of which loop 1 contains the input port, and loop 2 the output port. Let there be no emf's in the network. If there are, they can be taken care of by superposition, and we can set them all to zero for our purposes. The loop currents are I_k, k = 1 to N. If emf V is at port 1 and port 2 is shorted, then Kirchhoff's laws give Σz_1kI_k = V, and Σz_jkI_k = 0, j = 2 to N. The current I₂ can then be expressed in terms of determinants as I₂ = -V det A/Δ, where the determinant in the numerator has been expanded by minors of the second column, which is all zeros except for V in the first position. A is the matrix of the z's with the first row and second column taken away. Δ is the determinant of the matrix of the z_jk. Write out this solution in detail if it is not clear from this description.

Now connect V in port 2, and short port 1. The solution by determinants for I₁ is -V det B/Δ, where B is the matrix of the z's with the second row and first column taken away. If we compare matrices A and B, we see that they are transposes of each other, provided that z_jk = z_kj. However, the off-diagonal z's are just the mutual impedances of the current loops (impedances common to a pair of loops), and do not depend on the order of the subscripts. (They are not the coefficients of an impedance model here, simply actual impedances.) Since the determinant of the transpose of a matrix is equal to the determinant of the matrix, the two solutions are the same, and I₁ = I₂. This proves the reciprocity theorem for networks made from linear, bilateral elements, Q.E.D.

Fig. 11.6

We have proved the reciprocity theorem for a relatively limited class of networks, but it is possible to extend the theorem more widely. The figure at the right shows that reciprocity holds for ideal transformers at least. Since real transformers may be modeled by networks containing only impedances and ideal transformers, real transformers must also be reciprocal, and this extends the utility of the theorem further. Reciprocity crops up in many unexpected places, such as in electromagnetic fields and microwaves.

 

References

1. J. D. Irwin, Basic Engineering Circuit Analysis, 2nd ed. (New York: Macmillan, 1987). Chapter 15.

2. R. M. Kerchner and G. F. Corcoran, Alternating-Current Circuits, 3rd ed. (New York: John Wiley & Sons, 1951). p. 137f.

3. R. P. Ward, Introduction to Electrical Engineering (New York: Prentice-Hall, 1952). p. 93, Problem 24-IV.