Reciprocity Theorem

Electrical Circuit Analysis > Network Theorems

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What is reciprocity theorem?

The reciprocity theorem is applicable only to single-source networks. It is, therefore, not a theorem employed in the analysis of multisource networks described thus far. The reciprocity theorem states the following:

The current $I$ in any branch of a network, due to a single voltage source $E$ anywhere else in the network, will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current $I$ was originally measured.

In other words, the location of the voltage source and the resulting current may be interchanged without a change in current. The reciprocity theorem requires that the polarity of the voltage source have the same correspondence with the direction of the branch current in each position.

Fig. 1: Demonstrating the impact of the reciprocity theorem.

In the representative network of [Fig. 1(a)], the current I due to the voltage source E was determined. If the position of each is interchanged as shown in [Fig. 1(b)], the current I will be the same value as indicated. To demonstrate the validity of this statement and the reciprocity theorem, consider the network of [Fig. 2], in which values for the elements of [Fig. 1(a)] have been assigned.

Fig. 2: Demonstrating the impact of the reciprocity theorem.

The total resistance is

$$\begin{split} R_T &= R_1+ R_2 || (R_3 + R_4)\\ &= 12Ω +6Ω || (2Ω + 4Ω )\\ &= 12Ω +6Ω || 6Ω \\ &= 12Ω +3Ω = 15Ω\\ \end{split}$$

and

$$ I_s = {E \over R_T} = {45V \over 15Ω} = 3A$$

with

$$ I = {3A \over 2} = 1.5A$$

Fig. 3: Interchanging the location of E and I of [Fig. 2] to demonstrate the validity of the reciprocity theorem.

For the network of Fig. 3, which corresponds to that of Fig. 1(b), we find

$$\begin{split} R_T &= R_4 + R_3 + R_1 || R_2\\ &= 4Ω + 2Ω + 12Ω || 6Ω\\ &= 10Ω\\ \end{split}$$

and

$$ I_s = {E \over R_T} = {45V \over 10Ω} = 4.5A$$

using current divider rule for I:

$$ I = {I_s R_2 \over R_1+R_2} = {4.5V)(6) \over 18Ω} = 1.5A$$

which agrees with the above. The uniqueness and power of such a theorem can best be demonstrated by considering a complex, single-source network such as the one shown in [Fig. 4].

Fig. 4: Demonstrating the power and uniqueness of the reciprocity theorem.

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