2
\$\begingroup\$

I know the result is h12 = -h21, but it's strange since h12 = V1/V2 and h21 = I2/I1.

Reciprocity theorem says that ratio of excitation to result must remain same if they are interchanged, how do you arrive at that result from the statement?

\$\endgroup\$
  • \$\begingroup\$ What is the context? \$\endgroup\$ – Bradman175 May 28 '16 at 6:07
2
\$\begingroup\$

Reciprocity theorem says that ratio of excitation to result must remain same if they are interchanged

No, the reciprocity theorem does not say that. That is just the definition of reciprocity (actually, not well stated). The reciprocity theorem states, instead, that a certain specific class of networks, that of networks composed of resistors, inductors, capacitors, and transformers, is reciprocal.

First, let me start with a better definition of reciprocal network [1]:

A reciprocal network is one in which, for any pair of excitation and response points, here labeled 1 and 2, \$I_1 = I_2\$ if \$V_1 = V_2\$.

Notice that when you set a current excitation you should look at the voltage response and viceversa.

Consider now the hybrid representation of a two-port network,

$$ \begin{pmatrix} V_1 \\ I_2 \end{pmatrix} = \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix} \begin{pmatrix} I_1 \\ V_2 \end{pmatrix}.$$

Set, e.g, a current excitation at port 1, \$I_1 = I\$, and look at the open circuit voltage response at port 2, that is, find \$V_2\$ when \$I_1 = I\$ and \$I_2 = 0\$. Plugging these conditions into the matrix equation above, yields

$$ \begin{pmatrix} V_1 \\ 0 \end{pmatrix} = \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix} \begin{pmatrix} I \\ V_2 \end{pmatrix},$$

from which we obtain

$$V_2 = -\frac{h_{21}}{h_{22}}I.$$

Now exchange the position of the excitation and the response: set \$I_2 = I\$, \$I_1 = 0\$ and find \$V_1\$. From

$$ \begin{pmatrix} V_1 \\ I \end{pmatrix} = \begin{pmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{pmatrix} \begin{pmatrix} 0 \\ V_2 \end{pmatrix}$$

we get

$$V_1 = \frac{h_{12}}{h_{22}}I.$$

The reciprocity condition requires the two responses to be equal, \$V_1 = V_2\$, that is,

$$\frac{h_{12}}{h_{22}}I = -\frac{h_{21}}{h_{22}}I$$

from which \$h_{12} = -h_{21}\$.

[1] N. Balabanian, T. A. Bickart, Electrical network theory, John Wiley & Sons, Inc., 1969.

| improve this answer | |
\$\endgroup\$

Not the answer you're looking for? Browse other questions tagged or ask your own question.