Image Impedance, Propagation Constant & Matched Two-Port Operation

Image parameters answer a cascade question: what impedance must terminate port 2 so that port 1 presents its own designated image impedance, and vice versa? They are most natural for reciprocal filters and transmission sections whose repeated behavior matters more than arbitrary open- or short-circuit tests.

Mutual termination definition #

$Z_{i1}$ and $Z_{i2}$ satisfy both conditions:

• terminating port 2 in $Z_{i2}$ makes the input at port 1 equal $Z_{i1}$;
• terminating port 1 in $Z_{i1}$ makes the input at port 2 equal $Z_{i2}$.

For a reciprocal ABCD matrix with $AD-BC=1$:

$$Z_{i1}=\sqrt{\frac{AB}{CD}},\qquad Z_{i2}=\sqrt{\frac{BD}{AC}},$$

with square-root branches selected for the passive physical solution. For a symmetric network $A=D$, both reduce to

$$Z_i=\sqrt{\frac{B}{C}}.$$

Propagation constant #

A reciprocal two-port can be parameterized as

$$A=\sqrt{\frac{Z_{i1}}{Z_{i2}}}\cosh\gamma,$$

$$B=\sqrt{Z_{i1}Z_{i2}}\sinh\gamma,$$

$$C=\frac{1}{\sqrt{Z_{i1}Z_{i2}}}\sinh\gamma,$$

$$D=\sqrt{\frac{Z_{i2}}{Z_{i1}}}\cosh\gamma.$$

Thus $\gamma=\alpha+j\beta$ combines attenuation $\alpha$ and phase shift $\beta$ per section. For cascaded identical image-matched sections, the propagation constants add.

What matching means here #

An image-matched load suppresses the repeated reflection implied by the section model. It does not mean the same impedance is optimal for power, noise, bandwidth, or every external source. Image impedance is also distinct from the characteristic impedance of an arbitrary physical medium, though they coincide for appropriate uniform line sections.

Terminating the output in $100\,\Omega$ gives $Z_{in}=(1.25\cdot100+75)/(0.0075\cdot100+1)=100\,\Omega$, which directly checks the fixed-point definition rather than relying only on the closed-form square root.