Derive reflection coefficient for terminated LC ladder

Transmission coefficient

The transmission coefficient $t(s)$ is the ratio (in power terms) of the power actually delivered to the load $R_2$ to the maximum available power of the source $P_{\max} = |V_1|^2/4R_1$:

$$|t(j\omega)|^2 = \frac{P_2}{P_{\max}} = \frac{4R_1}{R_2}\left|\frac{V_2(j\omega)}{V_1(j\omega)}\right|^2 , \qquad t(s) = 2\sqrt{\frac{R_1}{R_2}}\,\frac{V_2(s)}{V_1(s)}$$

Information obtained: $|t|^2$ is the fraction of available power reaching the load, so it is the filter's power response. $|t| = 1$ means perfect matching and maximum power transfer; $|t| = 0$ marks a transmission zero; and it bounds the gain of a passive ladder ($|t| \le 1$).

Derivation of reflection coefficient

   R1        I1 -->
 o-/\/\----+-------------+------+
 |         |             |      |
(~) V1   Z1(s) ->  Lossless LC  R2  V2
 |         |      ladder  |      |
 o---------+-------------+------+

1. Maximum available power of the source (delivered to a matched load $R_1$):

$$P_{\max} = \frac{|V_1|^2}{4R_1}$$

2. Input power to the ladder with $Z_1 = R_{in} + jX_{in}$ and $I_1 = \dfrac{V_1}{R_1+Z_1}$:

$$P_1 = |I_1|^2 R_{in} = \frac{|V_1|^2\,\text{Re}(Z_1)}{|R_1+Z_1|^2}$$

3. Power not delivered ("reflected") is $P_{\max} - P_1$:

$$\begin{aligned} P_{\max} - P_1 &= \frac{|V_1|^2}{4R_1}\left[1 - \frac{4R_1\,\text{Re}(Z_1)}{|R_1+Z_1|^2}\right] \\ &= \frac{|V_1|^2}{4R_1}\cdot\frac{|R_1+Z_1|^2 - 4R_1\,\text{Re}(Z_1)}{|R_1+Z_1|^2} \end{aligned}$$

4. Using $|R_1+Z_1|^2 - 4R_1 R_{in} = (R_1+R_{in})^2 + X_{in}^2 - 4R_1R_{in} = (R_1-R_{in})^2 + X_{in}^2 = |R_1-Z_1|^2$:

$$\frac{P_{\max}-P_1}{P_{\max}} = \left|\frac{R_1 - Z_1}{R_1+Z_1}\right|^2$$

5. This ratio is defined as $|\rho(j\omega)|^2$, so the reflection coefficient is

$$\rho(s) = \pm\frac{R_1 - Z_1(s)}{R_1 + Z_1(s)}$$

6. Since the ladder is lossless, $P_1 = P_2$, so $|t|^2 = P_1/P_{\max}$ and

$$|\rho(j\omega)|^2 + |t(j\omega)|^2 = 1 \quad \text{(Feldtkeller equation)}$$

7. Solving for $Z_1$ gives the synthesis formula:

$$Z_1(s) = R_1\frac{1 \mp \rho(s)}{1 \pm \rho(s)}$$

Use: given $t(s)$, find $|\rho|^2 = 1-|t|^2$, factor $\rho(s)\rho(-s)$, pick $\rho(s)$ with a Hurwitz denominator, find $Z_1$ and expand it into an LC ladder.