Terminated ladder with finite transmission zeros

Definitions

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

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

The reflection coefficient $\rho(s)$ of a resistively terminated lossless ladder compares the power reflected back towards the source with the maximum power the source can give. Looking into the ladder (input impedance $Z_1$) from a source of resistance $R_1$:

$$\rho(s) = \frac{R_1 - Z_1(s)}{R_1 + Z_1(s)}, \qquad |\rho(j\omega)|^2 = \frac{P_{\text{reflected}}}{P_{\max}}$$

Since the ladder is lossless, all power entering it reaches the load, so the two coefficients are linked by Feldtkeller's equation:

$$|\rho(j\omega)|^2 + |t(j\omega)|^2 = 1$$

Realizing finite transmission zeros in a terminated ladder

A transmission zero is a frequency where no signal reaches the load ($t = 0$, $|\rho| = 1$). All-pole filters (Butterworth, Chebyshev) have all zeros at $s = \infty$, made by series inductors and shunt capacitors. Filters such as inverse Chebyshev and elliptic (Cauer) also need finite zeros on the $j\omega$ axis. In a ladder a transmission zero is produced when either:

• a series arm becomes an open circuit: a parallel LC tank in the series arm, resonant at $\omega_z = 1/\sqrt{LC}$; or
• a shunt arm becomes a short circuit: a series LC branch from line to ground, resonant at $\omega_z$.

 series tank (open at wz)    shunt series-LC (short at wz)
 o--+--UUU--+--o             o-----+-----o
    |       |                      |
    +--||---+                     UUU L
                                   |
                                  === C
                                   |
 o-------------o             o-----+-----o

Procedure (zero shifting by partial removal):

1. From $t(s)$ find $\rho(s)$ and $Z_1(s) = R_1\dfrac{1+\rho}{1-\rho}$ as usual. The finite zeros of $t(s)$ are known, say $\pm j\omega_z$.
2. Expand $Z_1$ (or $Y_1$) step by step. Before each finite zero, partially remove a pole at infinity (a series $L$ or shunt $C$), choosing its value so that the remaining function has a zero exactly at $s = j\omega_z$: e.g. $C = Y(s)/s$ evaluated at $s^2 = -\omega_z^2$.
3. The reciprocal of the remainder now has a pole at $\pm j\omega_z$; remove it fully as a series tank (or shunt series-LC branch). This creates the transmission zero.
4. Continue with the next zero; the final remainder must equal the load $R_2$.

Example. For $y = s(s^2+4)/(s^2+1)$ with a required zero at $\omega = 3$: remove shunt $C = y/s|_{s^2=-9} = 5/8$ F, leaving $y' = 3s(s^2+9)/[8(s^2+1)]$, which is zero at $j3$. Then $1/y'$ has a pole at $j3$, removed as a series tank $C = 27/64$ F, $L = 64/243$ H that blocks the signal at $\omega = 3$ rad/s.

The same idea gives the elliptic (Cauer) ladder, where each series inductor has a capacitor in parallel to produce a finite stopband zero.