1.6 Synthesis of One-Port and Two-Port Networks

1.6 Synthesis of One-Port and Two-Port Networks

Introduction to Network Synthesis

Network analysis determines the behavior of a given circuit. Network synthesis is the inverse problem: given a desired electrical behavior (specified as a transfer function, impedance, or admittance), find a physical circuit that realizes it. This is the fundamental design process for filters, impedance matching networks, and equalizers. Synthesis bridges the gap between mathematical specifications and practical, buildable circuits using standard passive (R, L, C) and active components. This section covers the mathematical foundations (Hurwitz polynomials, positive real functions) and the canonical realization techniques (Foster, Cauer, ladder networks) for both one-port and terminated two-port networks.

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1. Mathematical Foundations for Synthesis

1.1 Hurwitz Polynomials

1. Definition: A polynomial $P(s)$ with real coefficients is a Hurwitz polynomial (or strictly Hurwitz) if all its roots (zeros) lie in the open left half of the s-plane (i.e., have negative real parts). It is the characteristic polynomial of a stable system.

2. Properties:

   • All coefficients must be positive and non-zero. (A necessary but not sufficient condition).

   • For a polynomial $P(s) = a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0$, all $a_i > 0$.

   • The polynomials formed by its even and odd parts must have alternating real roots on the negative real axis.

3. Significance in Synthesis: The denominator of any driving-point impedance/admittance $Z(s)$ or $Y(s)$ of a passive network must be a Hurwitz polynomial to ensure the network's stability and realizability with passive components.

1.2 Positive Real (PR) Functions

1. Definition: A rational function $F(s)$ of the complex variable $s = \sigma + j\omega$ is a Positive Real (PR) function if it satisfies: a. $F(s)$ is real when $s$ is real (i.e., it has real coefficients). b. $\operatorname{Re}[F(s)] \ge 0$ whenever $\operatorname{Re}[s] \ge 0$. This is equivalent to the following set of necessary and sufficient conditions (for rational functions):

   • $F(s)$ is real for real $s$.

   • $F(s)$ has no poles in the right half s-plane ($\operatorname{Re}[s] > 0$).

   • Any poles on the $j\omega$-axis (including $s=0$ and $s=\infty$) must be simple and have real positive residues.

   • $\operatorname{Re}[F(j\omega)] \ge 0$ for all $\omega$.

2. Physical Interpretation: The driving-point impedance $Z(s)$ or admittance $Y(s)$ of any linear, time-invariant, passive (R, L, C, M) network is a positive real function. It guarantees the network cannot produce energy, only dissipate or store it.

3. Tests for PR-ness:

   • Pole-Zero Test: Poles and zeros must be in the LHP or on the $j\omega$-axis. $j\omega$-axis poles/zeros must be simple.

   • Real Part Test on $j\omega$-axis: Plot $\operatorname{Re}[F(j\omega)]$ vs. $\omega$; it must be non-negative for all $\omega$.

   • Sturm's Test or Routh-Hurwitz Criterion on related polynomials.

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2. Canonical Forms for One-Port LC Network Synthesis

For lossless networks (containing only inductors L and capacitors C), the impedance $Z_{LC}(s)$ is a special type of PR function: it is an odd function of $s$ (or the ratio of an even to odd polynomial, or vice-versa) and its real part on the $j\omega$-axis is zero. Two canonical realization methods exist.

2.1 Properties of LC Impedance/Admittance

For $Z_{LC}(s)$ or $Y_{LC}(s)$:

1. It is the ratio of an even to odd polynomial, or odd to even polynomial. $Z_{LC}(s) = \frac{m_1(s) + n_1(s)}{m_2(s) + n_2(s)} = \frac{\text{Either } m_1(s)}{n_2(s)} \text{ or } \frac{n_1(s)}{m_2(s)}$ where $m(s)$ are even polynomials and $n(s)$ are odd polynomials.

2. Poles and zeros are simple and lie only on the $j\omega$-axis.

3. They alternate (a pole is always followed by a zero, and vice-versa).

4. There is a pole or zero at both $s=0$ and $s=\infty$.

2.2 Foster Canonical Forms

Foster forms realize the given LC function by a direct partial fraction expansion of $Z(s)$ or $Y(s)$.

1. Foster I Form (First Foster Form):

   • Applied to an LC impedance $Z_{LC}(s)$.

   • Expand $Z_{LC}(s)$ into partial fractions with respect to $s$: $Z_{LC}(s) = k_\infty s + \frac{k_0}{s} + \sum_{i=1}^{n} \frac{2k_i s}{s^2 + \omega_i^2}$

   • Circuit Realization: A series connection of:

     • An inductor $L_\infty = k_\infty$ (from pole at infinity).

     • A capacitor $C_0 = 1/k_0$ (from pole at zero).

     • A series LC tank for each finite $j\omega$-axis pole: $L_i = 2k_i/\omega_i^2$, $C_i = 1/(2k_i)$.

2. Foster II Form (Second Foster Form):

   • Applied to an LC admittance $Y_{LC}(s) = 1/Z_{LC}(s)$.

   • Expand $Y_{LC}(s)$ into partial fractions: $Y_{LC}(s) = k'_\infty s + \frac{k'_0}{s} + \sum_{i=1}^{n} \frac{2k'_i s}{s^2 + \omega_i^2}$

   • Circuit Realization: A parallel connection of:

     • A capacitor $C_\infty = k'_\infty$.

     • An inductor $L_0 = 1/k'_0$.

     • A parallel LC tank for each finite pole: $C_i = 2k'_i/\omega_i^2$, $L_i = 1/(2k'_i)$.

2.3 Cauer Canonical Forms

Cauer forms realize the given LC function by a continued fraction expansion (synthetic division) of $Z(s)$ or $Y(s)$ about $s = \infty$ or $s = 0$, yielding a ladder network.

1. Cauer I Form (First Cauer Form):

   • Expansion about $s = \infty$ (removing poles at infinity).

   • For $Z_{LC}(s)$, perform successive long divisions, extracting the highest powers of $s$.

   • Circuit Realization: Results in a ladder network starting with a series inductor (from the first extraction).

   • Topology: Series L, shunt C, series L, shunt C, ...

2. Cauer II Form (Second Cauer Form):

   • Expansion about $s = 0$ (removing poles at zero).

   • For $Z_{LC}(s)$, perform successive long divisions, extracting the lowest powers of $s$.

   • Circuit Realization: Results in a ladder network starting with a shunt capacitor.

   • Topology: Shunt C, series L, shunt C, series L, ...

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3. Synthesis of RL and RC One-Port Networks

RL and RC networks are lossy. Their impedance/admittance functions have distinct pole-zero patterns on the negative real axis (not the $j\omega$-axis).

3.1 Properties of RL and RC Impedance/Admittance

1. For an RL Impedance $Z_{RL}(s)$:

   • Poles and zeros are simple, lie on the negative real axis, and alternate.

   • The critical frequency nearest the origin is a zero.

   • The critical frequency nearest to infinity is a pole.

   • $Z_{RL}(0) \le Z_{RL}(\infty)$.

2. For an RC Impedance $Z_{RC}(s)$:

   • Poles and zeros are simple, lie on the negative real axis, and alternate.

   • The critical frequency nearest the origin is a pole.

   • The critical frequency nearest to infinity is a zero.

   • $Z_{RC}(0) \ge Z_{RC}(\infty)$.

3. For RC Admittance $Y_{RC}(s)$: It has the same properties as $Z_{RL}(s)$.

3.2 Synthesis Procedures

The Foster and Cauer techniques can be adapted for RL and RC networks by recognizing the location of poles/zeros on the negative real axis.

1. Foster Forms for RL/RC:

   • Use a partial fraction expansion in $s$ for Foster I (for impedance).

   • Each term corresponds to a simple RL or RC combination (e.g., a term $k/(s+\sigma)$ for RC impedance corresponds to a parallel RC branch with $R=1/k$ and $C=1/(k\sigma)$).

2. Cauer Ladder Forms for RL/RC:

   • Cauer I: Continue to remove poles at $s = \infty$. For an RC impedance, this alternately removes a constant (resistor) and a pole at infinity (capacitor), yielding a ladder starting with a resistor.

   • Cauer II: Continue to remove poles at $s = 0$. For an RC impedance, this alternately removes a pole at zero (resistor) and a constant (capacitor), yielding a different ladder.

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4. Synthesis of Resistively Terminated Two-Port Networks

Often, a network must be designed to provide a specific voltage transfer function $H(s) = V_2(s)/V_1(s)$ between a source with resistance $R_s$ and a load resistance $R_L$.

4.1 The Problem Setup

• Given: A desired transfer function magnitude $|H(j\omega)|$ (e.g., Butterworth, Chebyshev response), source resistance $R_s$, and load resistance $R_L$.

• Find: A passive (LC, or RLC) two-port network inserted between $R_s$ and $R_L$ that realizes the given $H(s)$.

4.2 Active vs. Passive Synthesis

1. Passive Synthesis (Lossless Ladder):

   • The two-port is typically a lossless LC network (no resistors inside).

   • The only losses are in the terminating resistors $R_s$ and $R_L$.

   • Design methods include image parameter and insertion loss methods. The insertion loss method is more modern and precise.

   • Procedure: a. From the given $|H(j\omega)|^2$, determine the associated characteristic function and reflection coefficient. b. Find the input impedance $Z_{in}(s)$ looking into the lossless two-port when terminated in $R_L$. c. Synthesize $Z_{in}(s)$ as a one-port LC network using Cauer or Foster forms. This yields the lossless ladder network.

2. Active RC Synthesis:

   • Employs active elements (op-amps) along with resistors and capacitors to realize transfer functions, especially when inductors are undesirable (bulky, non-ideal).

   • Enables realization of right-half-plane zeros and high-Q poles not possible with passive RLC networks alone.

   • Common Configurations: Sallen-Key, Multiple Feedback (MFB), state-variable filters.

4.3 Ladder Networks

1. Definition: A ladder network is a series of series and shunt branches (impedances) connected alternately. It is the most common topology arising from Cauer synthesis and for terminated LC filters.

2. Advantages:

   • Modular Structure: Easy to design and analyze section by section.

   • Low Sensitivity: The filter's performance (e.g., passband) has low sensitivity to variations in component values, which is a highly desirable property.

   • Canonical: For a given transfer function of order $n$, the ladder network uses exactly $n$ reactive elements (the minimum number).

3. Types:

   • All-Pole Ladders (e.g., for Butterworth, Chebyshev responses): Only series and shunt branches containing a single L or C.

   • Ladders with Transmission Zeros: Include resonant branches (series or parallel LC) to create zeros of transmission at finite frequencies (e.g., for elliptic/Cauer filters).

4.4 The Insertion Loss Method (Key Steps)

This is the primary method for designing practical passive LC filters.

1. Power Ratio: Define the insertion loss function in terms of power: $\frac{P_{\text{max}}}{P_L} = \frac{|V_1|^2/(4R_s)}{|V_2|^2/R_L} = \frac{1}{|H(s)|^2} \cdot \frac{4R_s}{R_L}$ for $s=j\omega$. This leads to $|H(j\omega)|^2 = \frac{4R_s}{R_L} \cdot \frac{1}{1 + |K(j\omega)|^2}$, where $K(s)$ is the characteristic function.

2. Identification: Choose a polynomial form for $|H(j\omega)|^2$ (e.g., Butterworth: $1/(1+\omega^{2n})$).

3. Input Impedance: Using the reflection coefficient $\rho(s)$ related to $H(s)$, compute the input impedance: $Z_{in}(s) = R_s \frac{1 + \rho(s)}{1 - \rho(s)}$

4. Synthesis: Synthesize $Z_{in}(s)$ as a lossless one-port (LC network) terminated in $R_L$ using continued fraction expansion (Cauer). The resulting ladder is the desired filter.

Conclusion: Network synthesis transforms a mathematical specification into a realizable circuit. The journey from a positive real function to a physical Foster or Cauer canonical form, or from a desired filter response to a resistively-terminated LC ladder, embodies the core of passive filter design. Mastery of these concepts enables the design of networks that meet precise frequency-selective criteria, a fundamental task in communication, control, and signal processing systems.