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Practice MCQ

Unit 6 - Notes

ELE205

Unit 6: Positive Real Function and Network Synthesis

1. Introduction to Network Synthesis

1.1 Network Analysis vs. Network Synthesis

To understand the need for synthesis, one must distinguish it from analysis.

Feature	Network Analysis	Network Synthesis
Input	Network topology + Excitation	Excitation + Desired Response
Output	Response (Voltage/Current)	Network topology + Element Values
Solution	Unique solution exists	Solution is not unique (multiple circuits can produce the same response)
Process	Deduction	Design / Inverse Problem

1.2 The Need for Network Synthesis

Network synthesis is the foundation of circuit design, particularly in:

Filter Design: Creating circuits that pass specific frequencies while rejecting others (Low pass, High pass, Band pass).
Impedance Matching: Designing networks to match the source impedance to the load impedance for maximum power transfer.
Equalizers: Correcting signal distortion in communication channels.
Control Systems: Designing compensators to stabilize systems.

2. Mathematical Foundations

2.1 Hurwitz Polynomials

A polynomial $P(s)$ is a Hurwitz polynomial if all its roots (zeros) lie in the left half of the s-plane or on the $j\omega$ axis (boundary). This is a prerequisite for system stability.

Properties of Hurwitz Polynomials:

$P(s)$ is real when $s$ is real.
All coefficients must be positive.
There should be no missing terms in powers of $s$ between the highest and lowest degree (unless the polynomial is purely even or purely odd).
The continued fraction expansion of the ratio of the odd to even parts (or vice versa) yields all positive quotient terms.

Testing a Polynomial (Routh-Hurwitz Criterion):
To test $P(s) = M(s) + N(s)$ (where M is even part, N is odd part):

Perform continued fraction expansion of $\psi(s) = \frac{M(s)}{N(s)}$ or $\frac{N(s)}{M(s)}$ .
If all quotients are strictly positive, $P(s)$ is Hurwitz.

2.2 Positive Real Function (PRF)

A function $F(s)$ (usually Impedance $Z(s)$ or Admittance $Y(s)$ ) is Positive Real if it represents the driving point function of a passive network (containing R, L, C, but no independent sources).

Definition:
A function $F(s)$ is PRF if:

$F(s)$ is real for real $s$ .
$Re[F(s)] \ge 0$ for all $Re[s] \ge 0$ .

Necessary and Sufficient Conditions for PRF:

$F(s)$ is a rational function (ratio of two polynomials).
Poles and zeros of $F(s)$ cannot lie in the Right Half Plane (RHP).
Poles and zeros on the $j\omega$ axis must be simple (multiplicity of 1) and have real, positive residues.
The degrees of the numerator and denominator polynomials may differ by at most 1.
$Re[F(j\omega)] \ge 0$ for all $\omega$ (from $0$ to $\infty$ ).

3. Synthesis of One-Port Networks

Synthesis involves realizing a physical circuit from a given impedance function $Z(s)$ or admittance $Y(s)$ . We utilize two primary methods: Foster (Partial Fraction Expansion) and Cauer (Continued Fraction Expansion).

3.1 Synthesis of LC Networks (Lossless)

LC networks contain only Inductors and Capacitors. They are lossless systems.

Properties of LC Impedance $Z_{LC}(s)$ :

Poles and zeros lie strictly on the $j\omega$ axis.
Poles and zeros are interlaced (alternate).
The slope $\frac{dZ(\omega)}{d\omega}$ is always strictly positive.
$Z(0)$ and $Z(\infty)$ are either 0 or $\infty$ .
Standard form: $Z(s) = K \frac{(s^2 + \omega_1^2)(s^2 + \omega_3^2)...}{s(s^2 + \omega_2^2)...}$

A. Foster Form I (LC)

Method: Partial Fraction Expansion of impedance $Z(s)$ .
Structure: Series connection of parallel LC tanks (and potentially a lone L or C).
Expansion:
$Z(s) = K_0 s + \frac{K_\infty}{s} + \sum \frac{2K_i s}{s^2 + \omega_i^2}$
Elements:
- $K_0 s \rightarrow$ Inductor $L = K_0$
- $K_\infty / s \rightarrow$ Capacitor $C = 1/K_\infty$
- Summation terms $\rightarrow$ Parallel LC tank where $L = \frac{2K_i}{\omega_i^2}$ and $C = \frac{1}{2K_i}$ .

B. Foster Form II (LC)

Method: Partial Fraction Expansion of admittance $Y(s)$ .
Structure: Parallel connection of series LC branches.
Expansion:
$Y(s) = K_0 s + \frac{K_\infty}{s} + \sum \frac{2K_i s}{s^2 + \omega_i^2}$
Elements:
- Terms represent admittances in parallel.
- $K_0 s \rightarrow$ Capacitor $C = K_0$ .
- Summation terms $\rightarrow$ Series LC branch.

C. Cauer Form I (LC)

Method: Continued Fraction Expansion (CFE) of $Z(s)$ or $Y(s)$ by arranging polynomials in descending order of powers of $s$ (removing poles at $\infty$ ).
Structure: Ladder Network starting with Series Inductor.
Process: Divide Numerator by Denominator to get , invert remainder, divide again to get , etc.
- $Z(s) = L_1 s + \cfrac{1}{C_2 s + \cfrac{1}{L_3 s + \dots}}$

D. Cauer Form II (LC)

Method: Continued Fraction Expansion (CFE) by arranging polynomials in ascending order of powers of $s$ (removing poles at origin).
Structure: Ladder Network starting with Series Capacitor.
Process:
- $Z(s) = \frac{1}{C_1 s} + \cfrac{1}{\frac{1}{L_2 s} + \cfrac{1}{\frac{1}{C_3 s} + \dots}}$

3.2 Synthesis of RC Networks

RC networks contain Resistors and Capacitors.

Properties of RC Impedance $Z_{RC}(s)$ :

Poles and zeros lie on the negative real axis.
Poles and zeros are interlaced.
The critical frequency nearest to the origin ( $s=0$ ) must be a pole.
The critical frequency nearest to $\infty$ must be a zero.
Slope $\frac{dZ(\sigma)}{d\sigma} < 0$ .
$Z(\infty) < Z(0)$ .

A. Foster Form I (RC)

Method: Partial Fraction Expansion of $Z(s)$ .
Structure: Series connection of (Parallel R-C) blocks.
Expansion:
$Z(s) = K_\infty + \frac{K_0}{s} + \sum \frac{K_i}{s + p_i}$
Elements:
- $K_\infty \rightarrow$ Series Resistor.
- $K_0/s \rightarrow$ Series Capacitor.
- $\frac{K_i}{s+p_i} \rightarrow$ Parallel R-C tank where $R = K_i/p_i$ and $C = 1/K_i$ .

B. Foster Form II (RC)

Method: Partial Fraction Expansion of $\frac{Y(s)}{s}$ . (Crucial Step: Divide by s first).
Structure: Parallel connection of (Series R-C) branches.
Expansion:
$\frac{Y(s)}{s} = K_\infty + \frac{K_0}{s} + \sum \frac{K_i}{s + p_i}$
Then multiply by $s$ :
$Y(s) = K_\infty s + K_0 + \sum \frac{K_i s}{s + p_i}$
Elements:
- $K_\infty s \rightarrow$ Parallel Capacitor.
- $K_0 \rightarrow$ Parallel Resistor (Conductance).
- Summation terms $\rightarrow$ Series R-C branch.

C. Cauer Form I (RC)

Method: CFE of $Z(s)$ in descending order (removing poles at $\infty$ ).
Structure: Ladder network with Series R and Shunt C.
Note: Since $Z_{RC}$ generally does not have a pole at $\infty$ , the first term is a constant (Resistance), and the next term (admittance) has a pole at $\infty$ (Capacitance).

D. Cauer Form II (RC)

Method: CFE of $Z(s)$ in ascending order (removing poles at origin).
Structure: Ladder network with Series C and Shunt R.

3.3 Synthesis of RL Networks

RL networks contain Resistors and Inductors. They are the dual of RC networks.

Properties of RL Impedance $Z_{RL}(s)$ :

Poles and zeros lie on the negative real axis.
Poles and zeros are interlaced.
The critical frequency nearest to the origin ( $s=0$ ) must be a zero.
The critical frequency nearest to $\infty$ must be a pole.

A. Foster Form I (RL)

Method: Partial Fraction Expansion of $\frac{Z(s)}{s}$ . (Crucial Step: Divide by s first).
Structure: Series connection of (Parallel R-L) blocks.
Expansion:
$\frac{Z(s)}{s} = K_\infty + \frac{K_0}{s} + \sum \frac{K_i}{s + p_i}$
Multiply by $s$ :
$Z(s) = K_\infty s + K_0 + \sum \frac{K_i s}{s + p_i}$
Elements:
- $K_\infty s \rightarrow$ Series Inductor.
- $K_0 \rightarrow$ Series Resistor.
- Summation terms $\rightarrow$ Parallel R-L tank.

B. Foster Form II (RL)

Method: Partial Fraction Expansion of $Y(s)$ .
Structure: Parallel connection of (Series R-L) branches.
Expansion:
$Y(s) = K_\infty + \frac{K_0}{s} + \sum \frac{K_i}{s + p_i}$
Elements:
- $K_\infty \rightarrow$ Parallel Resistor.
- $K_0/s \rightarrow$ Parallel Inductor.
- Summation terms $\rightarrow$ Series R-L branch.

C. Cauer Form I (RL)

Method: CFE of $Z(s)$ in descending order.
Structure: Ladder network with Series L and Shunt R.

D. Cauer Form II (RL)

Method: CFE of $Z(s)$ in ascending order.
Structure: Ladder network with Series R and Shunt L.

4. Summary Table for Synthesis Forms

Feature	LC Network	RC Network	RL Network
Critical Frequencies	On $j\omega$ axis	On $-\sigma$ axis	On $-\sigma$ axis
Alternation Pattern	Pole-Zero-Pole...	Pole-Zero-Pole...	Zero-Pole-Zero...
Lowest Freq ( $s=0$ )	Pole or Zero	Pole (or finite value)	Zero (or finite value)
Highest Freq ( $s=\infty$ )	Pole or Zero	Zero (or finite value)	Pole (or finite value)
Foster I Strategy	PFE of $Z(s)$	PFE of $Z(s)$	PFE of $Z(s)/s$
Foster II Strategy	PFE of $Y(s)$	PFE of $Y(s)/s$	PFE of $Y(s)$
Cauer I Strategy	CFE Descending (Pole @ $\infty$ )	CFE Descending	CFE Descending
Cauer II Strategy	CFE Ascending (Pole @ 0)	CFE Ascending	CFE Ascending

Legend:

PFE = Partial Fraction Expansion
CFE = Continued Fraction Expansion