Communication Network and Transmission Lines (CNTL) (EC-503 (A)) - Unit 4 Short Notes

UNIT 4: Advanced Network Synthesis, Filter Design & Two-Port Networks

I. Advanced Two-Port Network Analysis

Symmetrical vs. Asymmetrical Networks

• Symmetrical Network:

  • Image impedances equal: $$\displaystyle Z_{i1} = Z_{i2} $$

  • Parameters: $$\displaystyle A = D $$, $$\displaystyle AD - BC = 1 $$

• Asymmetrical Network:

  • Image impedances differ: $$\displaystyle Z_{i1} \neq Z_{i2} $$

• Image Impedance: Impedance that, when terminating a port, makes the network appear infinite at the other port.

[!TIP] For asymmetrical T-networks, use the two equations above; substitute one into the other to solve.

Lattice and Bridged-T Networks

• Lattice Network:

  • Structure: Crossed series impedances $$\displaystyle Z_a $$, $$\displaystyle Z_b $$.

  • Symmetrical, $$\displaystyle Z_i = \sqrt{Z_a Z_b} $$.

  • Applications: Equalizers, filters.

• Bridged-T Network:

  • T-network with a bridge resistor across series arms.

  • Used for attenuation and impedance matching.

T and π Equivalent Circuits

• T → π (T: $$\displaystyle Z_1 $$, $$\displaystyle Z_2 $$ series, $$\displaystyle Z_3 $$ shunt):

$$ \begin{aligned} Z_{\pi1} &= \frac{Z_1 Z_3}{Z_1+Z_2+Z_3} \\ Z_{\pi2} &= \frac{Z_2 Z_3}{Z_1+Z_2+Z_3} \\ Z_{\pi3} &= \frac{Z_1 Z_2 + Z_2 Z_3 + Z_3 Z_1}{Z_3} \end{aligned} $$

• π → T (π: $$\displaystyle Z_{\pi1} $$, $$\displaystyle Z_{\pi2} $$ shunt, $$\displaystyle Z_{\pi3} $$ series):

$$ \begin{aligned} Z_1 &= \frac{Z_{\pi1} Z_{\pi3}}{Z_{\pi1} + Z_{\pi2} + Z_{\pi3}} \\ Z_2 &= \frac{Z_{\pi2} Z_{\pi3}}{Z_{\pi1} + Z_{\pi2} + Z_{\pi3}} \\ Z_3 &= \frac{Z_{\pi1} Z_{\pi2}}{Z_{\pi3}} \end{aligned} $$

Characteristic Impedance of Symmetric Networks

• T-section: $$\displaystyle Z_0 = \sqrt{Z^2 + 2 Z Z'} $$

  ($Z$ = each series arm, $Z'$ = shunt)

• π-section: $$\displaystyle Z_0 = \sqrt{\dfrac{Z_s^2}{4} + Z_s Z_p} $$

  ($$\displaystyle Z_s $$ = series impedance, $$\displaystyle Z_p $$ = each shunt impedance)

II. Filter Design & Approximation Techniques

Constant-K Filters

• Design: Half-section with $$\displaystyle Z Z' = K^2 $$, cut-off $$\displaystyle \omega_c = 1/\sqrt{LC} $$.

• Limitations: Stopband attenuation only 6 dB/octave; poor roll-off.

m-Derived Filters

• T-section design (low-pass example):

  • Series: $$\displaystyle Z_s = m Z $$

  • Shunt: $$\displaystyle Z_p = \dfrac{Z'}{1-m^2} $$

  • Infinite attenuation frequency: $$\displaystyle \omega_{in} = \dfrac{\omega_c}{\sqrt{1-m^2}} $$ ($$\displaystyle m < 1 $$)

• π-section: Similar transformation.

• Role: Sharper cutoff than constant-K; used in composite filters.

Chebyshev Approximation

• Passband: Equiripple (equal amplitude variations).

• Stopband: Monotonic.

• Roll-off: Steeper than Butterworth.

• Design: Use Chebyshev polynomials; apply frequency transformation for HPF/BPF/BSF.

Composite Filters

• Combine constant-K and m-derived sections.

• Use different $m$ values to balance stopband attenuation and cutoff sharpness.

Frequency Transformation (from LPF prototype)

• LPF → HPF: $$\displaystyle s \rightarrow \dfrac{\omega_c}{s} $$

• LPF → BPF: $$\displaystyle s \rightarrow \dfrac{s^2 + \omega_0^2}{B s} $$

  ($$\displaystyle \omega_0 $$ = center freq, $B$ = bandwidth)

• LPF → BSF: $$\displaystyle s \rightarrow \dfrac{B s}{s^2 + \omega_0^2} $$

Reactance Curves

• LPF: $$\displaystyle X_L = \omega L \uparrow $$, $$\displaystyle X_C = 1/(\omega C) \downarrow $$; intersect at $$\displaystyle \omega_c $$.

• HPF: $$\displaystyle X_L \downarrow $$, $$\displaystyle X_C \uparrow $$; intersect at $$\displaystyle \omega_c $$.

[!TIP] Reactance curves indicate passband (where $$\displaystyle |X_L| \neq |X_C| $$) and stopband (where $$\displaystyle |X_L| = |X_C| $$).

III. Network Synthesis Methods

Fundamentals

• Positive Real (PR) Function: $\text{Re}[F(s)] \geq 0$ for $$\displaystyle \text{Re}(s) > 0 $$.

  • Conditions:

    1. No poles in $$\displaystyle \text{Re}(s) > 0 $$.

    2. Poles on $j\omega$ axis are simple with positive residues.

    3. At $$\displaystyle s=\infty $$, if pole exists, residue positive.

• Hurwitz Polynomial: All roots in $\text{Re}(s) \leq 0$.

  Check: coefficients positive, all Hurwitz determinants positive.

Foster Synthesis

• Foster I (for $Z(s)$ PR, no pole at $\infty$):

$$ Z(s) = R_0 + sL_\infty + \sum_{k=1}^m \frac{R_k}{s + a_k} + \sum_{k=1}^n \frac{2 R_k' s}{s^2 + \omega_k^2} $$

Realization: Series connection of $R$, $L$, and parallel $RC$ circuits.

• Foster II (for $Y(s)$ PR):

$$ Y(s) = G_0 + sC_\infty + \sum \frac{G_k}{s + a_k} + \sum \frac{2 G_k' s}{s^2 + \omega_k^2} $$

Realization: Parallel connection of $G$, $C$, and series $RL$ circuits.

Cauer Synthesis

• Cauer I (ladder):

  Start with $Z(s)$ → remove pole at $\infty$ (subtract $$\displaystyle sL_\infty $$) → take reciprocal → remove pole at $\infty$ (subtract $$\displaystyle G_0 $$) → repeat.

  Yields: series $L$, shunt $C$.

• Cauer II:

  Start with $Y(s)$ → remove pole at $\infty$ (subtract $$\displaystyle sC_\infty $$) → reciprocal → etc.

  Yields: shunt $C$, series $L$.

Brune's Method

• For functions with complex poles.

• Remove a pair of complex conjugate poles using a Brune section (parallel $RLC$).

• Brune coefficient $k$ determines component values.

• Continue with Foster/Cauer for remaining PR function.

Bott-Duffin Method

• Synthesizes minimum positive real functions.

• Principle: Remove one pole at a time using a specific network configuration.

Realization of Network Functions

1. Given $F(s)$, ensure it is PR/Hurwitz.

2. Expand in partial fractions.

3. Implement using Foster or Cauer forms.

   • Example: $$\displaystyle F(s) = \dfrac{(s+2)(s+6)}{(s+3)(s+9)} $$ → partial fractions → realize.

IV. Attenuators

Types and Design

• Symmetrical: Input impedance = output impedance = $$\displaystyle Z_0 $$.

• Asymmetrical: Input impedance ≠ output impedance.

• T-Attenuator (symmetrical):

$$ \begin{aligned} R_1 &= R_2 = Z_0 \frac{k-1}{k+1} \\ R_3 &= \frac{2 Z_0 k}{k^2-1} \end{aligned} $$

where $$\displaystyle k = 10^{A/20} $$, $A$ = attenuation in dB.

• π-Attenuator (symmetrical):

$$ \begin{aligned} R_1 &= R_2 = Z_0 \frac{k+1}{k-1} \\ R_3 &= \frac{2 Z_0 k}{k^2-1} \end{aligned} $$

• Bridged-T Attenuator:

  T-attenuator with a bridge resistor across the series arms.

  Used for impedance matching networks.

[!TIP] For asymmetrical attenuators, design equations depend on given input/output impedances and loss.

V. Advanced Transmission Line Topics

Microstrip Lines

• Structure: Dielectric substrate ($$\displaystyle \varepsilon_r $$), ground plane, conducting strip (width $W$, thickness $t$).

• Effective dielectric constant ($W/h \geq 1$):

$$ \varepsilon_{\text{eff}} = \frac{\varepsilon_r+1}{2} + \frac{\varepsilon_r-1}{2} \frac{1}{\sqrt{1+12h/W}} $$

• Characteristic impedance ($W/h \geq 1$):

$$ Z_0 = \frac{120\pi}{\sqrt{\varepsilon_{\text{eff}}} \left[ \frac{W}{h} + 1.393 + 0.667 \ln\left(\frac{W}{h}+1.444\right) \right]} $$

• Losses:

  • Conductor loss: Increases with $\sqrt{f}$ due to skin effect.

  • Dielectric loss: $G \propto f \tan\delta$.

High-Frequency Line Parameters

• Skin effect: $R \propto \sqrt{f}$.

• Dielectric losses: $G \propto f \tan\delta$.

• At high $f$, both $R$ and $G$ become significant.

Equalizers

• Compensate for frequency-dependent attenuation on lines.

• Full Series Equalizer: Series inductor $L$ and shunt capacitor $C$.

  Design for flat attenuation over band $$\displaystyle [f_1, f_2] $$:

$$ \begin{aligned} L &= \frac{Z_0}{\pi (f_2 - f_1)} \\ C &= \frac{\pi (f_2 - f_1)}{Z_0 \cdot (2\pi f_c)^2} \quad \text{(where $$\displaystyle f_c $$ is center frequency)} \end{aligned} $$

Exam Focus & Quick Reference

[!TIP] Always check realizability: PR function for synthesis, Hurwitz for denominator of stable systems. For filter design, remember $$\displaystyle m < 1 $$ for m-derived LPF to have $$\displaystyle \omega_{in} > \omega_c $$.