Network Transfer Function and Frequency Response
Complex frequency representation, transfer function definitions, s-plane pole-zero dynamics, step-by-step asymptotic Bode plotting, resonance selectivity, and passive filter designs.
4.1 & 4.2 Complex Frequency & Network Transfer Functions
Concept of Complex Frequency \(s = \sigma + j\omega\)
In classical AC analysis, the steady-state sinusoidal signal has a constant amplitude: \(v(t) = V_m \cos(\omega t)\). In real dynamic networks, waveforms can grow or decay exponentially while oscillating. The generalized complex frequency \(s\) unifies both damping and oscillation into a single variable:
\[s = \sigma + j\omega\]
Where:
• \(\sigma\) = Neper frequency or damping factor [Nepers/sec] (controls exponential envelope \(e^{\sigma t}\)).
• \(\omega\) = Radian frequency [rad/sec] (controls sinusoidal oscillation \(\cos(\omega t)\)).
Types of Network Transfer Functions
A transfer function \(H(s)\) is defined as the ratio of the Laplace transform of the output response \(Y(s)\) to the Laplace transform of the input excitation \(X(s)\), with all initial conditions set to zero: \[H(s) = \frac{Y(s)}{X(s)}\]
| Classification | Symbol & Definition | Physical Meaning |
|---|---|---|
| Driving-Point Impedance | \(Z_{\text{in}}(s) = \frac{V_1(s)}{I_1(s)}\) | Input impedance looking into port 1 with all output terminations specified. |
| Driving-Point Admittance | \(Y_{\text{in}}(s) = \frac{I_1(s)}{V_1(s)}\) | Input admittance looking into port 1. |
| Voltage Transfer Function | \(G_{21}(s) = \frac{V_2(s)}{V_1(s)}\) | Voltage gain from input port 1 to output port 2. |
| Current Transfer Function | \(\alpha_{21}(s) = \frac{I_2(s)}{I_1(s)}\) | Current gain from input port 1 to output port 2. |
| Transfer Impedance | \(Z_{21}(s) = \frac{V_2(s)}{I_1(s)}\) | Ratio of output open-circuit voltage to input current. |
| Transfer Admittance | \(Y_{21}(s) = \frac{I_2(s)}{V_1(s)}\) | Ratio of output short-circuit current to input voltage. |
4.3 & 4.4 Poles, Zeros & Frequency Response from the s-Plane
Definition of Poles and Zeros
Express the rational transfer function in factored form:
\[H(s) = K \frac{(s - z_1)(s - z_2)\dots(s - z_m)}{(s - p_1)(s - p_2)\dots(s - p_n)}\]
• Zeros (\(z_k\)): Values of \(s\) for which \(H(s) = 0\). Represented by circles (\(\circ\)) on the s-plane. Zeros block transmission at that complex frequency.
• Poles (\(p_k\)): Values of \(s\) for which \(H(s) \to \infty\). Represented by crosses (\(\times\)) on the s-plane. Poles represent the natural resonance modes of the circuit.
For a linear passive network to be Bounded-Input Bounded-Output (BIBO) stable, all poles of \(H(s)\) must lie strictly in the Left Half of the s-Plane (LHP) (\(\text{Re}(p_k) < 0\)). If any pole lies in the Right Half Plane (RHP), the natural response contains \(e^{+\sigma t}\), causing unbounded exponential explosion. Simple poles on the imaginary \(j\omega\) axis give marginally stable, sustained oscillations.
4.5 Asymptotic Bode Diagrams (Magnitude & Phase)
The 5-Step Universal Procedure for Bode Plots
- Convert \(H(s)\) to Bode Standard Time-Constant Form: Factor each term as \(\left(1 + \frac{s}{\omega_c}\right)\). Identify the constant Bode gain \(K_B\).
- Identify All Critical Frequencies:
• Number of poles at origin \(s^N\) or zeros at origin \(s^M\).
• Corner frequencies for simple poles/zeros: \(\omega_c = 1/\tau\).
• Resonant corner frequencies for quadratic poles: \(\omega_n\).
Arrange all corner frequencies in ascending numerical order: \(\omega_{c1} < \omega_{c2} < \dots < \omega_{ck}\). - Construct the Magnitude Slope Table:
• For \(\omega < \omega_{c1}\): Slope is \(-20N\text{ dB/decade}\) (from poles at origin) or \(+20M\text{ dB/decade}\) (from zeros at origin).
• At each corner frequency, update the net slope:
- Zero adds \(+20\text{ dB/decade}\)
- Pole adds \(-20\text{ dB/decade}\)
- Quadratic pole adds \(-40\text{ dB/decade}\) - Calculate Decibel Magnitude at Key Points: Low-frequency asymptote passes through \(20\log_{10}|K_B|\) at \(\omega = 1\text{ rad/s}\). Use formula \(\Delta M_{\text{dB}} = \text{Slope} \times \log_{10}\left(\frac{\omega_{\text{next}}}{\omega_{\text{prev}}}\right)\) to calculate the gain at each corner point.
- Plot the Phase Angle Curve: \[\phi(\omega) = -90^\circ \times N + \sum \tan^{-1}\left(\frac{\omega}{\omega_{zi}}\right) - \sum \tan^{-1}\left(\frac{\omega}{\omega_{pi}}\right)\]
Master Worked Example: 2082 Bhadra Q6 Bode Plot
Zero: \(s = -5\).
Put into Bode form: \[T(s) = \frac{100 \times 5 \left(1 + \frac{s}{5}\right)}{s^2 \left(1 + \frac{s}{1}\right) \times 20 \left(1 + \frac{s}{20}\right)} = \frac{25 \left(1 + \frac{s}{5}\right)}{s^2 (1 + s)\left(1 + \frac{s}{20}\right)}\] Bode gain: \(K_B = 25 \implies 20\log_{10}(25) \approx 27.96\text{ dB}\).
2. \(\omega_{z1} = 5\text{ rad/s}\) (Simple zero)
3. \(\omega_{p2} = 20\text{ rad/s}\) (Simple pole)
| Frequency Interval (rad/s) | Factor Involved | Slope Change | Net Segment Slope |
|---|---|---|---|
| \(\omega < 1\) | Double pole at origin \(1/s^2\) | - | \(-40\text{ dB/decade}\) |
| \(1 \le \omega < 5\) | Pole at \(\omega = 1\) | \(-20\text{ dB/dec}\) | \(-60\text{ dB/decade}\) |
| \(5 \le \omega < 20\) | Zero at \(\omega = 5\) | \(+20\text{ dB/dec}\) | \(-40\text{ dB/decade}\) |
| \(\omega \ge 20\) | Pole at \(\omega = 20\) | \(-20\text{ dB/dec}\) | \(-60\text{ dB/decade}\) |
At \(\omega = 1\): \(\phi = -180^\circ - 45^\circ + 11.3^\circ - 2.86^\circ \approx -216.6^\circ\).
At \(\omega \to \infty\): \(\phi \to -180^\circ - 90^\circ + 90^\circ - 90^\circ = -270^\circ\).
4.6 & 4.7 Resonance, Bandwidth & Passive Filters
Series vs. Parallel Resonant Circuits
| Parameter | Series Resonant Circuit | Parallel Resonant Circuit |
|---|---|---|
| Resonant Frequency \(\omega_0\) | \(\omega_0 = \frac{1}{\sqrt{LC}}\) | \(\omega_0 = \frac{1}{\sqrt{LC}}\) |
| Impedance at Resonance | Minimum: \(Z_0 = R\) | Maximum: \(Z_0 = R\) (or \(L/CR\) for practical coil) |
| Current at Resonance | Maximum: \(I_0 = V/R\) | Minimum: \(I_0 = V/Z_0\) |
| Quality Factor \(Q\) | \(Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}}\) | \(Q = \frac{R}{\omega_0 L} = \omega_0 C R = R \sqrt{\frac{C}{L}}\) |
| Bandwidth \(BW\) | \(BW = \frac{\omega_0}{Q} = \frac{R}{L}\text{ rad/s}\) | \(BW = \frac{\omega_0}{Q} = \frac{1}{RC}\text{ rad/s}\) |
Passive Filter Configurations
| Filter Type | Typical RC / RL Configuration | Transfer Function \(H(s)\) | Cut-off Frequency \(\omega_c\) |
|---|---|---|---|
| Low-Pass (LPF) | Series \(R\), shunt \(C\) (output across \(C\)) | \(\frac{1}{1 + sRC}\) | \(\omega_c = \frac{1}{RC}\) |
| High-Pass (HPF) | Series \(C\), shunt \(R\) (output across \(R\)) | \(\frac{sRC}{1 + sRC}\) | \(\omega_c = \frac{1}{RC}\) |
| Band-Pass (BPF) | Series RLC network (output across \(R\)) | \(\frac{s(R/L)}{s^2 + s(R/L) + 1/LC}\) | Center \(\omega_0 = \frac{1}{\sqrt{LC}}\), \(BW = \frac{R}{L}\) |
| Band-Stop (Notch) | Series RLC network (output across \(L+C\)) | \(\frac{s^2 + 1/LC}{s^2 + s(R/L) + 1/LC}\) | Rejects \(\omega_0 = \frac{1}{\sqrt{LC}}\) |