Ch 4. Network Transfer Function and Frequency Response

8 hours · 8 marks

This chapter links Laplace-domain network functions to steady-state sinusoidal behaviour. Exam questions typically ask definitions (network function, poles/zeros), Bode asymptotic construction for given \(H(s)\), bandwidth/Q concepts, and filter classification.

4.1 Complex Frequency

A general exponential signal may be written \(e^{st}\) where the complex frequency is

\[ s = \sigma + j\omega \]

• \(\sigma\) (nepers/s) — damping / growth rate of the envelope
• \(\omega\) (rad/s) — angular frequency of oscillation

Special cases:

\(s\)	Signal	Interpretation
\(s = 0\)	DC / constant	steady value
\(s = \sigma\) (real)	\(e^{\sigma t}\)	pure exponential (no oscillation)
\(s = j\omega\)	\(e^{j\omega t}\)	steady-state sinusoid (phasor analysis)
\(s = \sigma + j\omega\)	\(e^{\sigma t}e^{j\omega t}\)	damped / growing sinusoid

Replacing \(s\) by \(j\omega\) in a network function yields the sinusoidal steady-state frequency response.

4.2 Transfer Functions of Two-Port Networks

Network function (definition)

A network function (driving-point or transfer) is the Laplace-domain ratio of a response transform to an excitation transform, with zero initial conditions:

\[ H(s) = \frac{\text{Response}(s)}{\text{Excitation}(s)} \]

Common transfer functions of a two-port:

Name	Definition	Condition
Voltage transfer	\(G_v(s) = V_2(s)/V_1(s)\)	port 2 open or specified load
Current transfer	\(\alpha(s) = I_2(s)/I_1(s)\)	port 2 short or specified
Transfer impedance	\(Z_{21}(s) = V_2(s)/I_1(s)\)	\(I_2=0\)
Transfer admittance	\(Y_{21}(s) = I_2(s)/V_1(s)\)	\(V_2=0\)
Driving-point impedance	\(Z_{\text{in}}(s) = V_1(s)/I_1(s)\)	same port

In general a rational network function has the form

\[ H(s) = K\,\frac{(s-z_1)(s-z_2)\cdots(s-z_m)}{(s-p_1)(s-p_2)\cdots(s-p_n)} \] where \(K\) is a gain constant, \(z_i\) are zeros and \(p_k\) are poles.

4.3 Poles and Zeros

• Zeros: values of \(s\) that make \(H(s)=0\) (numerator roots). At a zero the response vanishes for that complex-frequency excitation.
• Poles: values of \(s\) that make \(H(s)\to\infty\) (denominator roots). Poles determine natural modes and stability.
• Pole-zero diagram: zeros marked ○, poles marked × in the \(s\)-plane.

Example — locate poles and zeros. Given

\[ H(s)=\frac{10(s^2+8s+15)}{s^2(s+3)(0.05s+1)} \]

Numerator: \(s^2+8s+15=(s+3)(s+5)\) ⇒ zeros at \(s=-3,-5\).

Denominator: \(s^2(s+3)(0.05s+1)=0\) ⇒ poles at \(s=0\) (double), \(s=-3\), \(s=-20\).

Note the zero at \(-3\) cancels one factor of \((s+3)\) if allowed, leaving an effective pole-zero set after cancellation (check before Bode drawing).

4.4 Magnitude and Phase Response

On the \(j\omega\) axis, \(H(j\omega)=|H(j\omega)|\,e^{j\phi(\omega)}\):

\[ |H(j\omega)| = \text{magnitude},\qquad \phi(\omega)=\angle H(j\omega)=\text{phase} \]

In decibels:

\[ |H|_{\text{dB}} = 20\log_{10}|H(j\omega)| \]

Useful identities for Bode construction:

\[ 20\log|AB|=20\log|A|+20\log|B|,\qquad 20\log\Bigl|\frac{A}{B}\Bigr|=20\log|A|-20\log|B| \] \[ \angle(AB)=\angle A+\angle B,\qquad \angle(A/B)=\angle A-\angle B \]

4.5 Bode Diagrams

A Bode diagram plots \(20\log|H|\) (dB) and \(\phi\) (degrees) vs \(\omega\) on a log frequency axis. Asymptotic Bode plots use straight-line approximations that are fast and exam-friendly.

Standard factors

Write \(H(s)\) in Bode (corner-frequency) form before plotting:

\[ H(s)=K\,\frac{\prod(1+s/\omega_{z})}{\prod(1+s/\omega_{p})\cdot\prod\bigl(1+2\zeta s/\omega_n + s^2/\omega_n^2\bigr)} \] (also include \(s^{\pm r}\) factors for poles/zeros at the origin).

Asymptotic magnitude rules

Factor	Asymptote
Constant \(K\)	\(20\log|K|\) dB horizontal
\(s\) (zero at origin)	\(+20\) dB/decade through \(0\) dB at \(\omega=1\)
\(1/s\) (pole at origin)	\(-20\) dB/decade through \(0\) dB at \(\omega=1\)
\(1+s/\omega_c\) (real zero)	\(0\) dB for \(\omega\ll\omega_c\); \(+20\) dB/dec for \(\omega\gg\omega_c\); corner at \(\omega_c\)
\(1/(1+s/\omega_c)\) (real pole)	\(0\) dB then \(-20\) dB/dec after \(\omega_c\)
Quadratic \(\omega_n,\zeta\)	\(0\) then \(\pm 40\) dB/dec after \(\omega_n\); peaking depends on \(\zeta\)

Asymptotic phase rules (real poles/zeros)

• Constant \(K>0\): \(0^\circ\); \(K<0\): \(\pm 180^\circ\)
• Pole at origin \(1/s\): \(-90^\circ\) constant; zero at origin \(s\): \(+90^\circ\)
• Real pole at \(\omega_c\): phase goes \(0^\circ\to -90^\circ\); asymptotic sketch often uses \(0^\circ\) until \(0.1\omega_c\), linear to \(-90^\circ\) at \(10\omega_c\)
• Real zero at \(\omega_c\): \(0^\circ\to +90^\circ\) over the same decade span
• Quadratic: \(0^\circ\to \pm 180^\circ\) centred at \(\omega_n\)

4.6 Bandwidth, High-Q and Low-Q Circuits

For a second-order band-pass or resonant circuit with centre (resonant) frequency \(\omega_0\) and half-power frequencies \(\omega_1,\omega_2\):

\[ \text{BW}=\omega_2-\omega_1=\frac{R}{L}\quad\text{(series RLC)},\qquad Q=\frac{\omega_0}{\text{BW}} \]

• High-Q (\(Q\gg 1\), typically \(Q>10\)): sharp resonance, narrow bandwidth, lightly damped; magnitude peak near \(\omega_0\).
• Low-Q (\(Q\) small, e.g. \(Q<0.5\)): overdamped / broad response; poles are real or widely spaced; little peaking.

For the standard quadratic factor \(s^2+2\zeta\omega_n s+\omega_n^2\), \(Q=1/(2\zeta)\). High-Q ⇔ small damping ratio \(\zeta\).

4.7 Filters: HP, LP, Band-Stop, Band-Pass

Type	Pass / stop behaviour	Typical first-order TF (RC)
Low-pass (LP)	passes low \(\omega\), attenuates high	\(\dfrac{\omega_c}{s+\omega_c}\)
High-pass (HP)	passes high \(\omega\), attenuates low	\(\dfrac{s}{s+\omega_c}\)
Band-pass (BP)	passes a band around \(\omega_0\)	\(\dfrac{(\text{BW})s}{s^2+(\text{BW})s+\omega_0^2}\)
Band-stop / notch	rejects a band around \(\omega_0\)	\(\dfrac{s^2+\omega_0^2}{s^2+(\text{BW})s+\omega_0^2}\)

Order equals degree of denominator (after proper reduction). Cascading stages raises order and steepens roll-off (\(\pm 20n\) dB/decade for order \(n\) real poles).

Solved Examples — Asymptotic Bode Plots

Example 1

Exam type

Sketch the asymptotic magnitude Bode plot of

\[ H(s)=\frac{10(s^2+8s+15)}{s^2(s+3)(0.05s+1)} \]

Example 2

Exam type

\[ T(s)=\frac{100(s+5)}{s^2(s^2+21s+20)} \]

Example 3

Exam type

\[ H(s)=\frac{100s(s+5)}{(s+1)(s^2+10s+100)} \]

Example 4

Exam type

\[ G(s)=\frac{500(s+2)}{s(s^2+15s+121)} \]

Most Probable Questions