Chapter 11 — Frequency Response and Bode Plots

Chapter Overview (click to expand)

Frequency response analysis reveals how a circuit treats signals of different frequencies, describing the gain and phase shift at every point in the spectrum through the transfer function H(jω). This chapter introduces Bode plots — logarithmic graphs of magnitude (in decibels) and phase versus frequency — as the standard engineering tool for visualizing and designing circuit frequency behavior. **Key Takeaways** 1. The transfer function H(jω) = V_out / V_in is a complex-valued function of frequency that fully characterizes a linear circuit's input-output relationship. 2. Bode magnitude plots use a decibel scale (20 log|H|) and a logarithmic frequency axis, allowing asymptotic straight-line approximations that make hand analysis practical. 3. Poles and zeros of the transfer function each contribute predictable ±20 dB/decade slopes and ±90° phase shifts to the Bode plot, making design by inspection possible.

11.1 Introduction: How Circuits Hear the World

Different musical instruments sound different even when playing the same note. A piano, a violin, and a synthesizer all playing middle C share the same fundamental frequency, but they sound completely different because of their harmonic content — the mix of frequencies above that fundamental.

Circuits behave the same way. They do not treat all frequencies equally. Some circuits amplify low frequencies and attenuate high ones (like a bass boost). Others pass high frequencies while blocking low ones (like a treble filter). Understanding how a circuit responds to different frequencies is the key to designing audio equalizers, radio receivers, noise filters, and countless other applications.

Frequency response analysis asks: "What does this circuit do to all frequencies?" The answer comes in the form of curves called Bode plots — named after Hendrik Bode, a Bell Labs engineer who developed these techniques in the 1930s for telephone network analysis.

Key insight for linear circuits: When a sinusoidal input at frequency \(f\) is applied to a linear circuit, the output is always a sinusoid at the same frequency \(f\) — only the amplitude and phase change. Frequency response tells us exactly how much the amplitude changes and how much the phase shifts at every frequency.

11.2 Transfer Function

The transfer function \(H(j\omega)\) is the ratio of the output phasor to the input phasor as a function of angular frequency \(\omega = 2\pi f\):

\[H(j\omega) = \frac{\mathbf{V}_{out}(j\omega)}{\mathbf{V}_{in}(j\omega)}\]

Because \(H(j\omega)\) is a complex number, it has both magnitude and phase:

\[H(j\omega) = |H(j\omega)|\,\angle\,\phi(\omega)\]

where:

\(|H(j\omega)|\) is the magnitude response (gain or attenuation at each frequency)
\(\phi(\omega) = \angle H(j\omega)\) is the phase response (phase shift introduced by the circuit)

How to find the transfer function: Replace each impedance with its phasor-domain form (\(Z_R = R\), \(Z_C = 1/(j\omega C)\), \(Z_L = j\omega L\)), then use voltage divider or mesh/node analysis to form \(V_{out}/V_{in}\).

Example — RC low-pass filter: For a series resistor \(R\) followed by a shunt capacitor \(C\) with output taken across the capacitor:

\[H(j\omega) = \frac{Z_C}{Z_R + Z_C} = \frac{\dfrac{1}{j\omega C}}{R + \dfrac{1}{j\omega C}} = \frac{1}{1 + j\omega RC}\]

Properties of the transfer function:

Property	Formula
Magnitude	\(\\|H\\| = \sqrt{(\mathrm{Re}\{H\})^2 + (\mathrm{Im}\{H\})^2}\)
Phase	\(\phi = \arctan\!\left(\dfrac{\mathrm{Im}\{H\}}{\mathrm{Re}\{H\}}\right)\)
DC gain	\(\\|H(0)\\|\) — evaluate at \(\omega = 0\)
High-frequency gain	\(\lim_{\omega\to\infty}\\|H(j\omega)\\|\)

11.3 Magnitude and Phase Response

Magnitude Response

The magnitude response tells us the gain (or attenuation) that the circuit applies to signals of each frequency. It is often expressed in decibels (dB):

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

The decibel scale is used because:

It compresses large dynamic ranges into manageable numbers.
Cascaded stages simply add their dB gains together.
The half-power point falls at a convenient \(-3\) dB.

Reference values to memorize:

Linear gain \(\\|H\\|\)	dB value	Interpretation
\(1\)	\(0\) dB	Unity gain — output equals input
\(0.707\)	\(-3\) dB	Half power — cutoff frequency
\(0.5\)	\(-6\) dB	Half voltage
\(0.1\)	\(-20\) dB	10× attenuation
\(0.01\)	\(-40\) dB	100× attenuation
\(10\)	\(+20\) dB	10× gain
\(100\)	\(+40\) dB	100× gain

Phase Response

The phase response \(\phi(\omega)\) tells us the phase shift that the circuit introduces at each frequency:

\[\phi(\omega) = \angle H(j\omega) = \theta_{\mathrm{out}} - \theta_{\mathrm{in}}\]

Sign conventions:

Negative phase: output lags input (common for low-pass filters at high frequency)
Positive phase: output leads input (common for high-pass filters at low frequency)
At the cutoff frequency of a first-order filter: \(\phi = \pm 45°\)

11.4 Bode Plots

A Bode plot is a pair of graphs that display the frequency response of a circuit:

Bode magnitude plot — \(|H|\) in dB on the vertical axis vs. \(\log_{10}(f)\) on the horizontal axis
Bode phase plot — \(\phi\) in degrees on the vertical axis vs. \(\log_{10}(f)\) on the horizontal axis

Why logarithmic frequency axis? Real-world circuits operate over many orders of magnitude in frequency — from DC to MHz or beyond. A log axis gives equal visual weight to each decade (10× change), making the full range readable on one plot.

Decades and Octaves

A decade is a 10× change in frequency. An octave is a 2× change in frequency.

\[1\ \mathrm{decade} \approx 3.32\ \mathrm{octaves} \qquad 1\ \mathrm{octave} \approx 0.301\ \mathrm{decades}\]

Frequency change	Decades	Octaves
\(2\times\)	\(0.301\)	\(1\)
\(10\times\)	\(1\)	\(3.32\)
\(100\times\)	\(2\)	\(6.64\)
\(1{,}000\times\)	\(3\)	\(9.97\)

Roll-off rates are quoted in dB/decade (more common in the US) or dB/octave (common in audio engineering):

\[-20\ \mathrm{dB/decade} = -6\ \mathrm{dB/octave}\]

Diagram: Filter Frequency Response

11.5 Cutoff Frequency and the Half-Power Point

The cutoff frequency \(f_c\) (also called the corner frequency or break frequency) is the frequency at which the magnitude response falls to \(1/\sqrt{2} \approx 0.707\) of its passband maximum — exactly \(-3\) dB:

\[\left|H(j\omega_c)\right| = \frac{\left|H\right|_{\max}}{\sqrt{2}} \approx 0.707\left|H\right|_{\max}\]

Why is this the "half-power" point? Power is proportional to voltage squared:

\[P \propto |V|^2 \propto |H|^2\]

When \(|H|\) drops to \(0.707\), power drops to \((0.707)^2 = 0.5\) — exactly half. Hence the cutoff frequency is also called the half-power point or \(-3\) dB point.

For the first-order RC filter, the cutoff frequency is found by setting \(|H(j\omega_c)| = 1/\sqrt{2}\):

\[f_c = \frac{1}{2\pi RC} \qquad \omega_c = \frac{1}{RC}\ \mathrm{rad/s}\]

At the cutoff frequency of a first-order filter:

\(|H| = 0.707 = -3\ \mathrm{dB}\) (magnitude drops 3 dB from passband)
\(\phi = -45°\) for a low-pass filter (or \(+45°\) for a high-pass filter)
Inductive reactance equals capacitive reactance (for RLC circuits)

Three Names, One Frequency

Cutoff frequency, corner frequency, break frequency, and half-power frequency all refer to the same point \(f_c\). In Bode plot construction, "corner" is most common because it is where the asymptotic approximation "corners" or bends.

11.6 Roll-Off Rate and Filter Order

The roll-off rate describes how quickly the magnitude response decreases beyond the cutoff frequency. It is determined by the filter order — the number of independent reactive elements (capacitors and inductors) in the filter circuit.

\[\text{Roll-off} = -20n\ \mathrm{dB/decade} = -6n\ \mathrm{dB/octave}\] where \(n\) is the filter order (number of poles).

Filter Order	Reactive Elements	Roll-off (dB/decade)	Roll-off (dB/octave)	Max Phase Shift
1st	1	\(-20\)	\(-6\)	\(-90°\)
2nd	2	\(-40\)	\(-12\)	\(-180°\)
3rd	3	\(-60\)	\(-18\)	\(-270°\)
4th	4	\(-80\)	\(-24\)	\(-360°\)

Higher filter order means:

Sharper transition from passband to stopband
Greater phase shift
More components (cost and complexity)
Potential for overshoot or ringing (underdamped designs)

Common design philosophies for higher-order filters:

Approach	Passband	Roll-off	Best Use
Butterworth	Maximally flat (no ripple)	Moderate	General purpose
Chebyshev	Equiripple	Steeper than Butterworth	Sharp cutoff needed
Bessel	Gentle roll-off	Gentle	Linear phase (pulse/data)
Elliptic	Equiripple in both bands	Steepest possible	Maximum selectivity

Diagram: Bandwidth and Selectivity

11.7 Asymptotic Approximation

Asymptotic approximation (also called the straight-line Bode approximation) replaces the smooth, curved frequency response with piecewise-linear segments. It makes Bode plots fast to sketch by hand and reveals key behavior at a glance.

Magnitude Asymptotes for a First-Order Low-Pass Filter

For \(H(j\omega) = \dfrac{1}{1 + j\omega/\omega_c}\):

Below cutoff (\(\omega \ll \omega_c\)):

\[\left|H\right| \approx 1 \quad \Rightarrow \quad 0\ \mathrm{dB}\ (\text{flat line})\]

Above cutoff (\(\omega \gg \omega_c\)):

\[\left|H\right| \approx \frac{\omega_c}{\omega} \quad \Rightarrow \quad -20\ \mathrm{dB/decade\ slope}\]

The two asymptotes meet at \(\omega = \omega_c\) (the corner frequency). The actual curve deviates from the asymptote by a maximum of \(-3\) dB at the corner and \(-1\) dB one decade away.

Phase Asymptotes for a First-Order Low-Pass Filter

The phase transitions from \(0°\) to \(-90°\) across roughly two decades centered on \(\omega_c\):

Frequency range	Asymptote
\(\omega < 0.1\,\omega_c\)	\(\phi = 0°\) (flat)
\(0.1\,\omega_c < \omega < 10\,\omega_c\)	Linear slope of \(-45°/\text{decade}\)
\(\omega > 10\,\omega_c\)	\(\phi = -90°\) (flat)

The actual phase at the corner is exactly \(-45°\), and the asymptote error peaks at about \(5.7°\) at one decade above or below the corner.

Step-by-Step Bode Plot Construction

Factor \(H(j\omega)\) into standard pole/zero form (see Section 11.8).
Identify the DC gain \(K\) and all corner frequencies \(\omega_{p1}, \omega_{p2}, \ldots, \omega_{z1}, \ldots\)
Magnitude plot:
Start at \(20\log_{10}|K|\) dB for \(\omega \to 0\).
At each pole frequency: slope decreases by \(20\) dB/decade.
At each zero frequency: slope increases by \(20\) dB/decade.
Connect segments with straight lines; round corners by \(\pm 3\) dB if needed.
Phase plot:
Start at \(0°\) (or \(\pm 180°\) if \(K < 0\)).
Each pole contributes a \(-90°\) transition spanning one decade on each side of \(\omega_p\).
Each zero contributes a \(+90°\) transition spanning one decade on each side of \(\omega_z\).

11.8 Poles and Zeros

Poles and zeros are the characteristic frequencies embedded in the transfer function. They come from factoring \(H(s)\) (the Laplace-domain transfer function, evaluated at \(s = j\omega\) for frequency response).

General factored form:

\[H(s) = K\,\frac{(s - z_1)(s - z_2)\cdots}{(s - p_1)(s - p_2)\cdots}\]

Zeros \(z_1, z_2, \ldots\) are the roots of the numerator (frequencies where \(H = 0\))
Poles \(p_1, p_2, \ldots\) are the roots of the denominator (frequencies where \(|H| \to \infty\) in theory, but real circuits limit this)
Filter order = number of poles

For passive RC circuits, poles appear at negative real values on the \(s\)-plane, and there are no right-half-plane poles (the circuit is stable).

Effect of each factor on the Bode plot:

Factor	Magnitude contribution	Phase contribution
Constant \(K\)	\(+20\log_{10}\\|K\\|\) dB (flat)	\(0°\) (or \(180°\) if \(K<0\))
Pole at origin \(1/s\)	\(-20\) dB/decade from \(\omega=0\)	\(-90°\) (constant)
Zero at origin \(s\)	\(+20\) dB/decade from \(\omega=0\)	\(+90°\) (constant)
Real pole \(1/(1+j\omega/\omega_p)\)	\(-20\) dB/decade corner at \(\omega_p\)	\(-90°\) transition at \(\omega_p\)
Real zero \((1+j\omega/\omega_z)\)	\(+20\) dB/decade corner at \(\omega_z\)	\(+90°\) transition at \(\omega_z\)

Example — RC low-pass: \(H(j\omega) = \dfrac{1}{1 + j\omega RC}\) has one pole at \(\omega_p = 1/RC\), no zeros, DC gain = 0 dB. The Bode magnitude plot is flat at 0 dB then falls at \(-20\) dB/decade after \(\omega_p\).

Example — RC high-pass: \(H(j\omega) = \dfrac{j\omega RC}{1 + j\omega RC}\) has one pole at \(\omega_p = 1/RC\) and one zero at the origin. The magnitude rises at \(+20\) dB/decade up to \(\omega_p\), then is flat.

Poles Determine Filter Order

The number of poles equals the filter order and equals the number of independent reactive elements. Count poles — not zeros — to determine the roll-off rate.

11.9 Filter Types

A filter is a circuit designed to pass signals in certain frequency ranges (the passband) while attenuating signals in other ranges (the stopband). There are four fundamental filter types:

Low-Pass Filter (LPF)

Passes frequencies below the cutoff; blocks frequencies above the cutoff.

\[H(j\omega) = \frac{1}{1 + j\omega/\omega_c} \qquad f_c = \frac{1}{2\pi RC}\ \text{(first-order RC)}\]

DC gain: \(1\) (0 dB)
Roll-off: \(-20\) dB/decade per order
Phase: \(0°\) at DC, \(-45°\) at \(f_c\), \(-90°\) at high frequency
Applications: Anti-aliasing before ADC, audio bass boost, power supply smoothing, noise reduction

High-Pass Filter (HPF)

Passes frequencies above the cutoff; blocks frequencies below the cutoff.

\[H(j\omega) = \frac{j\omega/\omega_c}{1 + j\omega/\omega_c} \qquad f_c = \frac{1}{2\pi RC}\ \text{(first-order RC)}\]

DC gain: \(0\) (\(-\infty\) dB) — blocks DC completely
Slope below cutoff: \(+20\) dB/decade per order
Phase: \(+90°\) at DC, \(+45°\) at \(f_c\), \(0°\) at high frequency
Applications: DC blocking (coupling capacitors), rumble filters, audio treble boost, subsonic filtering

Band-Pass Filter (BPF)

Passes a band of frequencies between \(f_L\) and \(f_H\); blocks frequencies outside this band.

\[f_0 = \sqrt{f_L \cdot f_H} \quad \text{(center frequency)} \qquad BW = f_H - f_L \quad \text{(bandwidth)}\]

For a series RLC band-pass filter:

\[f_0 = \frac{1}{2\pi\sqrt{LC}} \qquad BW = \frac{R}{2\pi L} \qquad Q = \frac{f_0}{BW}\]

The quality factor \(Q\) measures selectivity: high \(Q\) means a narrow, selective passband.

Applications: Radio tuning (AM/FM), audio graphic equalizer bands, signal selection in communications

Band-Reject Filter / Notch Filter

Blocks a narrow band of frequencies around a center frequency \(f_0\); passes all other frequencies. When the rejected band is extremely narrow, it is called a notch filter.

\[f_0 = \frac{1}{2\pi\sqrt{LC}} \qquad Q = \frac{f_0}{BW}\]

A high-\(Q\) band-reject filter is a notch filter.
Applications: 60 Hz hum removal from audio, interference rejection in instrumentation, acoustic feedback suppression

Summary of filter types:

Filter Type	Passband	Stopband	DC Gain	Key Equation
Low-pass	\(0\) to \(f_c\)	Above \(f_c\)	Maximum	\(f_c = 1/(2\pi RC)\)
High-pass	Above \(f_c\)	\(0\) to \(f_c\)	Zero	\(f_c = 1/(2\pi RC)\)
Band-pass	\(f_L\) to \(f_H\)	Outside band	Zero	\(f_0 = \sqrt{f_L f_H}\)
Band-reject	Outside band	\(f_L\) to \(f_H\)	Maximum	\(f_0 = 1/(2\pi\sqrt{LC})\)

Diagram: First-Order Filters

11.10 Practical Filter Circuits and Worked Examples

First-Order RC Low-Pass Filter

Circuit: Series \(R\), shunt \(C\), output across \(C\).

\[H(j\omega) = \frac{1}{1 + j\omega RC} \qquad f_c = \frac{1}{2\pi RC}\] \[\left|H(j\omega)\right| = \frac{1}{\sqrt{1 + (\omega/\omega_c)^2}} \qquad \phi(\omega) = -\arctan\!\left(\frac{\omega}{\omega_c}\right)\]

Worked Example: Design an RC low-pass filter with \(f_c = 1\) kHz using \(C = 10\) nF.

\[R = \frac{1}{2\pi f_c C} = \frac{1}{2\pi \times 1{,}000 \times 10 \times 10^{-9}} = 15{,}915\ \Omega \approx 15.9\ \mathrm{k\Omega}\]

At \(f = 10\) kHz (one decade above cutoff), the asymptotic approximation gives:

\[\left|H\right| \approx -20\ \mathrm{dB}\]

The exact value: \(|H| = 1/\sqrt{1+10^2} = 1/\sqrt{101} \approx 0.0995 = -20.04\ \mathrm{dB}\) — nearly identical to the asymptote.

First-Order RC High-Pass Filter

Circuit: Series \(C\), shunt \(R\), output across \(R\).

\[H(j\omega) = \frac{j\omega RC}{1 + j\omega RC} \qquad f_c = \frac{1}{2\pi RC}\] \[\left|H(j\omega)\right| = \frac{\omega/\omega_c}{\sqrt{1 + (\omega/\omega_c)^2}} \qquad \phi(\omega) = 90° - \arctan\!\left(\frac{\omega}{\omega_c}\right)\]

Phase transitions from \(+90°\) at DC to \(0°\) at high frequency, passing through \(+45°\) at \(f_c\).

Series RLC Band-Pass Filter

Circuit: Series \(R\)–\(L\)–\(C\), output taken across \(R\).

\[H(j\omega) = \frac{R}{R + j\left(\omega L - \dfrac{1}{\omega C}\right)}\] \[f_0 = \frac{1}{2\pi\sqrt{LC}} \qquad BW = \frac{R}{2\pi L} \qquad Q = \frac{f_0}{BW} = \frac{1}{R}\sqrt{\frac{L}{C}}\]

Worked Example: A series RLC circuit has \(L = 10\ \mu\mathrm{H}\), \(C = 100\ \mathrm{pF}\), \(R = 10\ \Omega\). Find center frequency, bandwidth, and Q.

\[f_0 = \frac{1}{2\pi\sqrt{10\times10^{-6} \times 100\times10^{-12}}} = \frac{1}{2\pi\times10^{-8}} \approx 15.9\ \mathrm{MHz}\]

\[BW = \frac{10}{2\pi \times 10\times10^{-6}} \approx 159\ \mathrm{kHz}\]

\[Q = \frac{15.9\ \mathrm{MHz}}{159\ \mathrm{kHz}} = 100\]

This is a highly selective filter — useful for AM radio receiver tuning.

Second-Order Low-Pass Filter

For a second-order low-pass with natural frequency \(\omega_0\) and damping ratio \(\zeta\):

\[H(j\omega) = \frac{\omega_0^2}{(j\omega)^2 + 2\zeta\omega_0(j\omega) + \omega_0^2}\]

\(\zeta = 1/\sqrt{2} \approx 0.707\): Butterworth (maximally flat)
\(\zeta < 0.707\): Underdamped — peak (resonance bump) in passband
\(\zeta > 0.707\): Overdamped — early roll-off begins before \(\omega_0\)
Roll-off: \(-40\) dB/decade above \(\omega_0\)
Maximum phase shift: \(-180°\)

Diagram: Second-Order Filter Response

11.11 Chapter Summary

Frequency response analysis is one of the most powerful and widely applied tools in electrical engineering. The key ideas of this chapter:

Transfer function \(H(j\omega) = V_{out}/V_{in}\) captures the complete frequency behavior of any linear circuit as a complex function of frequency.
Magnitude response \(|H|\) gives gain or attenuation at each frequency; expressed in dB as \(20\log_{10}|H|\).
Phase response \(\phi(\omega)\) gives the phase shift introduced by the circuit at each frequency.
Bode plots use logarithmic frequency axes to reveal magnitude (in dB) and phase over wide frequency ranges; they are the standard graphical tool for frequency response.
Decades and octaves measure frequency ratios on a log scale; roll-off is quoted in dB/decade or dB/octave.
Cutoff frequency \(f_c = 1/(2\pi RC)\) is the \(-3\) dB half-power point where \(|H| = 0.707\) and (for a first-order filter) phase is \(\pm 45°\).
Roll-off rate is \(-20n\) dB/decade for an \(n\)th-order filter; each pole adds \(-20\) dB/decade.
Asymptotic approximation replaces smooth curves with piecewise-linear segments for fast Bode plot sketching; maximum error is 3 dB at the corner frequency.
Poles are roots of the denominator of \(H(s)\): each adds \(-20\) dB/decade roll-off and \(-90°\) phase shift. Zeros are roots of the numerator: each adds \(+20\) dB/decade and \(+90°\) phase contribution.
Four filter types serve different signal-processing needs:
- Low-pass: passes DC to \(f_c\), blocks above
- High-pass: blocks DC to \(f_c\), passes above
- Band-pass: passes \(f_L\) to \(f_H\), blocks outside band
- Band-reject / notch: blocks \(f_L\) to \(f_H\), passes outside band

✅ Ready to test your knowledge? Go to Chapter 11 Quiz →