Frequency Response#

Every analog circuit has a frequency response — how its gain and phase change with frequency. Even circuits that aren’t designed as filters have bandwidth limits, resonances, and roll-off characteristics that determine how they handle signals. Understanding frequency response is understanding what a circuit actually does to a signal across its entire operating range.

Gain vs. Frequency#

The magnitude response shows how much gain (or attenuation) the circuit provides at each frequency.

Key features:

  • Passband — The frequency range where gain is approximately flat (within some tolerance, like ±3 dB)
  • Cutoff frequency (f_c or f_3dB) — Where gain has dropped 3 dB from the passband level. This is the conventional boundary of “useful” bandwidth
  • Roll-off rate — How fast gain decreases outside the passband. Measured in dB/decade or dB/octave. First-order = -20 dB/decade. Second-order = -40 dB/decade. Each additional pole adds another -20 dB/decade
  • Stopband — The frequency range where the circuit provides significant attenuation
  • Gain peaking — An unintended rise in gain near the cutoff frequency, caused by underdamped poles (Q > 0.707 for second-order systems). A warning sign for potential instability

Phase vs. Frequency#

The phase response shows how much the circuit delays the signal (in terms of phase angle) at each frequency.

Key features:

  • Each pole contributes up to -90 degrees of phase shift. A first-order system can shift 0 to -90 degrees. A second-order system: 0 to -180 degrees
  • Each zero contributes up to +90 degrees
  • At the cutoff frequency of a first-order system, the phase shift is -45 degrees
  • Phase shift is cumulative through cascaded stages

Why phase matters:

  • In feedback systems, excessive phase shift at the crossover frequency (where loop gain = 1) causes oscillation. This is the stability criterion — see Stability & Oscillation
  • Phase distortion (non-constant group delay) distorts waveforms even without any amplitude distortion. Bessel filters minimize this; Chebyshev filters are worst
  • In audio, phase shifts between channels create spatial distortion

Bode Plots#

The standard visualization of frequency response: gain (in dB) and phase (in degrees) plotted against frequency on a logarithmic scale.

Reading a Bode plot:

  • Flat horizontal line = constant gain. The circuit is in its passband
  • Downward slope of -20 dB/decade = one pole dominating. Single RC roll-off
  • Downward slope of -40 dB/decade = two poles. Second-order roll-off
  • Upward slope = a zero providing gain that increases with frequency
  • A peak in the magnitude plot = resonance or underdamped poles. The height of the peak indicates Q

Sketching approximate Bode plots by hand:

For many circuits, a reasonable Bode plot can be sketched by identifying poles and zeros:

  1. At each pole frequency, the slope changes by -20 dB/decade
  2. At each zero frequency, the slope changes by +20 dB/decade
  3. Connect the segments with straight lines (asymptotic approximation)
  4. The actual curve is -3 dB from the asymptote at each break frequency

This asymptotic approximation is surprisingly useful for quick analysis and sanity-checking simulation results.

Bandwidth#

-3 dB bandwidth is the standard measure: the frequency range over which gain stays within 3 dB of the maximum. For a low-pass system, bandwidth extends from DC to f_3dB.

Gain-bandwidth product (GBW): For most amplifiers, the product of gain and bandwidth is approximately constant. An amplifier with GBW = 10 MHz can provide gain of 100 up to 100 kHz, or gain of 10 up to 1 MHz. The tradeoff is gain for bandwidth, but the product stays fixed.

Rise time and bandwidth: For a first-order system, t_rise ≈ 0.35 / BW. An amplifier with 10 MHz bandwidth has a rise time of about 35 ns. This connects frequency-domain and time-domain behavior.

Unintended Frequency Behavior#

Circuits that aren’t designed as filters still have frequency-dependent behavior:

  • Amplifier bandwidth — Every amplifier rolls off at some frequency. The open-loop gain of an op-amp drops at 20 dB/decade above its dominant pole (often a few Hz)
  • Parasitic resonances — Stray capacitance and lead inductance create unintended LC resonances. A decoupling capacitor plus its trace inductance forms a series resonant circuit that can amplify noise at the resonant frequency
  • Cable capacitance — A cable driving a high-impedance input forms an RC low-pass filter. A 100 pF/meter cable at 3 meters into a 10 kohm load has a cutoff around 50 kHz
  • Power supply impedance — The output impedance of a power supply rises with frequency (the regulation loop has finite bandwidth). Above the loop bandwidth, supply impedance may resonate with decoupling capacitors

Measuring Frequency Response#

Swept sine (network analyzer or Bode plot analyzer):

  • Apply a sine wave at each frequency, measure output amplitude and phase
  • The gold standard for accurate frequency response measurement
  • Can be done manually with a signal generator and oscilloscope, but it’s tedious

Step response (oscilloscope):

  • Apply a square wave and observe the output
  • Overshoot indicates gain peaking. Ringing indicates underdamped resonance. Slow rise time indicates limited bandwidth
  • Quick and qualitative, but doesn’t provide precise dB values at each frequency

Noise injection:

  • Inject broadband noise and compare input to output spectrum (FFT)
  • Gives the full frequency response in one measurement
  • Requires an FFT or spectrum analyzer and more setup

Tips#

  • Use the asymptotic Bode plot approximation for quick sanity checks before detailed simulation
  • Measure bandwidth with an oscilloscope or probe that has at least 3-5× the bandwidth being measured
  • For pulse circuits where waveform fidelity matters, choose Bessel response to minimize phase distortion
  • Estimate rise time from bandwidth using t_rise ≈ 0.35 / BW for first-order systems

Caveats#

  • -3 dB is not “half” — -3 dB is half power, but it’s 70.7% of voltage amplitude. -6 dB is half voltage amplitude. The distinction matters when specifying requirements
  • Gain-bandwidth product assumes single-pole roll-off — For amplifiers with more complex frequency responses (multiple poles, zeros), GBW is an approximation. Some amplifiers have gain-dependent bandwidth that doesn’t follow a simple GBW rule
  • Measuring bandwidth requires the right instrument bandwidth — The oscilloscope or probe must have significantly more bandwidth than what’s being measured (at least 3-5×). A 100 MHz scope measuring a 50 MHz signal shows a -3 dB error from the scope itself
  • Phase margin is hard to measure directly — In a closed-loop system, injecting a signal into the feedback loop without breaking it is difficult. Loop-breaking techniques (like Middlebrook’s method) exist but require careful setup
  • Frequency response changes with operating point — A transistor amplifier’s bandwidth depends on its bias current. An op-amp’s bandwidth depends on its closed-loop gain. The frequency response measured at one operating point may not apply at another

In Practice#

  • Overshoot on a step response indicates gain peaking — measure the frequency response to locate the resonance
  • Slow rise time compared to calculations suggests bandwidth is limited by the measurement equipment, not the circuit
  • Ringing on pulse waveforms indicates underdamped poles — the ringing frequency matches the resonance frequency
  • Frequency response that varies with signal amplitude suggests nonlinearity or clipping at extremes
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