Transfer Function and Bode Plot Formulas
Transfer functions describe LTI systems in the Laplace domain.
Bode plots show magnitude and phase response vs frequency on logarithmic axes.
The Formulas
First-order LPF: H(s) = 1 / (1 + sτ), ωc = 1/τ
Magnitude (dB): |H(jω)|dB = 20 log|H(jω)|
Phase: ∠H(jω) = arctan(ω/ωc)
A transfer function H(s) is the ratio of the Laplace transform of a system's output to its input, with zero initial conditions. It completely characterizes a linear time-invariant (LTI) system in the frequency domain. A Bode plot is a standard way to visualize H(s): it shows magnitude (in dB) and phase (in degrees) plotted against frequency (on a log scale).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| H(s) | Transfer function in complex frequency domain | dimensionless |
| s = jω | Complex frequency variable (on imaginary axis for sinusoids) | rad/s |
| τ = RC | Time constant | seconds |
| ωc | Corner (cutoff) frequency = 1/τ | rad/s |
| fc | Cutoff frequency = ωc/(2π) | Hz |
| −3 dB point | Frequency where gain drops to 1/√2 = 0.707 of DC value | Hz |
Asymptotic Bode approximations:
- Below corner frequency (ω << ωc): gain ≈ 0 dB (flat), phase ≈ 0°
- Above corner frequency (ω >> ωc): gain rolls off at −20 dB/decade per pole
- At the corner: gain is exactly −3.01 dB, phase is exactly −45°
Example — RC Low-Pass Filter
RC low-pass filter with R = 1 kΩ and C = 1 μF. Find the transfer function, corner frequency, and Bode plot key points.
H(s) = 1/(1 + sRC) = 1/(1 + s × 10−3)
τ = RC = 10³ × 10−6 = 10−3 s = 1 ms
ωc = 1/τ = 1000 rad/s
fc = ωc/(2π) = 1000/6.283 = 159.2 Hz
At 159 Hz: |H| = −3 dB, phase = −45°
At 1590 Hz (10× fc): |H| ≈ −20 dB, phase ≈ −84°
At 15,900 Hz (100× fc): |H| ≈ −40 dB — the filter attenuates high frequencies at 20 dB/decade
When to Use It
Use transfer functions and Bode plots when:
- Designing audio filters, amplifiers, and equalizers
- Analyzing feedback control systems (servo motors, PID controllers, op-amps)
- Checking stability margins — gain margin and phase margin from Bode plots
- Characterizing sensor frequency response (accelerometers, microphones)
- Designing switching power supply compensation networks