Transfer Function Gain Calculator Formula

Understand the math behind the transfer function gain calculator. Each variable explained with a worked example.

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Formulas Used

DC Gain (at f=0)

dc_gain = k

Magnitude at f

magnitude = k / sqrt(1 + pow(omega * tau, 2))

Magnitude (dB)

magnitude_db = 20 * log10(k / sqrt(1 + pow(omega * tau, 2)))

Phase Angle

phase_deg = -atan(omega * tau) * 180 / pi

Variables

VariableDescriptionDefault
kStatic Gain (K)10
tauTime Constant (tau)(s)0.5
freqFrequency (f)(Hz)1
omegaDerived value= 2 * pi * freqcalculated

How It Works

First-Order Transfer Function

A first-order system has the transfer function G(s) = K / (tau*s + 1). The magnitude and phase at any frequency characterize how the system attenuates and delays sinusoidal inputs.

Frequency Response

G(j*omega) = K / sqrt(1 + (omega*tau)²)

Phase = -arctan(omega*tau)

At the corner frequency omega = 1/tau, the magnitude drops by 3 dB from the DC gain and the phase is -45 degrees.

Worked Example

A system with K=10, tau=0.5 s, evaluated at f=1 Hz.

k = 10tau = 0.5freq = 1
  1. 01omega = 2*pi*1 = 6.283 rad/s
  2. 02omega*tau = 6.283 × 0.5 = 3.142
  3. 03|G| = 10 / sqrt(1 + 3.142²) = 10 / sqrt(10.87) = 10 / 3.297 = 3.033
  4. 04dB = 20*log10(3.033) = 9.64 dB
  5. 05Phase = -arctan(3.142) = -72.3°

Frequently Asked Questions

What is the DC gain?

DC gain is the output-to-input ratio when the input is constant (frequency = 0). For a first-order system G(s) = K/(tau*s+1), the DC gain equals K.

What happens at the corner frequency?

At omega = 1/tau, the magnitude is K/sqrt(2) (down 3 dB from DC gain) and the phase is -45 degrees. Above this frequency, the magnitude rolls off at -20 dB/decade.

How is this used in control design?

Understanding the frequency response helps engineers design controllers. The gain and phase at key frequencies determine stability margins and closed-loop bandwidth.

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