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Magnitude at f
3.0331
Magnitude at f vs Static Gain (K)
Formula
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)
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.
Esempio Risolto
A system with K=10, tau=0.5 s, evaluated at f=1 Hz.
- 01omega = 2*pi*1 = 6.283 rad/s
- 02omega*tau = 6.283 × 0.5 = 3.142
- 03|G| = 10 / sqrt(1 + 3.142²) = 10 / sqrt(10.87) = 10 / 3.297 = 3.033
- 04dB = 20*log10(3.033) = 9.64 dB
- 05Phase = -arctan(3.142) = -72.3°
Domande Frequenti
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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