  # Frequency Response (Bode Plots)

The figure below shows the Bode plot for a transfer function with a specified gain. The poles (x) and zeros (o) of the transfer function are plotted on the complex plane. In addition the system is multiplied by an adjustable gain (in green).

Use the controls to change the order of the numerator (number of zeros) and denominator (number of poles). Drag the poles and zeros on the complex plane to change their location. Complex poles or zeros are shown in the transfer function using the standard form s2 + 2ζ ωns + ωn2

Use the Final Value Theorem to calculate the steady-state response for a transfer function.
lim g(t) t→∞ = lim s→0 s(G(s))
G(s) = 1.0
1____________ s3 + 10 s2+ 12s + 24
poles: 2   zeros: 1

### Questions:

1. How much total phase shift is associated with each pole or zero? (answer)
Each pole contributes -90° of phase shift. Each zero contributes +90° of phase shift.
2. What is the DC gain of 1/s? (answer)
A pole at zero (an integrator) has an infinite gain at zero frequency (DC). This should be intuitive. If you integrate a constant (frequency = 0 r/s)the value continues to grow and is infinite when time → ∞
3. How does the damping ratio of a complex pair affect the shape of the gain plot? (answer)
Complex poles have a bump at their natural frequency. The height of the bump is inversely proportional to the damping ratio (ζ).
4. How does the addition of poles affect the stability the system? (answer)
Each pole add -90° of phase shift. Once the system surpasses -180° of phase shift (three poles) the system can become unstable.