Frequency Response Plots - Polar Plot

Example

Tip: Use magnitude and phase

Most Famous Reponses

Nyquist = continue polar in the same direction

Integrator

• Polar: negative y axis from $-\mathrm{\infty }$ to $0$
• Nyquist: y axis from $-\mathrm{\infty }$ to $+\mathrm{\infty }$

Differentiator

• Polar: positive y axis from $0$ to $+\mathrm{\infty }$
• Nyquist: y axis from $-\mathrm{\infty }$ to $+\mathrm{\infty }$

First Order

• Polar: downward half circle from (1,0) to (0,0) tangent to y-axis
• Nyquist: full circle from (1,0) to (0,0) tangent to y-axis then back to (1,0)

Reciprocal of First Order

• Polar: from (1,0) to (1, $\mathrm{\infty }$)
• Nyquist: from (1, $-\mathrm{\infty }$) to (1, $\mathrm{\infty }$)

Second Order

• Polar: overshooting downward half circle from (1,0) to (0,0) tangent to x-axis
• Nyquist: overshooting upward half circle that starts from (0,0) tangent to x-axis, passes through (0,0) parallel to y-axis, continuous as an overshooting downward half circle to (0,0) tangent to x-axis
• Small $\zeta$ = Large amplitude
• Largest amplitude at ${\omega }_r$

Reciprocal of Second Order

• Polar: from (1,0) to top left exponentially
• Nyquist: see the graph easier :>
• Small $\zeta$ = Large amplitude
• Largest amplitude at ${\omega }_r$

Time Delay

Golden ratio!!