Chapter 14

# 14.2 Polar Plots Revisited

The frequency response of a system is described by a complex frequency function, $G(j \omega)$. Any complex function can be represented in two different ways, using polar coordinates or rectangular coordinates. In general, consider the function to be represented in polar coordinates:

 $G(j \omega)=$ $\mid G(j \omega) \mid \cdot e^{j \angle G(j\omega)}$ $M( \omega)= \mid G(j \omega) \mid$ $\Phi ( \omega) = \angle G(j \omega)$ $G(j \omega) = M( \omega) \cdot e^{j \Phi( \omega )}$ Equation 14-1

In short-hand notation:

 $G(j \omega)= \mid G(j \omega) \mid \angle G(j \omega)$ $G(j \omega)=M(\omega) \angle \Phi( \omega)$ Equation 14-2

The two functions of frequency, magnitude function $M( \omega )$, and phase function $\Phi( \omega )$ , can be computed and plotted, resulting in a familiar frequency response plot, also referred to as a Bode plot. The phase function $\Phi ( \omega )$ is usually plotted using degrees vs. radian/sec scale. The magnitude function $M( \omega )$, is usually plotted using the standard dB vs. radian/sec scale. However, for some purposes, it may be more convenient to plot $M( \omega )$ using Volt/Volt vs. radian/sec scale.

The advantage of using the magnitude-phase representation of the frequency response is that both functions can be measured experimentally. This allows an empiric identification of the system transfer function $G(s)$ based on the measured magnitude and phase plots.

The same frequency response function $G(j \omega )$ can be represented in rectangular coordinates:

 $G(j \omega) =$ $Re \{ G(j \omega) \} +jIm \{ G(j \omega) \}$ $Re( \omega)= Re \{ G(j \omega) \}$ $Im( \omega )=Im \{ G(j \omega ) \}$ $G(j \omega)=Re( \omega)+jIm( \omega)$ Equation 14-3

The two functions of frequency,$Re ( \omega )$ and  $Im( \omega )$, can be computed and plotted, but they cannot be measured experimentally. The relationship between $Re ( \omega )$ , $Im( \omega )$ functions and $M( \omega )$ ,$\Phi( \omega )$ functions, based on complex numbers algebra, is as follows:

 $G(j \omega)= M( \omega) \cdot e^{j \Phi( \omega)}=$ $Re( \omega )+$ $jIm( \omega )$ $M( \omega )=$ $\sqrt{Re( \omega )^2+Im( \omega )^2}$ $\Phi ( \omega ) = tan^-1( \frac{Im( \omega )}{Re( \omega )})$ Equation 14-4

Inversely:

 $G(j \omega )=M( \omega ) \cdot e^{j \Phi ( \omega )} = Re( \omega ) + jIm( \omega )$ $Re( \omega ) = M( \omega ) \cdot cos( \Phi ( \omega ))$ $Im( \omega ) = M( \omega ) \cdot sin( \Phi ( \omega ))$ Equation 14-5

Note that in the above equations, the magnitude function is expressed in Volt/Volt units, not in decibels. Functions $Re( \omega )$,$Im( \omega )$ can be plotted in rectangular coordinates (using Volt/Volt units on both $Re, Im$ axis) with frequency $\omega$ being a parameter along the curve, resulting in the Polar Plot.

Polar Plots cannot be directly obtained from an experiment, and have to be computed based on magnitude-phase plots. Their application is mainly in determining the system stability in frequency domain (Gain and Phase Margin concepts and Nyquist Stability Criterion).