Chapter 14
14.2 Polar Plots Revisited
The frequency response of a system is described by a complex frequency function, [latex]G(j \omega)[/latex]. 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:
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[latex]G(j \omega)=[/latex] [latex]\mid G(j \omega) \mid \cdot e^{j \angle G(j\omega)}[/latex] [latex]M( \omega)= \mid G(j \omega) \mid[/latex] [latex]\Phi ( \omega) = \angle G(j \omega)[/latex] [latex]G(j \omega) = M( \omega) \cdot e^{j \Phi( \omega )}[/latex] |
Equation 14-1 |
In short-hand notation:
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[latex]G(j \omega)= \mid G(j \omega) \mid \angle G(j \omega)[/latex] [latex]G(j \omega)=M(\omega) \angle \Phi( \omega)[/latex] |
Equation 14-2 |
The two functions of frequency, magnitude function [latex]M( \omega )[/latex], and phase function [latex]\Phi( \omega )[/latex] , can be computed and plotted, resulting in a familiar frequency response plot, also referred to as a Bode plot. The phase function [latex]\Phi ( \omega )[/latex] is usually plotted using degrees vs. radian/sec scale. The magnitude function [latex]M( \omega )[/latex], is usually plotted using the standard dB vs. radian/sec scale. However, for some purposes, it may be more convenient to plot [latex]M( \omega )[/latex] 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 [latex]G(s)[/latex] based on the measured magnitude and phase plots.
The same frequency response function [latex]G(j \omega )[/latex] can be represented in rectangular coordinates:
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[latex]G(j \omega) =[/latex] [latex]Re \{ G(j \omega) \} +jIm \{ G(j \omega) \}[/latex] [latex]Re( \omega)= Re \{ G(j \omega) \}[/latex] [latex]Im( \omega )=Im \{ G(j \omega ) \}[/latex] [latex]G(j \omega)=Re( \omega)+jIm( \omega)[/latex] |
Equation 14-3 |
The two functions of frequency,[latex]Re ( \omega )[/latex] and [latex]Im( \omega )[/latex], can be computed and plotted, but they cannot be measured experimentally. The relationship between [latex]Re ( \omega )[/latex] , [latex]Im( \omega )[/latex] functions and [latex]M( \omega )[/latex] ,[latex]\Phi( \omega )[/latex] functions, based on complex numbers algebra, is as follows:
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[latex]G(j \omega)= M( \omega) \cdot e^{j \Phi( \omega)}=[/latex] [latex]Re( \omega )+[/latex] [latex]jIm( \omega )[/latex] [latex]M( \omega )=[/latex] [latex]\sqrt{Re( \omega )^2+Im( \omega )^2}[/latex] [latex]\Phi ( \omega ) = tan^-1( \frac{Im( \omega )}{Re( \omega )})[/latex] |
Equation 14-4 |
Inversely:
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[latex]G(j \omega )=M( \omega ) \cdot e^{j \Phi ( \omega )} = Re( \omega ) + jIm( \omega )[/latex] |
Equation 14-5 |
Note that in the above equations, the magnitude function is expressed in Volt/Volt units, not in decibels. Functions [latex]Re( \omega )[/latex],[latex]Im( \omega )[/latex] can be plotted in rectangular coordinates (using Volt/Volt units on both [latex]Re, Im[/latex] axis) with frequency [latex]\omega[/latex] 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).
