Question 12.5: Prepare a polar plot for a first-order system having the sys...

The frequency-response transfer function is

T( j\omega )=\frac{10}{2 j\omega  + 1}  .    (12.60)

On the one hand, the polar plot represents graphically the relationship between the magnitude T(ω) and the phase angle \phi ω{T} (\omega ) ; on the other hand, it shows the relationship between the real part T\cos \phi _{T} and the imaginary part T\sin \phi _{T} of the transfer function T(jω).When the latter is chosen, the expressions for the real and imaginary parts of T(jω) are found to be

Re[T( j\omega )]=\frac{10 }{4\omega ^{2} + 1} ,    (12.61)

Im[T( j\omega )]=-\frac{20\omega }{4\omega ^{2} + 1}  .    (12.62)

The numerical results for computations carried out at several frequencies are given in Table 12.2, which also includes corresponding values of {T} (\omega ) and \phi _{T} (\omega ) .The polar plot prepared from Table 12.2 is shown in Fig. 12.12. It can be shown analytically that this curve is a semicircle with its center at the 5.0 point on the real axis.

Note that the sign convention for phase angle is counterclockwise positive. Thus the phase angle \phi _{T} (\omega ) for this system is always negative. Although each computed point for this polar plot corresponds to a certain frequency, the frequency ω does not appear explicitly as an independent variable (unless successive tick marks are labeled for the curve, as was done in Fig. 12.12).

For complicated transfer functions, the process for computing the values of the data points is straightforward but becomes tedious. The next section shows how MATLAB easily generates both Bode diagrams and polar plots with a few simple commands.

Polar plot diagrams display most of the same information as Bode diagrams, especially if the frequency tick marks are included, but in a more compact form. Although they are not as easy to sketch by hand as the straight-line asymptotes of the Bode diagrams, they are especially useful in determining the stability of feedback control systems, which are discussed in Chap 13.

Table 12.2. Numerical data for plot of T( j\omega ) = 10/(2 j\omega  + 1)

A few comments and hints are in order here to aid in the preparation and/or verification (especially with computer-generated data) of polar plots. First, the points for zero frequency and infinite frequency are readily determined. The point for zero frequency is obtained by use of ω = 0 in T( jω). The point for infinite frequency is always at the origin because of the inability of real physical systems to have any response to very high frequencies. Stated mathematically,

\underset{\omega \rightarrow \infty }{\lim } T( j\omega ) = 0 .    (12.63)

Second, as the frequency approaches infinity, the terminal phase angle is (π/2)(m-n), where m is the order of the numerator and n is the order of the denominator (assuming that the transfer function is a ratio of polynomials). For example, if

T(s) =\frac{b_{0} + b_{1}s + ·· ·+b_{m} s^{m} }{a_{0}+ a_{1}s + ·· ·+a_{n} s^{n} } , m ≤ n, ,

the limit of the phase angle for frequency approaching infinity is given by

\underset{\omega \rightarrow \infty }{\lim } \phi _{T} ( \omega ) = (m− n)\frac{\pi }{2} .    (12.64)

Suggestion: Use this rule to check the validity of the curve shown in Fig. 12.12.