Question 12.4: Prepare asymptotic magnitude and phase characteristics for t...

The numerator has a single root r=-({b_{0} }/{b_{1}}) so that its time constant is

\tau _{1} =-(1/r)=({b_{1} }/{b_{0}}) .

The denominator has three roots, or poles, p_{1}, p_{2}, and p_{3}, with p_{1} = 0. Assuming that the second-order factor in the denominator is overdamped (in contrast to the underdamped case in Example 12.3), the damping ratio \xi  =a_{1}/2\sqrt{a_{0}a_{1}}   is greater than 1.

Thus the poles associated with this second-order factor are real, and the factor may be expressed in terms of its time constants\tau _{2} and\tau _{3}:

a_{2}s^{2}+a_{1}s+a_{0} =a_{0} (\tau _{2}s+1 )(\tau _{3}s+1 ) ,

where

\tau _{2} =-\frac{1}{p_{2} }=\frac{a_{1} }{2a_{0} }\left(1+\sqrt{1-\frac{4a_{0}a_{2}  }{a^{2}_{1} } } \right)    ,

\tau _{3} =-\frac{1}{p_{3} }=\frac{a_{1} }{2a_{0} }\left(1-\sqrt{1-\frac{4a_{0}a_{2}  }{a^{2}_{1} } } \right)    .

Therefore the transfer function may be written as

T(s)=\frac{b_{0} }{a_{0}}\frac{(\tau _{1}s+1 )}{s(\tau _{2}s+1 )(\tau _{3}s+1 )}  ,   (12.53)

and the sinusoidal transfer function is

T(j\omega )=\frac{b_{0} }{a_{0}}\frac{(j\omega \tau _{1}+1 )}{j\omega (j\omega \tau _{2}+1 )(j\omega \tau _{3}+1 )}   ,   (12.54)

which can be expressed in terms of five individual transfer functions:

T(j\omega )=\frac{N_{1}(j\omega )N_{2}(j\omega ) }{D_{1}(j\omega )D_{2}(j\omega )D_{3}(j\omega )}  ,   (12.55)

where N_{1} =b_{0} /a_{0}, N_{2}=j\omega \tau _{1}+1 , D_{1}=j\omega  ,   D_{2}=j\omega \tau _{2}+1 , and   D_{3}=j\omega \tau _{3}+1  .

Now, the individual magnitudes and phase angles may be used to obtain

T(j\omega )=\frac{N_{1}(\omega )e^{j\phi _{N_{1}}} N_{2}(\omega )e^{j\phi _{N_{2}}} }{D_{1}(\omega )e^{j\phi _{D_{1}}}  D_{2}(\omega )e^{j\phi _{D_{2}}}  D_{3}(\omega )e^{j\phi _{D_{3}}}}  ,

or

T(j\omega )=\frac{N_{1} N_{2}}{D_{1}  D_{2} D_{3}}e^{j(\phi _{N_{1}}+\phi _{N_{2}}+\phi _{D_{1}}+\phi _{D_{2}}+\phi _{D_{3}} )}  , (12.56)

The magnitude of the overall transfer function is

T(\omega )=\frac{N_{1}(\omega ) N_{2}(\omega )}{D_{1}(\omega )  D_{2}(\omega ) D_{3}(\omega )} , (12.57)

where N_{1} =b_{0} /a_{0}, N_{2}=\sqrt{\omega ^{2}\tau ^{2}_{1} +1 } , D_{1}=\omega  ,   D_{2}=\sqrt{\omega ^{2}\tau ^{2}_{2} +1 }  ,and D_{3}=\sqrt{\omega ^{2}\tau ^{2}_{3} +1 } , and the phase angle of the overall transfer function is

\phi _{T}=\phi _{N_{1}}+\phi _{N_{2}}-\phi _{D_{1}}-\phi _{D_{2}}-\phi _{D_{3}}  , (12.58)

where \phi _{N_{1}}=0 , \phi _{N_{2}}=\tan ^{-1}(\omega \tau _{1} ), \phi _{D_{1}}=\pi /2 ,  \phi _{D_{2}}=\tan ^{-1}(\omega \tau _{2} ) and \phi _{D_{3}}=\tan ^{-1}(\omega \tau _{3} ) .

The magnitude and phase curves for the overall transfer function are then prepared by summation of the ordinates of the individual transfer functions, by use of the straightline asymptotic approximations developed earlier. Usually the small departures at the corner frequencies are not of interest and thus can be ignored. Also, it is very easy to implement these procedures on a digital computer and use an x–y plotter to bypass a lot of tedious calculation and plotting by hand, in which case the results will be accurate at all frequencies.

The magnitude and phase curves obtained with straight-line asymptotic approximations are presented in Fig. 12.7, where \tau _{1}\gt \tau _{2}\gt  \tau _{3}  .

The preceding examples show how a general picture of the frequency response of even complicated systems can be easily sketched with just a few computations. On the other hand, simple spreadsheet applications can be used to develop highly accurate plots with minimal effort. Also, specialized programs such as MATLAB allow the user to develop very accurate plots with single commands, as will be demonstrated in Section 12.6.