Frequency Response OBJECTIVE - Bode and Nyquist plots for control analysis - Determination of transfer function - Gain and Phase margins - Stability in frequency response ~.~.

Magnitude and Phase Angle Transfer function is R(s)R(s) Y(s)Y(s) By replacing we can find its magnitude and its phase angle Higher order transfer function Can be presented in magnitude-phase form as

A system with transfer function of G(s) is subjected to a sinusoidal input. Determine the time response of the system. Solution The input in phasor form (manitude-phase form) can be presented as As the transfer function is Hence the output is by replacing In time domain which give the time response of the system, the output is Example

First order Frequency response Its magnitude and phase angle Transfer function by replacing

Second order Transfer function Frequency response by replacing Its magnitude and phase angle

Higher Order Cascade form Frequency response by replacing Or in phasor form Example: Find the frequency response of the following transfer function

Where and Respective magnitude and phase angle and

Bode Plot Consider higher order system Logarithmic form In dB. Phase angle

Bode Plot for constant gain Log magnitude in dB Its phase angle >> bode([200],[1]);grid Let K=200

Bode Plot of pole on the origin Log magnitude in dB

Slope of –20dB/decade or –6dB/octave. Its phase angle >>bode([1],[1 0]);grid

Bode Plot for Real Pole Frequency response Logarithmic magnitude in dB Phase angle

For low frequency, and For high frequency, and Can also be defined as a tenth of the corner frequency i.e. The corner frequency is Can also be defined as a tenth of the corner frequency i.e.

A straight line approximation for a first order system of transfer function

The actual Bode plot can be obtained by using the exact equation for the log magnitude and phase angle. There are small differences between the actual and approximate as shown by the following table Using MATLAB we can display the Bode plot of an open-loop transfer function of And the MATLAB command used is given by » nr=[1]; » dr=[1 1]; » sys=tf(nr,dr) » bode(sys)

Actual and approximate value of log magnitude for an open-loop transfer function of for a frequency range between rad/s

Actual and approximate value of phase angle for an open-loop transfer function of for a frequency range between rad/s

Bode Plot for Complex poles

For low frequency, and For high frequency and Is the corner frequency We define the low frequency as a tenth of the corner frequency i.e. While the high frequency as ten time of the corner frequency i.e.

Based on straight line approximation the Bode plots for LM and phase angle are shown

We can observe the Bode plots in MATLAB by considering In MATLAB >>zeta=0.25;wn=1;num=[1];den1=[1 2*zeta*wn wn*wn]; sys1=tf(num,den1); >>zeta=0.5;wn=1;num=[1];den2=[1 2*zeta*wn wn*wn]; sys2=tf(num,den2); >>zeta=0.75;wn=1;num=[1];den3=[1 2*zeta*wn wn*wn];sys3=tf(num,den3); >> bode(sys1,sys2,sys3);grid for and

Example Obtain the bode plot of the following block diagram if + _

500 1/s s+4 By replacing with s=j  and making the open-loop transform function into corner freqeuncy form

The intial LM and phse at strat frequency is given by

For the magnitude polot of the Bode plot, we build a table to tshow the contribution of gradient/slope by the zero and poles at respective corner frequencies. The pole at origin will provide a slope of -20db/decade for all range of frequency

- For the phase plot, we need to know the change of gradient of the phase at low and high frequencies of respective corner frequency - The pole at the origin does not contribute to gradient of the phase angle as its phase angle is a constant 90 o

The LM vs freq. plot can be determined the LM value at the corner frequency, this can be obtained by simple trigonometry (dB/dec) Using the trigonometry’s formula, we tabulate the values at the initial, corner frequencies and final value.

As for the LM slope, we apply a trignometry’s formula to obtain the phase angle slope at the low and high frequency for each corner frequency as in table below (deg/decade)

Bode plot for the magnitude and phase using straight line approximations

We can obtain the actual Bode plot using MATLAB as >>bode([ ],[ ],{10^-1,10^4});

Determination of transfer function Constant gain Example: If dB, determine the ransfer function. dB  (rad.s-1) - dB

Pole/zero at origin Example: If rad.s -1, dB and slope of dB/decade. dB  (rad.s-1) dB/dekad - We know rad.s -1, dB

Real pole/zero Example: dB 45 dB/dekad  (rad.s-1)

Pair of complex poles Example: Assume damping ratio of 0.5. dB dB/dekad  (rad.s-1) dB/dekad

Nyquist Plot and phase angle Nyquist plot is a plot of magnitude, for frequency on s-plane.  (rad.s-1) 0  However we can obtain the sketch of the plot by obtaining the following vectors: (i) at (ii) at (iii), crossing on the real axis at, crossing on the imaginary axis (iv) at

First order Frequency response Magnitude dan Phase angle

(i) At,,, and (ii) At, dan. i.e. or (iii) No crossing in the real axis as, (iv) No crossing in the imaginary axis as, is a circle.

Example: Nyquist plot of Frequency response and At and At, and,. >> nyquist([5],[.25 1])

Second Order Frequency response Rearrange

Magnitude Phase angle and (i) at, (ii) At, and. (iii) No crossing on the real axis. (iv) Crossing of the imaginary axis when,. and

Example: Frequency response Magnitude Phase angle

(i) At, (ii) At, and (iii) Real axis crossing at,,, dan rad.s -1 or Magnitude (iv) Imaginary crossing at =0.6 rad.s -1, Magnitude »dr1=[2 1]; » dr2=[1 1 1]; » dr=conv(dr1,dr2); » nr=1; » nyquist(nr,dr)

Nyquist Path For Nyquist path  jj r s-plane Contour of which will give the closed loop poles for till by replacing to become

However Hence on F(j  )-plane, we represent GH(j  )-plane to plot jj  GH(j  )-plane -1+j0.

Nyquist Stability Criterion The criteria states that the number of closed poles of a system is equal to the number of open-loop, G(s)H(s), zeros plus the number of encirclements in the clockwise direction by the Nyquist plot of the open-loop, G(s)H(s). Or, it can be represented in mathematic form as where Z = Number of zeros on the right-half plane of j  -axis, where. N = Number of clockwise encirclement of P = Number of open-loop poles on the right-half plane of j  -axis.

Consider, applying the Nyquist stability criteria Z=N+P=1+0=1 Hence the system has 1 closed-loop pole on the right of the imaginary axis

Is the measure of instability from on the-plane Phase crossover frequency, Frequency where the phase angle of is –180 o Gain crossover frequency, Frequency where the magnitude of Gain margin,

Gain margin margin in Nyquist Plot GH(j  )-plane jj  -1+j0 1/GM

Phase margin in Nyquist Plot jj  -1+j0 GH(j  )-plane Unit circle will give phase angle of, hence the phase margin Phase margin,

Negative gain and phase margins in Nyquist Plot -1+j0  jj GH(j  )-plane As, the gain margin As, the phase margin

For Example:, shows the Nyquist plot and its respective gain and phase margin + - Open loop transfer function Frequency response

Magnitude Phase angle (i) At, and (ii) At, and

Real crossing, when. (iii) then Imaginary crossing, when (iv) then » nr=40; » dr1=[ 1 3]; » dr2=[1 4 7]; » dr=conv(dr1,dr2); » nyquist(nr,dr)

Positive gain and Phase margins in Bode plot LM (dB)  (rad.s-1) 0 GM  0 o PM -180 o GM – gain margin PM – phase margin - Gain crossover frequency - Phase crossover frequency

Negative gain and Phase margins in Bode plot LM (dB) GM  (rad.s-1 ) PM 0  -180 o 0 o Note that, negative gain or phase margin means that the system is not stable

Example: If dan K=96 determine gain and phase crossover frequencies. Consequently what is the system gain and phase margin. + -

Frequency response and

kutub orijin kutub hakiki (rad.s -1 ) (dB/dekad) -20 (dB/dekad) 0-20 (dB/dekad) Total slope dB/decade Table of LM slope for the 3 poles

kutub hakiki  (rad.s -1 ) kecerunan (darjah/dekad) kecerunan (darjah/dekad) jumlah kecerunan (darjah/dekad) Table for phase angle slope from the two poles, notes that the pole at origin does not contribute to the slope as the angle is constant -90 o

(rad.s -1 ) Total slope (dB/dec) LM (dB) To obtain the LM vs freq. plot, we determine the LM value at the corner frequency, this can be obtained by simple trignometry (dB/dekad) Using the trignometry’s formula,

As for the LM slope, we apply a trignometry’s formula to obtain the phase angle slope at the low and high frequency for each corner frequency as in table below  (rad.s -1 ) Total slope (deg/dec) (deg) (deg/decade)

>> bode([96],[ ])

>>[GM,PM,Wg,Wp] = margin(sys)

>> num=[96];den=[ ]; >> sys=tf(num,den); >> [GM,PM,Wg,Wp] = margin(sys) Gm = Pm = Wg = Wp = >> Gm_dB = 20*log10(Gm) Gm_dB =