FREQUENCY RESPONSES Signals and System Analysis Transfer function G(s) s =jω Frequency response G(jω) Frequency (strictly angular frequency) ω Magnitude |G(jω)| Phase ∠G(jω) Signals and System Analysis Significance of the frequency response It indicates how much the system responds to a sinusoidal input at different frequencies Frequency response G(jω) Mmagnitude |G(jω)| Phase ∠G(jω) Signals and System Analysis There are several ways of plotting frequency responses but the two most common are: Direct plotting |G|-ω ∠G-ω Bode diagrams |G| - ω in log scale ∠G- ω in log scale Nyquist (or polar) plot Real(G) -Imag(G) Signals and System Analysis Any transfer function can be plotted 1. using the Control Kit in MATLAB. or 2. write a program using MATLAB codes Signals and System Analysis The Bode diagram consists of two graphs: 1. |G(jω)| against ω on a log scale The gain in dB, that is 20log10 |G(jω)|, is plotted on a linear scale 2. ∠G(jω) against ω on a log scale The Nyquist plot is a locus with ω as a parameter showing |G(jω)| and ∠G(jω as a curve as ω varies. Signals and System Analysis Bode plots of the following transfer functions K – a gain 1/sT – an integrator 1/(1+sT) – a time constant or first order system 2 2 2 ω0 /(s +2ζsω0+ω0 ) – a complex pole pair or second order system. Signals and System Analysis K – a gain G(s)=K G(jω)=K G( jω) = K 0 ∠G( jω) = tan −1 = 0 K Signals and System Analysis - K – a negative gain G(s)=-K G(jω)=-K G( jω) = K 0 ∠G( jω) = tan −1 =180o − K Signals and System Analysis Plot Frequency Response for K gain 2 1.5 1 |G| 0.5 0 0 10 20 30 40 50 60 70 80 90 100 1 K=1; 0.5 omega=0:0.1:100; 0 G(1:1001)=K; phase -0.5 G_mag=abs(G); G_phase=atan(imag(G)./ -1 0 10 20 30 40 50 60 70 80 90 100 real(G)); omega subplot(2,1,1); plot(omega, G_mag); ylabel('|G|'); title('Plot Frequency Response for K gain'); subplot(2,1,2); plot(omega, G_phase); ylabel('phase'); xlabel('omega'); Signals and System Analysis Plot Frequency Response for K gain 1 0.5 0 -0.5 20*log10(|G|) (db) 20*log10(|G|) -1 -1 0 1 2 10 10 10 10 1 0.5 0 phase K=1; -0.5 omega=0:0.1:100; G(1:1001)=K; -1 -1 0 1 2 G_mag=abs(G); 10 10 10 10 G_phase=atan(imag(G)./ real(G)); omega subplot(2,1,1); semilogx(omega, 20*log10(G_mag)); ylabel('20*log10(|G|) (db)'); grid; title('Plot Frequency Response for K gain'); subplot(2,1,2); semilogx(omega, G_phase); ylabel('phase'); grid; xlabel('omega'); Signals and System Analysis 1/sT – an integrator G(jω) = 1/jωT |G(jω)| = 1/ωT ∠G(jω = –90º. In dB the gain 20log10|G(jω)| = 20log10(1/ωT) On a graph with frequency on a log scale this will be a straight line with a negative slope. Signals and System Analysis Plot Frequency Response for integrator 10 5 |G| 0 0 10 20 30 40 50 60 70 80 90 100 2 0 omega=0.1:0.1:100; phase G=1 ./(j .*omega); -2 G_mag=abs(G); G_phase=atan(imag(G)./0 10 20 30 40 50 60 70 80 90 100 (real(G)+0.0000001)); omega subplot(2,1,1); plot(omega, G_mag); ylabel('|G|'); title('Plot Frequency Response for integrator'); subplot(2,1,2); plot(omega, G_phase); ylabel('phase'); axis([0 100 -pi pi]); xlabel('omega'); Signals and System Analysis Plot Frequency Response for integrator 20 0 -20 20*log10(|G|) (db) 20*log10(|G|) -40 -1 0 1 2 T=1; 10 10 10 10 omega=0.1:0.1:100; G=1 ./(j .*omega); 2 G_mag=abs(G); G_phase=atan(imag(G)./ 0 (real(G)+0.0000001)); subplot(2,1,1); phase semilogx(omega, -2 20*log10(G_mag)); -1 0 1 2 ylabel('20*log10(|G|) 10 10 10 10 (db)'); omega grid; title('Plot Frequency Response for integrator'); Slope is –6dB/octave subplot(2,1,2); –20dB/decade semilogx(omega, G_phase); axis([0 100 -pi pi]); ∠G(jω = –90º = -π/2 =-1.57 rad ylabel('phase'); grid; xlabel('omega'); Signals and System Analysis An octave is a frequency change of 2. A decade is a frequency change of 10. Geometric mean of two frequencies ωa and ωb is the ωaωb . Signals and System Analysis 1/(1+sT) – a time constant (first order system) G(jω) = 1/(1+jωT) = 1 for ω small = 1/jωT an integrator for ω large. Signals and System Analysis Plot Frequency Response for 1st-order system 1 0.5 |G| 0 0 10 20 30 40 50 60 70 80 90 100 0 T=1; -1 omega=0.1:0.1:100; G=1 ./(1+T*j .*omega); phase -2 G_mag=abs(G); G_phase=atan(imag(G)./ (real(G))); -3 subplot(2,1,1); 0 10 20 30 40 50 60 70 80 90 100 plot(omega, G_mag); omega ylabel('|G|'); title('Plot Frequency Response for 1st-order system'); subplot(2,1,2); plot(omega, G_phase); ylabel('phase'); axis([0 100 -pi 0]); xlabel('omega'); Signals and System Analysis Plot Frequency Response for 1st order system 0 -10 T=1; -20 omega=0.1:0.1:100; G=1 ./(1+T*j .*omega); -30 20*log10(|G|) (db) 20*log10(|G|) G_mag=abs(G); -40 -1 0 1 2 G_phase=atan(imag(G)./ 10 10 10 10 real(G)); subplot(2,1,1); 0 semilogx(omega, 20*log10(G_mag)); -1 ylabel('20*log10(|G|) phase -2 (db)'); grid; -3 title('Plot Frequency -1 0 1 2 Response for 1st order 10 10 10 10 omega system'); subplot(2,1,2); semilogx(omega, 0dB line to ωT = 1, G_phase); axis([0 100 -pi 0]); `Integrator’ line has slope –6dB/octave ylabel('phase'); The phase shift from 0 to –90º grid; xlabel('omega'); –45º at ωT = 1. Signals and System Analysis breakpoint ωT 0.1 0.125 0.25 0.5 1.0 2.0 4.0 8.0 10 dB -0.05 -0.1 -0.3 -1 -3 -7 -12.3 -18.1 -20.05 Approx 0 0 0 0 0 -6 -12 -18 -20.0 dB ϕ -6° -7° -14° -26° -45° -64° -76° -83° -84° Table for 1/(1+sT) Signals and System Analysis In DB scale 20log10 (|G(jω)|) • If |G(jω)|=1 20log10 (|G(jω)|) =0 (db) • If 2 × 20log10(2|G(jω)|) =20log10 (2) +20log10 (|G(jω)|)=6+20log10 (|G(jω)|) (db) • If 10 × 20log10 (10|G(jω)|) =20log10 (10) +20log10 (|G(jω)|)=20+20log10 (|G(jω)|)(db) Signals and System Analysis 2 2 2 ω0 /(s +2ζsω0+ω0 ) – a complex pole pair or second order system. At low frequencies s->0 the gain of this transfer function is unity (0dB). 2 2 At high frequencies G(jω)> -ω0 /ω ; this is two integrators with 0dB gain at ω = ω0. The asymptotic approximation is therefore two straight lines, 0dB for ω < ω0 and –12dB/oct for ω > ω0. The shape of both the gain and phase curves near to ω0 change significantly with the value of ζ. Signals and System Analysis Plot Frequency Response for 2st-order system 6 4 |G| 2 omega0=1; 0 zeta=0.1; 0 1 2 3 4 5 6 7 8 9 10 omega=0.1:0.1:10; G=omega0^2 ./(- omega 0 .^2 +j* 2*zeta*omega0 .*omega +omega0^2); -1 G_mag=abs(G); G_phase=atan2(imag(G), phase -2 (real(G+0.000001))); subplot(2,1,1); plot(omega, G_mag); -3 ylabel('|G|'); 0 1 2 3 4 5 6 7 8 9 10 title('Plot Frequency omega Response for 2st-order system'); subplot(2,1,2); plot(omega, G_phase); ylabel('phase'); axis([0 10 -pi 0]); xlabel('omega'); Signals and System Analysis Plot Frequency Response for 2st order system 20 0 -20 omega0=1; 20*log10(|G|) (db) 20*log10(|G|) zeta=0.1; -40 -1 0 1 omega=0.1:0.1:10; 10 10 10 G=omega0^2 ./(- omega .^2 +j* 2*zeta*omega0 .*omega 0 +omega0^2); G_mag=abs(G); -1 G_phase=atan2(imag(G), (real(G+0.000001))); subplot(2,1,1); phase -2 semilogx(omega, 20*log10(G_mag)); -3 ylabel('20*log10(|G|) -1 0 1 (db)'); 10 10 10 grid; omega title('Plot Frequency Response for 2st order system'); 0dB for ω < 1 subplot(2,1,2); semilogx(omega, G_phase); –12dB/oct for ω > 1. axis([0 10 -pi 0]); ylabel('phase'); grid; xlabel('omega'); Signals and System Analysis Plot Frequency Response for 2st order system 20 ζ=0.05, 0 ζ=0.1 -20 ζ=0.25 -40 ζ=0.707 20*log10(|G|) (db) -60 ζ=1 -1 0 1 10 10 10 0 -1 phase -2 -3 -1 0 1 10 10 10 omega Phase symmetrical about the –90º phase shift at ω = 1 for any ζ. Signals and System Analysis Nyquist (or polar) plot %Nyquist and Bode Plot clear all; Example: w=logspace(-1,1,500); s=j*w; g=1 ./(s.^2+0.6 .*s+1); 1 Re=real(g); G(s) = Im=imag(g); s2 + 0.6s +1 figure(1); plot(Re, Im,'*'); axis([-2 2 -2 2]); 1 grid; G( jω) = title ('Nyquist Plot'); −ω 2 + 0.6 jω +1 figure(2); subplot(2,1,1); semilogx(w,20*log10(abs(g))); ylabel('20log10(|G|)'); title ('Bode Plot'); grid; subplot(2,1,2); semilogx(w,atan2(Im,Re)); ylabel('Phase (rad)'); grid; Signals and System Analysis Nyquist Plot 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Bode Plot 10 0 -10 -20 20log10(|G|) -30 -40 -1 0 1 10 10 10 0 -1 -2 Phase (rad) Phase -3 -4 -1 0 1 10 10 10 Signals and System Analysis FILTERS For many signal processing situations we need to design filters to remove or amplify specific frequency components.