Overview of Bode Plots • Transfer function review • Piece-wise linear approximations • First-order terms • Second-order terms (complex poles & zeros) J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 1
Transfer Function Review H ( s ) x ( t ) y ( t ) Recall that if H ( s ) is known and x ( t ) = A cos( ωt + φ ) , then we can find the steady-state solution for y ( t ) : y ss ( t ) = A | H ( jω ) | cos ( ωt + φ + ∠ H ( jω )) J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 2
Bode Plots H ( s ) x ( t ) y ( t ) • Bode plots are standard method of plotting the magnitude and phase of H ( s ) • Both plots use a logarithmic scale for the x -axis • Frequency is in units of radians/second (rad/s) • The phase is plotted on a linear scale in degrees • Magnitude is plotted on a linear scale in decibels H dB ( jω ) � 20 log 10 | H ( jω ) | J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 3
Decibel Scales It is important to become adept at translating between amplitude, | H ( jω ) | , and decibels, H dB ( jω ) . Amplitude ( | H ( jω ) | ) Decibels ( 20 log 10 | H ( jω ) | ) 1 20 log 10 1 = 10 20 log 10 10 = 100 20 log 10 100 = 1000 20 log 10 1000 = 0.1 20 log 10 0 . 1 = 0.01 20 log 10 0 . 01 = 0.001 20 log 10 0 . 001 = 1 1 20 log 10 = -6.0206 2 2 2 20 log 10 2 = � � 1 1 20 log 10 = 2 2 J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 4
Example 1: Bode Plots 20 nF 10 k Ω 1 k Ω + v s ( t ) v o ( t ) R L - 1. Find the transfer function of the circuit shown above. 2. Generate the bode plot. J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 5
Example 1: Workspace J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 6
Example 1: Bode Plot Active Lowpass RC Filter 20 |H(j ω )| (dB) 10 0 1 2 3 4 5 10 10 10 10 10 180 ∠ H(j ω ) (degrees) 160 140 120 100 1 2 3 4 5 10 10 10 10 10 Frequency (rad/s) J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 7
Example 1: MATLAB Code w = logspace(1,5,500); H = -50e3./(j*w + 5e3); subplot(2,1,1); h = semilogx(w,20*log10(abs(H))); set(h,’LineWidth’,1.4); ylabel(’|H(j\omega)| (dB)’); title(’Active Lowpass RC Filter’); set(gca,’Box’,’Off’); grid on; set(gca,’YLim’,[-5 25]); subplot(2,1,2); h = semilogx(w,angle(H)*180/pi); set(h,’LineWidth’,1.4); ylabel(’\angle H(j\omega) (degrees)’); set(gca,’Box’,’Off’); grid on; set(gca,’YLim’,[85 185]); xlabel(’Frequency (rad/s)’); J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 8
Example 2: Bode Plots 1 µ F 2 µ F 1 k Ω 1 k Ω + v s ( t ) R L - 1. Find the transfer function of the circuit shown above. 2. Generate the bode plot. J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 9
Example 2: Workspace J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 10
Example 2: Bode Plot Active Lead/Lag RC Filter 8 6 |H(j ω )| (dB) 4 2 0 −2 1 2 3 4 5 10 10 10 10 10 −160 ∠ H(j ω ) (degrees) −165 −170 −175 −180 1 2 3 4 5 10 10 10 10 10 Frequency (rad/s) J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 11
Example 2: MATLAB Code w = logspace(1,5,500); H = -2*(j*w+500)./(j*w + 1000); subplot(2,1,1); h = semilogx(w,20*log10(abs(H))); set(h,’LineWidth’,1.4); ylabel(’|H(j\omega)| (dB)’); title(’Active Lead/Lag RC Filter’); set(gca,’Box’,’Off’); grid on; set(gca,’YLim’,[-2 8]); subplot(2,1,2); h = semilogx(w,angle(H)*180/pi); set(h,’LineWidth’,1.4); ylabel(’\angle H(j\omega) (degrees)’); set(gca,’Box’,’Off’); grid on; set(gca,’YLim’,[-180 -160]); xlabel(’Frequency (rad/s)’); J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 12
Bode Plot Approximations • Until recently (late 1980’s) bode plots were drawn by hand • There were many rules-of-thumb, tables, and template plots to help • Today engineers primarily use MATLAB, or the equivalent • Why discuss the old method of plotting by hand? – It is still important to understand how the poles, zeros, and gain influence the Bode plot – These ideas are used for transfer function synthesis, analog circuit design, and control systems • We will discuss simplified methods of generating Bode plots • Based on asymptotic approximations J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 13
Alternate Transfer Function Expressions There are many equivalent expressions for transfer functions. N ( s ) H ( s ) = D ( s ) b m s m + b m − 1 s m − 1 + · · · + b 1 s + b 0 = a n s n + a n − 1 s n − 1 + · · · + a 1 s + a 0 s ± ℓ ( s − z 1 )( s − z 2 ) . . . ( s − z m ) b m = ( s − p 1 )( s − p 2 ) . . . ( s − p n ) a n � � � � � � 1 − s 1 − s s 1 − . . . z 1 z 2 z m k s ± ℓ = � � � � � � 1 − s 1 − s s 1 − . . . p 1 p 2 p n • This last expression is called standard form • The first step in making bode plots is to convert H ( s ) to standard form J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 14
Magnitude Components Consider the expression for the transfer function magnitude: | H dB ( jω ) | = 20 log 10 | H ( jω ) | � � s ± ℓ (1 − s s z 1 ) . . . (1 − z m ) � � = 20 log 10 � k � � (1 − s s p 1 ) . . . (1 − p n ) � � � s = jω 20 log 10 | k | · | jω | ± ℓ | 1 − jω z 1 | . . . | 1 − jω z m | = | 1 − jω p 1 | . . . | 1 − jω p n | = 20 log 10 | k | ± ℓ 20 log 10 ω � � � � � 1 − jω � 1 − jω � � � � +20 log 10 � + · · · + 20 log 10 � � � � z 1 z m � � � � � � 1 − jω � 1 − jω � � � � − 20 log 10 � − · · · − 20 log 10 � � � � p 1 p n � J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 15
Magnitude Components Comments | H dB ( ω ) | = 20 log 10 | k | ± ℓ 20 log 10 ω � � � � � 1 − jω � 1 − jω +20 log 10 � + · · · + 20 log 10 � � � � z 1 z m � � � � � � 1 − jω � 1 − jω − 20 log 10 � − · · · − 20 log 10 � � � � p 1 p n � • Thus, | H dB ( ω ) | can be written as a sum of simple functions • This is similar like using basis functions { δ ( t ) , u ( t ) ,& r ( t ) } to write an expression for a piecewise linear signal • We will use this approach to generate our piecewise linear approximations of the bode plot • Note that there are four types of components in this expression – Constant – Linear term – Zeros – Poles J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 16
Magnitude Components: Constant (dB) | H ( jω ) | 40 20 0 (rad/sec) ω -20 -40 The constant term, 20 log 10 | k | , is a straight line on the Bode plot. J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 17
Magnitude Components: Linear Term (dB) | H ( jω ) | 40 20 0 (rad/sec) ω -20 -40 The linear term, ± ℓ 20 log 10 | ω | , is a line on the magnitude plot with a slope equal to ± ℓ 20 dB per decade. The x -axis intercept occurs at ω = 1 rad/s. Plot the bode magnitude plots for H ( s ) = s , 1 1 s , s 2 , s 2 . J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 18
Magnitude Components: Real Zeros Consider two limiting conditions for a term containing a zero, � 1 − jω � � 20 log 10 � z First condition: ω ≪ | z | � = 0 � 1 − jω � � z → 0 20 log 10 lim z ω � ≈ 0 . � 1 − jω ω � � Thus, if | z | ≪ 1 , then 20 log 10 z Second condition: ω ≫ | z | � = 20 log 10 | − jω � 1 − jω � � z →∞ 20 log 10 lim z | = 20 log 10 | ω | − 20 log 10 | z | z ω ω Thus, if | z | ≫ 1 , then this term is linear (on a log scale) with a slope of 20 dB per decade and an x -axis intercept at ω = | z | . J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 19
Magnitude Components: Real Zeros Continued (dB) | H ( jω ) | 40 20 0 (rad/sec) ω -20 -40 Our piecewise approximation joins these two linear asymptotic approximations at ω = | z | . � 1 − jω � � Plot the piecewise approximation of the term 20 log 10 � . z J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 20
Magnitude Components: Real Zeros Continued 2 Bode Magnitude Real Zero: z = ± 1 rad/s 50 40 30 Mag (dB) 20 10 0 −10 −2 −1 0 1 2 10 10 10 10 10 Frequency (rad/sec) The approximation is least accurate at ω = | z | . The true magnitude is 3 dB higher than the approximation at this corner frequency. J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 21
Magnitude Components: Real Poles (dB) | H ( jω ) | 40 20 0 (rad/sec) ω -20 -40 • Consider two limiting conditions for a term containing a pole, � � � 1 − jω − 20 log 10 � � p � • This is just the negative of the expression for a zero • The piecewise approximation is the mirror image of that for a zero J. McNames Portland State University ECE 222 Bode Plots Ver. 1.19 22
Recommend
More recommend
