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H ( s ) x ( t ) y ( t ) 1 1 1 c c c x ( t ) = A cos( t + - PowerPoint PPT Presentation

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Analog Filters Overview To Do List Discuss time-domain versus frequency domain characteristics Ideal Filters Overshoot First-Order Filters Settling time Active & Passive Filters Show examples of filters applied to

  1. Analog Filters Overview To Do List • Discuss time-domain versus frequency domain characteristics • Ideal Filters – Overshoot • First-Order Filters – Settling time • Active & Passive Filters – Show examples of filters applied to signals • Second-Order Filters • Discuss how to view time-domain characteristics • Resonance – Explain why impulse response is only appropriate for low-pass • RLC Filters and bandpass filters – Show step response of all filters • Impedance & Frequency Scaling • Use problem from Winter 2002 final as an example • Go over other popular second-order active filters discussed in Franco’s text Portland State University ECE 222 Analog Filters Ver. 1.19 1 Portland State University ECE 222 Analog Filters Ver. 1.19 2 Introduction to Filters Ideal Filters Lowpass Highpass Notch H ( s ) x ( t ) y ( t ) 1 1 1 ω c ω ω c ω ω c ω x ( t ) = A cos( ωt + φ ) Bandpass Bandstop y ss ( t ) = A | H ( jω ) | cos ( ωt + φ + ∠ H ( jω )) 1 1 • In general, H ( jω ) will vary with ω ω c 1 ω c 2 ω c 1 ω c 2 ω ω • Filters attenuate ranges of frequencies • There are five ideal filters • Used in many applications • Lowpass filters pass low frequencies: ω < ω c • Highpass filters pass high frequencies: ω > ω c • Bandpass filters pass a range of frequencies: ω c 1 < ω < ω c 2 • Bandpass filters pass two ranges: ω < ω c 1 and ω > ω c 2 • Notch filters pass all frequencies except ω ∼ = ω c Portland State University ECE 222 Analog Filters Ver. 1.19 3 Portland State University ECE 222 Analog Filters Ver. 1.19 4

  2. Ideal Filters Comments Example 1: Passive RC Filter Lowpass Highpass Notch R 1 1 1 � 1 H ( s ) = V o ( s ) � 1 + V s ( s ) = ω c ω c ω c ω ω ω 1 RC s + v s ( t ) C v o ( t ) RC Bandpass Bandstop 1 1 - ω c 1 ω c 2 ω ω c 1 ω c 2 ω 1. What is the DC ( ω = 0 ) response of this filter? 2. What is lim ω →∞ | H ( jω ) | ? • Phase is not shown 3. What type of filter to you think this is? • ω c is called the cutoff frequency • Generally, the ideal phase is 0 ◦ for all frequencies 1 4. Plot | H ( jω ) | and ∠ H ( jω ) for RC = 1 rad/s. • Can not build ideal filters in practice • Real filters appear as rounded versions of ideal filters • Most LTI systems can be thought of as non-ideal filters Portland State University ECE 222 Analog Filters Ver. 1.19 5 Portland State University ECE 222 Analog Filters Ver. 1.19 6 Example 1: Workspace Example 1: Bode Plot Passive Lowpass RC Filter 0 |H(j ω )| (dB) −10 −20 −30 −40 −2 −1 0 1 2 10 10 10 10 10 0 ∠ H(j ω ) (degrees) −20 −40 −60 −80 −2 −1 0 1 2 10 10 10 10 10 Frequency (rad/s) Portland State University ECE 222 Analog Filters Ver. 1.19 7 Portland State University ECE 222 Analog Filters Ver. 1.19 8

  3. Example 1: MATLAB Code Cutoff Frequency Defined The cutoff frequency is defined as the value of ω such that w = logspace(-2,2,1000); H = 1./(j*w + 1); | H ( jω ) | 2 1 subplot(2,1,1); = max ω | H ( jω ) | 2 2 h = semilogx(w,20*log10(abs(H))); set(h,’LineWidth’,1.5); | H ( jω ) | 2 ylabel(’|H(j\omega)| (dB)’); = title(’Passive Lowpass RC Filter’); H 2 set(gca,’Box’,’Off’); max grid on; where set(gca,’YLim’,[-45 5]); set(gca,’XLim’,[min(w) max(w)]) H max ≡ max | H ( jω ) | subplot(2,1,2); ω h = semilogx(w,angle(H)*180/pi); set(h,’LineWidth’,1.5); • Lowpass and highpass filters have a single cutoff frequency ylabel(’\angle H(j\omega) (degrees)’); set(gca,’Box’,’Off’); • Bandpass and bandstop filters have two cutoff frequencies grid on; set(gca,’YLim’,[-95 5]); • Notch filters have a single notch frequency (not formally defined) set(gca,’XLim’,[min(w) max(w)]) xlabel(’Frequency (rad/s)’); Portland State University ECE 222 Analog Filters Ver. 1.19 9 Portland State University ECE 222 Analog Filters Ver. 1.19 10 Example 2: Cutoff Frequency Example 2: Workspace R + v s ( t ) C v o ( t ) - Find the cutoff frequency for the circuit shown above. Portland State University ECE 222 Analog Filters Ver. 1.19 11 Portland State University ECE 222 Analog Filters Ver. 1.19 12

  4. Bode Plots & Cutoff Frequency Analog Filters Overview Lowpass Highpass Notch | H ( jω c ) | 2 1 1 1 1 = H 2 2 max ω c ω ω c ω ω c ω � | H ( jω c ) | 1 = Bandpass Bandstop 2 H max 1 1 � 1 20 log 10 | H ( jω c ) | − 20 log 10 H max = 20 log 10 ω c 1 ω c 2 ω c 1 ω c 2 2 ω ω H dB ( ω c ) − H dB , max = − 3 . 0103 dB • We will initially discuss first and second order filters H dB , max − H dB ( ω c ) = 3 . 0103 dB • These are poor filters • Cutoff frequency always occurs at ≈ 3 dB below H max • Poor filters are adequate for many applications • This is also known as the 3 dB frequency • These are the building blocks for very good filters • Passive filters consist only of passive elements: R , L , and C • Active filters also include active elements such as op amps or transistors Portland State University ECE 222 Analog Filters Ver. 1.19 13 Portland State University ECE 222 Analog Filters Ver. 1.19 14 Frequency Selective Impedance Frequency Selective Impedance Continued I ( s ) I ( s ) I ( s ) I ( s ) I ( s ) I ( s ) + + + + + + 1 1 V ( s ) R V ( s ) s L V ( s ) V ( s ) R V ( s ) s L V ( s ) s C s C - - - - - - • We can think of passive elements as frequency-selective • At high frequencies: s = jω → ∞ impedance sources – R is constant • At low frequencies: s = jω → 0 – L approximates an open circuit – R is constant – C approximates a short circuit – L approximates a short circuit • Inductors are rarely used directly because they are difficult to build – C approximates an open circuit compactly (in silicon) • This is the same as doing a DC analysis • In ECE 321 you will learn how to use op amps and capacitors to simulate inductors Portland State University ECE 222 Analog Filters Ver. 1.19 15 Portland State University ECE 222 Analog Filters Ver. 1.19 16

  5. First Order Lowpass First Order Lowpass Bode Plot Lowpass First−Order Filter C 0 |H(j ω )| (dB) −10 R 2 R R 1 −20 −30 + + −40 v s ( t ) C v o ( t ) v s ( t ) v o ( t ) R L −2 −1 0 1 2 10 10 10 10 10 - - 0 ∠ H(j ω ) (degrees) −20 −40 We derived the transfer functions for these circuits earlier. −60 1 1 kω c R 1 C RC H P ( s ) = H A ( s ) = − H ( s ) = −80 1 1 s + s + s + ω c RC R 2 C −2 −1 0 1 2 10 10 10 10 10 Frequency (rad/s) Portland State University ECE 222 Analog Filters Ver. 1.19 17 Portland State University ECE 222 Analog Filters Ver. 1.19 18 Example 3: First Order Highpass Example 3: Workspace C R 2 R 1 C + + v s ( t ) R v o ( t ) v s ( t ) v o ( t ) - - Find the transfer functions for the first-order highpass filters shown above. Portland State University ECE 222 Analog Filters Ver. 1.19 19 Portland State University ECE 222 Analog Filters Ver. 1.19 20

  6. Example 3: Bode Plot Example 3: MATLAB Code w = logspace(-2,2,5000); Highpass First−Order Filter H = j*w./(j*w + 1); 0 |H(j ω )| (dB) subplot(2,1,1); −10 h = semilogx(w,20*log10(abs(H))); set(h,’LineWidth’,1.5); −20 ylabel(’|H(j\omega)| (dB)’); −30 title(’Highpass First-Order Filter’); set(gca,’Box’,’Off’); −40 grid on; set(gca,’YLim’,[-45 5]); −2 −1 0 1 2 10 10 10 10 10 set(gca,’XLim’,[10^-2 10^2]) ∠ H(j ω ) (degrees) subplot(2,1,2); 80 h = semilogx(w,angle(H)*180/pi); set(h,’LineWidth’,1.5); 60 ylabel(’\angle H(j\omega) (degrees)’); 40 set(gca,’Box’,’Off’); grid on; 20 set(gca,’YLim’,[-5 95]); set(gca,’XLim’,[10^-2 10^2]) 0 xlabel(’Frequency (rad/s)’); −2 −1 0 1 2 10 10 10 10 10 Frequency (rad/s) Portland State University ECE 222 Analog Filters Ver. 1.19 21 Portland State University ECE 222 Analog Filters Ver. 1.19 22 Cascading Transfer Functions Example 4: Cascaded Circuits x ( t ) G ( s ) H ( s ) y ( t ) R L C H + v s ( t ) C L R H v o ( t ) F ( s ) x ( t ) y ( t ) - • In general, the transfer function of cascaded passive filters is not Lowpass Highpass the product: F ( s ) � = G ( s ) H ( s ) • For active filters, you can cascade the filters if the output of each Calculate the transfer function for the cascade of passive filters shown system is connected to the output of an op amp above. Generate the bode plots for the product of the transfer function of each stage and for the transfer function of the circuit. Use C L = 1 µ F, R L = 1 k Ω , C H = 1 µ F, R H = 10 k Ω . Portland State University ECE 222 Analog Filters Ver. 1.19 23 Portland State University ECE 222 Analog Filters Ver. 1.19 24

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