# ECE3050 Assignment 7

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1 ECE3050 Assignment 7. Sketch and label the Bode magnitude and phase plots for the transfer functions given. Use loglog scales for the magnitude plots and linear-log scales for the phase plots. On the magnitude plots, label the slopes of all asymptotes in dec/dec, label the break frequencies in rad/s, label the gain magnitude on all zero-slope asymptotes, and label the approximate gain magnitude for the actual plot at the break frequencies. You can check your plots with a convenient computer program, e.g. PSpice, Mathcad, MatLab, etc., to obtain computer generated plots. An example PSpice deck is given for one of the transfer functions. The output voltage is V(2), i.e. the voltage at node 2. It is obtained by the LAPLACE statement, which multiplies the voltage V() by the transfer function after the word LAPLACE. They-axisinthePSpice ProbegraphicsroutinemustbechangedtoalogscaletoseethecorrectslopesontheBode magnitude plot. The number of decades displayed on the y-axis should be changed to no more than 3 or 4 to get the best looking plot. T (s) =0 +s/00 T (s) =0 s/00 +s/00 0 T (s) = ( + s/00) ( + s/000) 0 T (s) = (s/00) 2 +2(s/00) + 2(s/00) T (s) = 00 (s/00) 2 +2(s/00) + T (s) =0 (s/00) 2 + 2(s/00) + 2(s/00) T (s) = 00 (s/00) 2 + 2(s/00) + T (s) =0 (s/00) (s/00) + 0.5(s/00) T (s) = 00 (s/00) (s/00) + T (s) =0 s/0 s/0 + (s/000) 2 T (s) =0 ( + s/00) ( + s/0000) (s/00) 2 T (s) =0 (s/00) 2 +2(s/00) + (s/00) 2 + T (s) =0 (s/00) 2 +2(s/00) + (s/00) 2 T (s) =0 (s/00) 2 + 2(s/00) + (s/00) 2 + T (s) =0 (s/00) 2 + 2(s/00) + (s/00) 2 T (s) =0 (s/00) (s/00) + (s/00) 2 + T (s) =0 (s/00) (s/00) + T (s) =0 (s/00)2 2(s/00) + (s/00) 2 + 2(s/00) + EXAMPLE TRANSFER FUNCTION BODE PLOT VS 0 AC E 2 0 LAPLACE {V()}={0*PWR(S/00,2)/(PWR(S/00,2)+SQRT(2)*S/00+)}.AC DEC 50 00K.PROBE.END 2. The figure shows an RLC circuit. Show that the voltage gain transfer function is of the form T (s) = (s/ω 0 ) 2 (s/ω 0 ) 2 + s/ (Qω 0 )+

2 where you must give the equations for ω 0 and Q. ForQ =0.5, show that the transfer function becomes T (s) = (s/ω 0) 2 (s/ω 0 +) 2 For Q<0.5, show that the transfer function becomes T (s) = s/ω s/ω + s/ω 2 s/ω 2 + where s µ ω,2 = ω 0 2 2Q ± 2Q Sketch the Bode magnitude and phase plots as a function of ω for the cases Q<0.5, Q =0.5, and Q>0.5. Label the slopes in dec/dec on the magnitude plot and label all break frequencies in the asymptotes. 3. Repeat problem 2 for the circuit given. Show that the transfer function is given by T (s) = s/ (Qω 0 ) (s/ω 0 ) 2 + s/ (Qω 0 )+ For Q =0.5, show that the transfer function becomes T (s) = s/ω 0 (s/ω 0 +) 2 For Q<0.5, show that the transfer function becomes r ω s/ω T (s) = ω 2 s/ω + s/ω 2 + where s µ ω,2 = ω 0 2 2Q ± 2Q Sketch the Bode magnitude and phase plots as a function of ω for the cases Q<0.5, Q =0.5, and Q>0.5. Label the slopes in dec/dec on the magnitude plot and label all break frequencies in the asymptotes. 2

3 4. Show that the voltage gain transfer function for the circuit is given by T (s) = R 2 k (R 3 + R 4 ) R + R 2 k (R 3 + R 4 ) +R 3 kr 4 Cs +(R kr 2 + R 4 ) kr 3 Cs Show that the input impedance is given by Z in =[R + R 2 k (R 3 + R 4 )] +[R 3k (R kr 2 + R 4 )] Cs +[R 3 k (R 2 + R 4 )] Cs 5. Show that the voltage gain transfer function for the circuit is given by T (s) = R 4 R k (R 2 + R 3 )+R 4 +(R + R 2 ) kr 3 Cs +(R kr 4 + R 2 ) kr 3 Cs Show that the input impedance is given by Z in =[R k (R 2 + R 3 )+R 4 ] +[R 3k (R kr 4 + R 2 )] Cs +[R 3 k (R + R 2 )] Cs 6. Solve for the transfer function for /V i for the circuit below. The short cut method we covered in class does not work with this circuit. However, the short cut method can be used to solve for V a /V i. Once this is obtained, superposition of V i and V a canbeusedtosolvefor. This eliminates writing node equations, but there is some algebra involved in combining terms to put the transfer function into the ratio of two polynomials. Sketch the Bode magnitude plot, label the break frequencies, and label the gain on the zero-slope asymptotes. Answer: = +[R 2R 3 / (R + R 2 + R 3 )] Cs V i +[(R + R 2 ) kr 3 ] Cs The element values are to be chosen so that the high-frequency asymptotic gain is 0.05 and the high-frequency asymptotic output resistance (with V i =0)is00 Ω. The frequency of the 3

4 zero in the transfer function is to be 00 Hz. If C = 220 µf, specify the element values in the circuit and calculate the pole frequency. Answers: R =2kΩ, R 2 =05.26 Ω, R 3 = Ω, f p =5Hz. 7. Solve for the transfer function for /V i for the circuit below. Sketch the Bode plot, label the break frequencies, and label the gain on the zero-slope asymptotes. Answer: V i = Z F R = R 2 R +R 3 Cs +(R 2 + R 3 ) Cs The circuit is to be designed as a lag-lead compensator for a motor control system. The specifications are low-frequency asymptotic gain: 2, input resistance: 0 kω, pole frequency: Hz, zero frequency: 0 Hz. Specify the element values. Answers: R =0kΩ, R 2 =20kΩ, R 3 = Ω, andc =7.620 µf. 8. Solve for /V i for the circuits below. Sketch and label the Bode magnitude plots. Answers: (a) The transfer function is a low-pass shelving function with a dc gain of K dc = R 2 + R 3 R + R 2 + R 3 4

5 and a high-frequency gain of The transfer function is V i = K = R 2 + R 3 R + R 2 + R 3 R 2 + R 3 kr 4 R + R 2 + R 3 kr 4 +(R 2 kr 3 + R 4 ) Cs +[(R + R 2 ) kr 3 + R 4 ] Cs (b) The transfer function is a high-pass shelving function. The zero-frequency gain is K dc =+ R + R 2 R 3 The high-frequency gain is K =+ R 2 R 3 The transfer function is given by µ V µ o Vf = = + R + R 2 +[(R2 + R 3 ) kr ] Cs V i R 3 +R Cs 9. It is desired to design a circuit that realizes the following impedance transfer function: Z = 000 ( + s/2πf 2)(+s/2πf 4 ) ( + s/2πf )(+s/2πf 3 ) where f =0Hz, f 2 = 00 Hz, f 3 =khz,andf 4 =0kHz. (a) Sketch the Bode magnitude plot for Z. Note that the impedance starts at 000 Ω, shelves at 00 Ω, then shelves again at 0 Ω. Label the break frequencies and label the magnitude of the impedance on each zero-slope asymptote. (b) A possible circuit realization is in the figure below. At low frequencies, the impedance starts at the value R + R + R 2. As frequency is increased, suppose that C becomes a short circuit well before C 2 becomes a short circuit. When C becomes a short, the impedance shelves at the value R + R 2. Therefore, C causes both a pole and a zero. As frequency is increased further, C 2 becomes a short and the impedance shelves at the value R. ThusC 2 also causes a pole and a zero. With this information show that the input impedance is approximately given by Z in (s) ' (R + R + R 2 ) +R k (R + R 2 ) C s +R C s +(R 2 kr) C 2 s +R 2 C 2 s where you consider C 2 to be an open in calculating the effect of C and you consider C to be a short in calculating the effect of C 2. 0Write the three equations for the resistors and solve for their values. Answers: R =0Ω, R = 900 Ω, andr 2 =90Ω. 5

6 (c) Solve for the two time constants for C assuming C 2 is an open circuit. What must be the value of C? Answer: The pole time constant is calculated with the inputs open circuited. The zero time constant is calculated with the inputs short circuited. The value of C is C =7.68 µf. (d) Solve for the two time constants for C 2 assuming C is a short circuit. What must be the value of C 2? Answer: The pole time constant is calculated with the inputs open circuited. The zero time constant is calculated with the inputs short circuited. The value of C 2 is C 2 =.77 µf. (e) Use SPICE to plot the magnitude of the impedance as a function of frequency. To do this, drive the circuit with an ac current source of A. The voltage across the terminals is the impedance. Use a log scale for the vertical axis to display the correct Bode plot. (.AC DEC 50 00K) 0. Use the inverting gain formula to solve for the voltage-gain transfer function for the circuit below Sketch and label the Bode magnitude plot. Answer: V i = R 3 Z = R 3 R + R 2 +R 2 Cs +(R kr 2 ) Cs. Using a single 00 µf capacitor, design a single op amp circuit which has the voltage-gain transfer function =0 +s/0 V i +s/00 Sketch and label the Bode magnitude plot. One possible answer is the circuit below. where µ V µ o Vf = = + R 3 +(R kr 3 + R 2 ) Cs V i R +R 2 Cs The element values are R =kω, R 2 = 00 Ω, R 3 =9kΩ, andc =5.92 µf. 6

7 2. For the circuit shown, show that = R 2 R V i V i2 +R 2 Cs 3. For the circuit shown, show that = +(R + R 2 ) Cs V i +R Cs 4. For the circuit shown, show that V i = R R 2 (s/ω 0 ) (s/ω 0 ) 2 +(/Q)(s/ω 0 )+ where ω 0 =/ LC and Q = ω 0 R C. Sketch and label the Bode magnitude and phase plots for Q =0.5, Q =,andq =2. 7

8 5. For the circuit shown, show that the impedance is real at the frequency s R 2 ω = L 2 + L C 6. For the circuit shown, show that = 2 V i RCs Show that the equivalent circuit for Z in is the resistor R = R in parallel with the negative inductor L = R 2 C. 7. For the potentiometer circuit shown, let the resistance below the wiper be xr p and the resistance above the wiper be ( x) R p.show that V i = x +x ( x) R p Cs 8

9 Show that the circuit has a worst-case minimum bandwidth when x = 0.5 and that the corresponding pole frequency is given by f pole = πr p C 8. For the circuit shown, show that V i = µ + R F R + R 2 +[R k (R 2 + R F )] Cs +(R kr 2 ) Cs 9

10 9. The figure shows a Schmidt trigger. It is given that V SAT =2Vand R F =0kΩ. Solvefor V REF and R for V A = 4V and V B =+2V.Answers:R =3.33 kω, V REF =.33 V. 20. The transfer function of a 4th order 0.5 db ripple Chebyshev low-pass filter is of the form T LP (s) =T (s) T 2 (s) where T (s) = T 2 (s) = (s/.033ω c ) 2 +(/2.9406) (s/.033ω c )+ (s/ ω c ) 2 +(/0.705) (s/ ω c )+ (a) Replace s/ω c with ω c /s to obtain the high-pass transfer functions. (b) For ω c = 00π, designsallen-keyfilters for the two transfer functions. In each filter, use C = C 2 =0. µf. (c) The low-frequency response of a closed-box loudspeaker can be modeled by a 2nd highpass transfer function. Suppose that T 2 (s) represents the transfer function of a particular closed-box loudspeaker. Use SPICE or a math program to obtain the Bode magnitude plot for T 2 (j2πf). Use the cursor or trace feature of the program to determine the lower 3dB cutoff frequency of the loudspeaker. (d) Let T (s) represent the transfer function of an active filter which precedes the power amplifier that drives the loudspeaker. Such a filter is called an equalizer. Use SPICE or a math program to obtain the Bode magnitude plot for T (j2πf). Use the trace feature of the program to determine the peak gain or peak lift of the filter and the frequency of the peak. (e) Use SPICE or a math program to obtain the Bode magnitude plot for T (j2πf) T 2 (j2πf). Use the trace feature of the program to determine the lower 3dB cutoff frequency of the loudspeaker plus equalizer. By what factor is it lower than that for the loudspeaker alone? (f) An audio amplifier is used to drive the loudspeaker. The loudspeaker can be assumed to have a resistive impedance of 8 Ω. The frequency response is to be measured using a sine wave source to drive the system. The amplitude of the sine wave source is set so that the power delivered to the loudspeaker at 500 Hz is 0 W. If the amplitude of the source is held constant as the frequency is decreased, determine the power delivered to the loudspeaker at the frequency where the equalizer exhibits its maximum peak lift. 0

11 2. If C and C 2 are specified for the second-order unity-gain Sallen-Key low-pass filter, it can be shown that R and R 2 are given by " r # R = ± 4Q R 2 2Qω 0 C 2 C 2 2 C where the plus sign can be used for either R or R 2 and the minus sign for the other, i.e. the values for R and R 2 are interchangeable. (a) Design the filter for ω 0 = 200π 0 and Q =.5. Use capacitor values having the ratio C /C 2 =2in the range from 000 pf and 0.22 µf. Specify C, C 2, R,andR 2. The capacitors should conform to standard 0% values. The resistors should conform to standard 5% values and should lie in the range from kω to 00 kω. You should attempt to come as close to the theoretical design as possible with the standard resistor and capacitor values. (The 5% values for one decade are 0,, 2, 3, 5, 6, 8, 20, 22, 24, 27, 30, 33, 36, 39, 43, 47, 5, 56, 62, 68, 75, 82, 9, and00. The0% values for onedecadeare0, 2, 5, 8, 22, 27, 33, 39, 47, 56, 68, 82, and00.) Answers: Pick C =2xµF and C 2 = xµf, wherex is any number that you can choose to obtain values for R and R 2 in the specified range. It follows that the calculated values for R and R 2 are 83.88/x Ω and 0.256/x kω. (b) For one choice of R and R 2, use SPICE to simulate the magnitude of the filter gain versus frequency over the range 0 Hz f 0 khz. Use an ac analysis with 50 points per decade and convert the y-axis to a log scale. Set the y-axis to display the gain from 0. to 0. (c) Repeat the previous part with the other choice for R and R 2. Note that the figure should be the same. 22. This problem illustrates how the table of 5% resistor values is determined. Let R and R 2 be two adjacent resistors in the 5% resistor table. If R is increased by 0%, itsnew value is.r. If this corresponds to the next resistor R 2 in the 5% table, it follows that R +5%R ' R 2 5%R 2.Inorderforthe5% table to repeat each decade, it is necessary for. n =0,wheren is an integer corresponding to the number of resistance values per decade. Solution for n yields n =24.6. In generating the 5% resistor table, this value is rounded off to n =24. It follows that R 2 = 24 0R for any two adjacent resistors in the the 5% table. Starting with R =0Ω, generate the 5% resistor table for 0 Ω R 00 Ω. When rounded off to 2 significant figures, verify that the values obtained correspond to those given in problem The frequency transformations for high-pass, band-pass, and band-reject transfer functions are defined as follows: Low-Pass to High-Pass p p µ Low-Pass to Band-Pass p B p + µ p Low-Pass to Band-Reject p B p + p where p = s/ω c is called the normalized frequency, ω c is the cutoff frequency, and the arrow is read is replaced by. The parameter B determines the 3 db bandwidth of the band-pass

12 and band-reject functions. For the following low-pass function T LP (s) =K +s/ (aω c ) where a is a constant such that a =for Butterworth filters and a 6= for Chebyshev filters, use the frequency transformations to show that (a) The high-pass function is given by (b) The band-pass function is given by (c) The band-reject function is given by T HP (s) =K as/ω c +as/ω c as/bω c T BP (s) =K (s/ω c ) 2 + as/ (Bω c )+ (s/ω c ) 2 + T BR (s) =K (s/ω c ) 2 + s/ (abω c )+ 24. A third-order low-pass transfer function is given by T LP (s) =K (s/ω c ) 3 + a 2 (s/ω c ) 2 + a (s/ω c )+ The magnitude squared function for a third-order Butterworth low-pass function is given by T LP (jω) 2 = K 2 +(ω/ω c ) 6 (a) Set s = jω, solvefor T LP (jω) 2, equate it to the Butterworth function, and solve for a and a 2 to make the transfer function Butterworth. Answers: a = a 2 =2. (b) Factor the transfer function into the product of a first-order function multiplied by a second-order function. What is the relationship between the break frequencies on the Bode plots of the first and second-order sections? What is the Q of the second-order section? Answers: T LP (s) =K +s/ω c (s/ω c ) 2 + s/ω c + The break frequency is the same for both functions and is equal to ω c. The second-order section has Q =. (c) Evaluate the transfer function for s = jω c and show that it is given by T LP (jω c )=K +j +j + = K For this problem, pertinent design data for Chebyshev filters is posted on the class web page at You can also find it in the ECE 3042 lab manual. 2

13 (a) A bandpass filter is to be designed that consists of the cascade of a high-pass filter and a low-pass filter. Each filter is to be a 3rd -order.25 db ripple Chebyshev filter. The high-pass filter is to have a cutoff frequency of f c = 00 Hz. The low-pass filter is to have a cutoff frequency of f c2 =0kHz. Use capacitor values in the range of 000 pf to 0.22 µf and resistors in the range of kω to 00 kω. The capacitors should conform to standard 0% values. The resistors should conform to standard 5% values. You should attempt to come as close to the theoretical design as possible with the resistor and capacitor values specified. (The 5% values for one decade are 0,, 2, 3, 5, 6, 8, 20, 22, 24, 27, 30, 33, 36, 39, 43, 47, 5, 56, 62, 68, 75, 82, 9, and00. The0% values for one decade are 0, 2, 5, 8, 22, 27, 33, 39, 47, 56, 68, 82, and00.). Answers: The low-pass function is T LP (s) = = h =0.427 θ =30 a 2 =0.453 a = b =2.57 +s/0.453ω c The high-pass transfer function is T HP (s) = 0.453s/ω c s/ω c2 (s/0.9774ω c ) 2 +(/2.57) (s/0.9774ω c )+ (0.9774s/ω c2 ) 2 (0.9774s/ω c2 ) 2 +(/2.57) (0.9774s/ω c2 )+ (b) Use SPICE to simulate the filter. Set the ac input voltage to V so that the output voltage will be equal to the gain. Display the gain versus frequency for frequencies between 0 Hz and 00 khz. Using a log scale, display the y-axis over the range from 0.00 to 0. Use the cursor feature of PROBE to flag the lower and upper cutoff frequencies, the lower and upper 3dB frequencies, the midband gain, and the amount of ripple at the low and high frequencies. Answer: Your plot should look like the following (which was made with Mathcad): 0 T f n f n 26. The circuit shows a Wein bridge oscillator. If R = R 2, C =0. µf, C 2 =0.22 µf, and R 4 =0kΩ, specify R, R 2,andR 3 for the circuit to have stable oscillations at f = 000 Hz. Answers: R = R 2 = 073 Ω and R 3 = 6875 Ω. 3

14 27. The figure shows a phase-shift oscillator with the feedback loop broken. By writing node equations, it can be shown that the loop-gain transfer function is given by Vo 0 = R F (RCs) 3 R (RCs) 3 +6(RCs) 2 +5(RCs)+ (a) To solve for the frequency of oscillation, what do you set Vo/V 0 o equal to? Answer: 6 0. (b) Use the transfer function to solve for the frequency of oscillation. Answer: f 0 = / 2π 6RC. (c) Use the transfer function to solve for value of R F /R in order for Vo/V 0 o = 6 0 at f = f 0.Answer:R F /R = The figure shows a phase shift oscillator. If C =0. µf, specifyr and R F for the circuit to have stable oscillations at f = 250 Hz. Answers:R =3249kΩ and R F =94.2 kω. 29. The loop-gain transfer function of a particular oscillator circuit is given by s Vo 0 = K (s/00) (s/00) + At what frequency does the circuit oscillate and what must be the value of K for stable oscillations? Answers: f =5.9Hz and K =

15 30. The figure shows a log converter. It is given that R =MΩ. (a)whenv I =0Vit is found that v O = V. Solve for the transistor saturation current I S. (b) What is v O if v I is increased to 20 V? To 40 V? (c) By how much does v O increase each time v I is doubled? Answers: (a) I S = A,(b)v O = 0.55 V and v O = V, (c) v O = 0.07 V. 3. The figure shows a precision rectifier. If R = R 2 = R 3 and the input resistance is to be 0 kω, specify the resistors in the circuit for the output voltage v O =5 v I.Forasinewave input, sketch the voltage waveforms at the two inputs to the inverting summer and show how they combine to produce the desired output. What effect does reversing the diodes have? Answers: R = R 2 = R 3 =20kΩ, R F =40kΩ, andr F 2 = 00 kω. Reversing the diodes causes the output voltage to be given by v O = 5 v I. 32. For the precision rectifier of problem 3: (a) Sketch v O and v O versus v I for R = R F = R 3 = 0 kω, R 2 =20kΩ, andr F 2 = 200 kω. (b) Sketch v O and v O versus t for v I =0.5sin(ωt) V. (c) For v I =0.5sin(ωt) Vand R 3 =20kΩ, sketchv O versus t. 33. Sketch the waveforms for v O and vo 0 for the circuit given. Identify which portions on the waveform where the diode is on and where the diode is off. Whatwouldbetheeffect of reversing the diode? How would you modify the circuit to prevent saturation in the first op amp? 5

16 34. The figure shows a feedback amplifier. (a) Identify the type of feedback. (b) Use signal tracing to verify that the feedback is negative. (c) Use signal tracing to identify whether the feedback increases or decreases the input resistance. (d) Use signal tracing to identify whether the feedback increases or decreases the output resistance. (e) If the gain before feedback is sufficiently large, show that v o ' R F + R F 2 v i R F The figure shows a feedback amplifier. (a) Identify the type of feedback. (b) Use signal tracing to verify that the feedback is negative. (c) Use signal tracing to identify whether the feedback increases or decreases the input resistance. (d) Use signal tracing to identify whether the feedback increases or decreases the output resistance. (e) If the gain before feedback is sufficiently large, show that v o ' R F i i. 36. The figure shows a feedback amplifier. (a) Identify the type of feedback. (b) Use signal tracing to verify that the feedback is negative. (c) Use signal tracing to identify whether the 6

17 feedback increases or decreases the input resistance. (d) Use signal tracing to identify whether the feedback increases or decreases the output resistance. (e) If the gain before feedback is sufficiently large and r 02 canbeassumedtobeinfinite, show that i o ' R C2 + R F + R F 2 v i αr C2 R F The figure shows a feedback amplifier. (a) Identify the type of feedback. (b) Use signal tracing to verify that the feedback is negative. (c) Use signal tracing to identify whether the feedback increases or decreases the input resistance. (d) Use signal tracing to identify whether the feedback increases or decreases the output resistance. To do this, you must draw r 02 as an external resistor from the collector to the emitter of Q 2. (e) If the gain before feedback is sufficiently large and r 02 canbeassumedtobeinfinite, show that i o ' α R F + R E2 i i R E2 7

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