R C C (8000 )( ) 20 log( k) 8 k 2.51

Size: px

Start display at page:

Download "R C C (8000 )( ) 20 log( k) 8 k 2.51"

Sharlene Cross
5 years ago
Views:

1 EE233 HW9 Solution Dec. 2 nd Due Date: Dec. 9 th 1. Consider the circuit below. (a) Design a HPF with a passband gain of 8dB and cutoff frequency of 4kHz using 25nF capacitor. Draw the circuit diagram and label all components c Rs R C C (8 )(251 ) s 2 log( k) 8 k 2.51 R f kr 399 s c (b) If the value of the feedback resistor in the filter is changed but the value of the resistor in the forward path is unchanged, what characteristic of the filter is changed? Briefly justify your answer. The passband gain will change. (c) Sketch the Bode magnitude and phase plots for the filter.

2 2. Consider the circuit in problem 1. (a) Design a HPF with a passband gain of 14dB and a cutoff frequency of 8kHz using a 3.9nF capacitor. Draw the circuit diagram and label all components c Rs 51 9 R C C (16 )(3.91 ) s 2log( k) 14 k 5 R kr 25.5 f s c

3 (b) Using only three components from Appendix H of the textbook (file posted on Canvas), design a HPF with a cutoff frequency and passband gain as close as possible to the specifications above in part (a). Draw the circuit diagram and label all component values. R 511 R R 25.5k R 27k 2 2 (c) Calculate the percent error in this new filter s cutoff frequency and passband gain when compared to the values specified in part (a). 1 c rad / s, fc 8683Hz RC K R R k 4.7k 5.74dB Freqeucny error : 14.8% (d) Sketch the Bode magnitude and phase plots for both filter designs on the same set of axes.

4 3. Using 1nF capacitors, design an active broadband first-order band-reject filter that has a lower cutoff frequency of 4Hz, an upper cutoff frequency of 4Hz, and a passband gain of db. Use prototype versions of the LPF and HPF in the design process (use scaling). (a) Draw the circuit diagram of the filter and label all the components. 1 1 R 39.8k (8 )(1 1 ) c1 L 9 RC L L 1 1 R 3.98k (8 )(1 1 ) c2 H 9 RC H H k 1 R R, so R R 1k s f s f

5 (b) Write the transfer function of the scaled filter s 8s 16 1 Hs () ( s4 )( s4 ) (c) Use the transfer function to find H( j ), where is the center frequency. H( j ) 16 1 j j44 (d) What is the passband gain (in db) of the filter at the center frequency? (16 1 ) 2 5 H( j ) dB 44 (16 1 ) (e) Sketch the Bode magnitude and phase plots for the filter.

6 4. Design an active LPF with a passband gain of 4dB, a cutoff frequency of 1kHz, and a gain rolloff rate of -6dB/decade. (a) Calculate the resistor and capacitor values used in the circuit. db rd order filter rad s 1/3 6 3 c / 2 K f 4 rad / s.5 Choose C 4 F in this case, 1 Km 19.9 R k 4 R 79.6 when R R 19.9 f (b) Draw the circuit diagram of the filter and label all the components. i

7 (c) Sketch the Bode magnitude and phase plots for the filter.

8 5. Your company has developed a new type of loud speaker system that uses very low power to reproduce sound. Like a home stereo system it users 3 independent speakers, the woofer, midrange, and tweeter, to reproduce the low, mid and high frequencies, respectively. The woofer is designed to reproduce frequencies up to 2Hz. The midrange delivers frequencies between 2Hz and 14Hz. The tweeter is designed for frequencies above 14Hz. You have been given the job of designing a filter system to separate the signal and deliver only the desired frequencies to each speaker. Others on your ream will take care of the rest (buffering of your input signal, level-adjustment to boost the filter output to the appropriate level, line conditioning to prevent RF interference, etc.) Also assume, for this exercise, that phasedistortion is not a problem. Each should filter should be designed separately. For the low frequency speaker uses an active low-pass filter with two cascaded first order filters to get sharper roll-off (a steeper cutoff in the frequency plot). The cutoff frequency should be 1Hz and should have unity gain in the passband. Make sure to take into account the shift in cutoff frequency that occurs when you cascade the two sections. For the midrange circuit implement a band-pass filter by cascading low-pass, high-pass, and inverting op-amp circuits. The resulting filter should pass frequencies from 1Hz to 14Hz, and have a maximum gain of 1 (2dB). The tweeters your company developed require more precise frequency selection. Implement a 4 th order high pass Butterworth filter with a cutoff frequency of 14Hz. For each of the three filters you should provide: - A schematic of the op-amp circuit - Specifications for all components used - Transfer function H(s) for your circuit - Magnitude only Bode Plot of the showing the frequency response of your circuit Use reasonable values for each component. For ease of grading, use 1 F capacitors when possible. Assume that you will use ideal op amps. Bonus: Simulate your circuits in Multisim and show the frequency response.

#7 (LPF) In[327]:= Clear@c, cp, R, Rp, fc, kf, km, wcd c = 1; H* Unscaled C *L cp = 1-6 ; H* Scaled C *L R = 1; H* Unscaled R *L fc = 1; H* Desired Corner Frequency *L wcp = 2 * p * fc; H* Desired

9 #7 (LPF) In[327]:= cp, R, Rp, fc, kf, km, wcd c = 1; H* Unscaled C *L cp = 1-6 ; H* Scaled C *L R = 1; H* Unscaled R *L fc = 1; H* Desired Corner Frequency *L wcp = 2 * p * fc; H* Desired Corner Frequencey radês *L n = 2; H* Filter order *L wc = SqrtA2 1ên - 1E H* Scaled Corner frequency *L kf = wcp êê N H* Frequency scaling factor *L wc km = 1 c êê N H* Magnitude scaling factor *L kf cp Rp = km * R H* Scaled resistor value *L Out[3213]= Out[3214]= Out[3215]= Out[3216]= In[3217]:= Clear@R, R1, R2, s, cd; R2 = Rp; H* LPF resistor R2 *L R1 = R2; H* LPF resistor R1 *L c = cp; H* LPF Capacitor *L h1 = 1 R1*c H* First order LPF transfer function *L s + 1 R2*c s = I * 2 * p * f; H* s=j*w *L h = h1 2 ; H* Transfer function in db *L hdb = 2 Log@1, Abs@hDD; H* *L LogLinearPlotA8-3, hdb<, 9f, 1-6, 1 3 =, PlotRange Ø 9911, 1 3 =, 8-1, <=, GridLines Ø AutomaticE Circuit Out[322]= s -2-4 Out[3223]= PSpice Sim

10 In[3682]:= cp, R, Rp, fc, kf, km, wcd c = 1; H* Unscaled C *L cp = 1-6 ; H* Scaled C *L R = 1; H* Unscaled R *L fc = 14; H* Desired Corner Frequency *L wcp = 2 * p * fc; H* Desired Corner Frequencey radês *L n = 1; H* Filter order *L wc = SqrtA2 1ên - 1E êê NH* Scaled Corner frequency *L kf = wcp H* Frequency scaling factor *L wc km = 1 c H* Magnitude scaling factor *L kf cp Rp = km * R H* Scaled resistor value *L Out[3686]= 1. #7 BPF Out[3687]= Out[3688]= Out[3689]= In[369]:= Clear@R, R1, R2, s, c, fd; R2 = Rp; H* LPF resistor R2 *L R1 = R2; H* LPF resistor R1 *L c = cp; H* LPF Capacitor *L hlp = 1 R1*c s + 1 R2*c H* First order LPF transfer function *L s = I * 2 * p * f; H* s=j*w *L hdb = 2 Log@1, Abs@hlpDD; H* Transfer function in db *L LogLinearPlotA8-3, hdb<, 9f, 1-6, 1 4 =, PlotRange Ø 99.1 * 1 3, 1 * 1 3 =, 8, -4<=, GridLines Ø AutomaticE Out[3692]= s µ Out[3693]= In[3694]:= Clear@c, cp, R, Rp, fc, kf, km, wcd c = 1; H* Unscaled C *L cp = 1-6 ; H* Scaled C *L R = 1; H* Unscaled R *L fc = 1; H* Desired Corner Frequency *L wcp = 2 * p * fc; H* Desired Corner Frequencey radês *L n = 1; H* Filter order *L wc = SqrtA2 1ên - 1E êê NH* Scaled Corner frequency *L kf = wcp H* Frequency scaling factor *L wc km = 1 c H* Magnitude scaling factor *L kf cp Rp = km * R H* Scaled resistor value *L Out[3698]= 1. Out[3699]= Out[37]= Out[371]= In[372]:= Clear@Rh, R1h, R2h, s, ch, fd; R2h = Rp; H* HPF resistor R2 *L R1h = R2h; H* HPF resistor R1 *L ch = cp; H* HPF Capacitor *L s h = H* First order HPF transfer function *L 1 s + R1h*ch s = I * 2 * p * f; H* s=j*w *L hhp = h H* Transfer function in db *L hdb = 2 Log@1, Abs@hhpDD; H* *L LogLinearPlotA8-3, hdb<, 9f, 1, 1 4 =, GridLines Ø Automatic, PlotRange Ø 991, 1 6 =, 8, -4<=E s Out[373]= s 2 Â f p Out[374]= Â f p Out[375]=

In[3754]:= H* design of inverting constant-gain amplifier stage *L Clear@G, Rf, RiD G = 1; Rf = 1 4 ; Ri = Rf G hinv = Rf Ri Out[3757]= 1 Out[3758]= 1 In[3751]:= hbp = hlp * hhp * hinv hbpdb = 2

11 In[3754]:= H* design of inverting constant-gain amplifier stage *L Rf, RiD G = 1; Rf = 1 4 ; Ri = Rf G hinv = Rf Ri Out[3757]= 1 Out[3758]= 1 In[3751]:= hbp = hlp * hhp * hinv hbpdb = 2 Log@1, Abs@hbpDD LogLinearPlotA82-3, hbpdb<, 9f, 1, 1 4 =, PlotRange Ø 991, 1 4 =, 82, <=, GridLines Ø AutomaticE Out[3751]= Out[3752]= Â f H Â f pl H Â f pl 2 LogA AbsA f EE H Â f pl H Â f pl Log@1D µ Out[3753]= 1 5 Circuit and PSpice simulation results.

12 #7 HPF - 4th order Butterworth In[4698]:= Clear@c, cp, R, Rp, fc, kf, km, wc, n, wcpd c1 = 2.61; H*Using values from example 15.9*L c2 =.38; c3 = 1.8; c4 =.924; fc = 14; H* Desired cutoff frequency *L wcp = 2 * p * fc; H* Desired Corner Frequencey radês *L n = 1; wc = SqrtA2 1ên - 1E êê NH* Scaled Corner frequency *L; kf = wcp H* Frequency scaling factor *L wc km = 1; c1 c1 = H* Scale example values by kf, km*l km * kf c2 c2 = km * kf c3 c3 = km * kf c4 c4 = km * kf R = 1 H* R is arbitrary *L Out[475]= Out[477]= µ 1-7 Out[478]= µ 1-8 Out[479]= µ 1-7 Out[471]= µ 1-7 Out[4711]= 1 In[4712]:= H* Deterimine Transfer function *L Clear@sD k1 = R * c1 k2 = R * c2 H* TF of first section *L hb1 = s 2 Is 2 + I 2 k1 M s + 1 k1*k2 M k3 = R * c3; k4 = R * c4; H* TF of second section *L s 2 hb2 = Is 2 + I 2 M s + I 1 MM k3 k3*k4 H* Optional gain stage *L hgain = 1; H* TF of complete filter *L hbutt = hb1 * hb2 * hgain Out[4713]= Out[4714]= Out[4715]= Out[4718]= Out[472]= s µ s + s 2 s µ s + s 2 s 4 I µ s + s 2 M I µ s + s 2 M In[4721]:= s = I * 2 * p * f hdb = 2 Log@1, Abs@hbuttDD; H* Transfer function in db *L LogLinearPlotA8-3, hdb<, 9f, 1, 1 4 =, GridLines Ø AutomaticE Out[4721]= 2 Â f p -2 Out[4723]= µ 1 4

13 In[4768]:= H* As an alternative design using c=1uf *L R1 = k1 ë 1-6 R2 = k2 ë 1-6 R3 = k3 ë 1-6 R4 = k4 ë 1-6 Out[4768]= Out[4769]= Out[477]= Out[4771]= 15.42

14 6. Textbook Problem 18.2: Find the z parameters for the circuit shown in Fig. P.18.2.

15 7. Textbook Problem 18.38: The z and y parameters for the resistive two-ports in Fig. P18.38 are given by: Calculate v o if v g is 3mVDC. z z z z z a a a ' ' ' ' ' a 21 a 22 y y y y y 41 a '' '' '' '' '' a 21 a 22 8 a a Using equations (18.74)-(18.77), a11 a a 4 a21 a Using Table 18.2, V VZ g L ( a a Z ) Z a a Z g L g V

EE233 Autumn 2016 Electrical Engineering University of Washington. EE233 HW7 Solution. Nov. 16 th. Due Date: Nov. 23 rd

EE233 Autumn 2016 Electrical Engineering University of Washington. EE233 HW7 Solution. Nov. 16 th. Due Date: Nov. 23 rd EE233 HW7 Solution Nov. 16 th Due Date: Nov. 23 rd 1. Use a 500nF capacitor to design a low pass passive filter with a cutoff frequency of 50 krad/s. (a) Specify the cutoff frequency in hertz. fc c 50000