Solutions to Problems in Chapter 4

Transcription

1 Solutions to Problems in Chapter 4 Problems with Solutions Problem 4. Fourier Series of the Output Voltage of an Ideal Full-Wave Diode Bridge Rectifier he nonlinear circuit in Figure 4. is a full-wave rectifier. It is often used as a first stage of a power supply to generate a constant voltage from the 6Hz sinusoidal line voltage for all inds of electronic devices. Here the input voltage is not sinusoidal. Figure 4.: Full-wave rectifier circuit he voltages are vin () t δ ( t ) A t u( t + ) u( t ), and vt () vin() t. Let be the fundamental period of the rectified voltage signal vt ( ) and let ω π be its fundamental frequency. Find the fundamental period. Setch the input and output voltages vin (), t v() t.

2 Let us first setch the input voltage, which is the periodic sawtooth signal shown in Figure 4.. hen the output voltage is simply the absolute value of the input signal which results in a triangular wave as shown in Figure 4.3. Figure 4.: Input voltage of full-wave rectifier Figure 4.3: Output voltage of full-wave rectifier We can see that the fundamental period of the output signal is the same as that of the input:. (b) Compute the Fourier series coefficients of vin ( t ). Setch the spectrum for A.

3 he DC component of the input is obviously : a tdt t / A A / A / / 4 4 for : / A jωt a te dt / / ja / jωt jωt ( te ) e dt π / / ja e e π ja cos π π jπ jπ + ja ( ) π he spectrum is imaginary, so we can have a single plot to represent it as in Figure 4.4. Figure 4.4: Line spectrum of input voltage, case A 3

4 (c) Compute the Fourier series coefficients of vt ( ). Write vt ( ) as a Fourier series. Setch the spectrum for A. Let us first compute the DC component of the output signal: / / A A A / 4 4 a t dt tdt A 8. For, the spectral coefficients are computed as follows: / A jωt a t e dt / / A jωt jωt te dt te dt / / A ( jωt jωt te e ) dt + / A t cos( ωt ) dt A π / ( ω ) tcos( ω t) dt / A / ( tsin( ωt) ) sin( ω t) dt π A sin π cos( ω t) π + ω A ( cos π ) ( π ) (( ) ) A ( π ) / 4

5 he spectrum of the triangular wave is real and even (because the signal is real and even), so we can have a single plot to represent it as in Figure 4.5. hus, Figure 4.5: Line spectrum of output voltage, case A jωt (( ) ) e A vt () ( π ) is the Fourier series expansion of the full-wave rectified voltage. (d) What is the average power of the input voltage vin( t) at frequencies higher than or equal to its fundamental frequency? Same question for the output voltage vt ( )? Discuss the difference in power. 5

6 Since a, and given the Parseval heorem, the average power of the input voltage vin( t) at frequencies higher than or equal to its fundamental frequency is equal to its total average power computed in the time domain: in / 4A / P a t dt 4A 3 / 3 t 3 / 4A A he average power in all harmonic components of vt ( ) excluding the DC component (call it P out ) is computed as follows: out / 4A / P a a t dt a A A A 3 4 Note that the input and output signals have the same total average power, but some of the power in the input voltage (namely A 4 ) was transferred over to DC by the nonlinear circuit. Problem 4. Given periodic signals with spectra x() t FS a and yt () b, show the following properties: FS FS (a) Multiplication: x() t y() t ab l l l 6

7 jωt jnωt n n xt () yt () ae be n p abe ab n p FS coeff. cp j( + n) ωt e ab p p jpω t e jpωt FS herefore, x( t) y( t) ab l l l. (b) Periodic convolution: x( τ) yt ( τ) dτ ab FS jωτ jpω( t τ) p p x( τ) y( t τ) dτ a e b e dτ jωτ jpωτ jpωt p p p p p p ae be e dτ abe abe jωt dτ e j( p) ωτ jpωt ab e dτ e j( p) ωτ jpωt herefore, x( τ) yt ( τ) dτ ab FS 7

8 Problem 4.3 Fourier Series of the Output of an LI System. t Consider the familiar rectangular waveform x( t) of period and duty cycle η. his signal is the input to an LI system with impulse response 5t ht () e sin( π tut ) (). Figure 4.6: Rectangular wave input to LI system (a) Find the frequency response H( jω) of the LI system. Give expressions for its magnitude H( jω ) and phase H( jω) as functions of ω. he frequency response of the system is given by 8

9 5t jωt H( jω) e sin( πt) e dt jπt j πt (5 + jω) t e e e dt ( ) j (5 + j( ω π)) t (5 + j( ω+ π)) t ( ) e e dt j j(5 + j( ω π)) j(5 + j( ω + π)) (5 + j( ω + π)) (5 + j( ω π)) j(5 + j( ω π))(5 + j( ω + π)) π 5 ω + π + jω π Magnitude: H( jω).5 (5 ω + π ) + ω ω Phase: H( jω ) arctan 5 ω + π (b) Find the Fourier series coefficients a of the input voltage x() t for s and a 6% duty cycle. he period given corresponds to a signal frequency of Hz, and the 6% duty cycle means that 3 η so that the spectral coefficients of the rectangular wave are given by a sinc( ). 5 5 (c) Compute the Fourier series coefficients b of the output signal y( t) (for the input described in (b) above), and setch its power spectrum. 9

10 π b H( jω ) a sinc( ) 5 ( ) ω + π + j ω 3 6π sinc( 5 ) 5 ( π) + π + jπ he power spectrum of the output signal is given by the expression below and shown in Figure 4.7. b 36π sinc ( ) ( π) + π + 4π Figure 4.7: Power spectrum of output signal (d) Using Matlab, plot an approximation to the output signal over three periods by summing the first harmonics of y( t), i.e., by plotting + π j t yt () be.

11 yt () + be j π t π sinc( ) e 5 ( π ) + π + j π π π b + t+ 5 ( π) + π + 4π j π t 6 3 5sinc( 5 ) cos π arctan 5 ( π ) + π 6π π sinc( ) π π cos π t+ arctan 5 ( π) + π + 4π 5 ( π ) + π Figure 4.8 shows a plot of ~ yt () from s to 3s. Figure 4.8: Approximation of output signal using truncated -harmonic Fourier series

12 Problem 4.4 Digital Sine Wave Generator A programmable digital signal generator generates a sinusoidal waveform by filtering the staircase approximation to a sine wave shown in Figure 4.9. Figure 4.9: Staircase approximation to a sinusoidal wave in Problem 4.4. (a) Find the Fourier series coefficients a of the periodic signal x( t). Show that the even harmonics vanish. Express x( t) as a Fourier series. First of all, the average over one period is, so a. For,

13 π j t a x() t e dt 3 π 6 π π j t j t j t A A A e dt e dt e dt 3 6 π 3 π 6 π j t j t j t A A A + e dt e dt e dt π π π π j t j t A j t j t 3 π π A A j t j t e e dt + e e dt + e e dt ja π ja π ja π sin t dt sin t dt sin t dt ja π ja π ja π cos t + cos t + cos t π π π ja π π π π cos cos cos cos( π ) cos π ja π π cos cos cos π ( π ) 6 3 Note that the coefficients are purely imaginary, which is consistent with our real, odd signal. he even spectral coefficients are for m, m,, : ja π 4π a am cos m cos m cos m π m ja π π cos m mπ cos m mπ π m ja π π cos m cos m π m 3 3 ( π ) 3

14 Figure 4. shows a plot of x( t ) computed using 5 harmonics in the Matlab script Fourierseries.m which can be found in the Chapter 4 folder on the CD-ROM. Figure 4.: runcated Fourier series approximation to staircase signal and first harmonic in Problem 4.4(a). he Fourier series representation of x() t is π π ja π π xt ( ) ae cos + cos + cos e π 3 3 j t j t ( π ). (b) Write x( t) using the real form of the Fourier series. xt ( ) a + [ B cos( ω t) C sin( ω t)] Recall that the C coefficients are the imaginary parts of the a 's. Hence A π π x( t) cos cos cos( π) sin( ωt) π

15 (c) Design an ideal lowpass filter that will produce the perfect sinusoidal waveform π yt () sin t at its output with x( t ) as its input. Setch its frequency response and specify its gain K and cutoff frequency ω c. he frequency response of the lowpass filter is shown in Figure 4.. Figure 4.: Frequency response of lowpass filter 3π he cutoff should be between the fundamental and the second harmonic, say ω c. he gain should be: K π π π cos cos A π π A 3A. (d) Now suppose that the first-order lowpass filter whose differential equation is given below is used to filter x( t ). dy() t τ + yt () Bxt () dt 5

16 where the time constant is chosen to be τ. Give the Fourier series representation of the π output yt ( ). Compute the total average power in the fundamental components P tot and in the third harmonic components P 3tot. Find the value of the DC gain B such that the output wt () produced by the fundamental harmonic of the real Fourier series of x( t ) has unit amplitude. H() s B τ s + B H( jω) τ jω + π π j t j πt B ja π π ( π ) j + π 3 3 yt ( ) ah( j ) e cos + cos + cos e Power: P tot B ja π π cos + cos + cos j + π 3 3 B j3a 9 AB j + π 4π ( π ) P 3tot B ja cos + cos + cos 3 j3+ π ( π) ( π) ( π) B ja 4A B A B [ ] j3+ π 4π 5π 6

17 For the filter's DC gain B, we found that the gain at ω should be: π B H( jω ) 3A τ jω + ( τω ) + π B ( τω ) 3A + B Exercises Problem 4.5 he output voltage of a half-wave rectifier is given by: vin (), t vin () t > vt ()., vin ( t) Suppose that the periodic input voltage signal is vin () t Asin( ωt), ω π. Find the fundamental period and the fundamental frequency ω of the half-wave rectified voltage signal vt ( ). Compute the Fourier series coefficients of vt ( ) and write the voltage as a Fourier series. he fundamental period of vt ( ) is and the fundamental frequency is ω ω. jωt A jωt A jωt jωt jωt a vte () dt sin( te ) dt ( e e ) e dt ω j π j( ) π j( + ) π A ( ) ( ) Aω e e e e dt + ω jω t jω + t ( ) j 4 π j j ω( ) jω( + ) jπ jπ jπ A e e A ( e + ) + 4 π ( ) ( + ) 4π jπ A ( e + ) A ( ) + π π 7

18 Note that we have to mae sure that this expression is finite for ± (L'Hopital's rule) a a d ( e jπ ) j A + d π A jπ e ja π d π 4 ( ) d jπ A jπ e ja π 4 Notice that these are imaginary whereas all the other coefficients are real! A ( ) + jωt hus, vt () e π is the Fourier series expansion of the half-wave rectified sinusoid. Problem 4.6 Suppose that the voltages in the full-wave bridge rectifier circuit of Figure 4. are v () t Asin( ω t), ω π, and vt () v () t. Let be the fundamental period of the in in rectified voltage signal vt ( ) and let ω π be its fundamental frequency. (a) Compute the Fourier series coefficients of vt ( ) and write vt ( ) as a Fourier series. (b) Express vt ( ) as a real Fourier series of the form vt ( ) a + [ Bcos( ω t) C sin( ω t)] 8

19 Problem 4.7 Fourier Series of a rain of RF Pulses Consider the following signal x( t) of fundamental frequency ω π, a periodic train of radio frequency (RF) pulses. Over one period from to, the signal is given by: xt () R S Acos( ω ct), < t <, < t <, < t < his signal could be used to test a transmitter-receiver radio communication system. Assume that the pulse frequency is an integer multiple of the signal frequency, i.e., ω Nω. Compute the Fourier series coefficients of x(). t c jωt A jωt () cos( ω ) a x t e dt N t e dt A jnωt jnωt jωt A jω( N ) t jω( N+ ) t ( e e ) e dt ( e e ) dt + ω ω ω ω A e e e e + jω( N ) jω( N + ) j ( N ) j ( N ) j ( N+ ) j ( N+ ) A sin[ ω( N ) ] sin[ ω( N + ) ] + ω( N ) ω( N + ) A { sinc[ f ( N )] + sinc[ f ( N+ )] } A { sinc[ ( fc f)] + sinc[ ( fc + f)] } A { sinc[ ( f fc)] + sinc[ ( f + fc)] } 9

20 Problem 4.8 (a) Compute and setch (magnitude and phase) the Fourier series coefficients of the sawtooth signal of Figure 4.. Figure 4.: Periodic sawtooth signal in Problem 4.8(a). (b) Express x( t ) as its real Fourier series of the form: x() t a + [ B cos( ω t) C sin( ω t)] (c) Use MALAB to plot, superimposed on the same figure, approximations to the signal over two periods by summing the first 5, and the first 5 harmonic components of x( t ), i.e., by plotting N xt () ae N π j t. Discuss your results. (d) he sawtooth signal x( t ) is the input to an LI system with impulse response t ht () e sin( π tut ) (). Let yt ( ) denote the resulting periodic output. Find the frequency response H( jω ) of the LI system. Give expressions for its magnitude H( jω ) and phase H( jω) as functions of ω. Find the Fourier series coefficients b of the output yt ( ). Use your

21 computer program of (c) to plot an approximation to the output signal over two periods by summing the first 5 harmonic components of yt ( ). Discuss your results. Problem 4.9 Consider the sawtooth signal yt ( ) in Figure 4.3. Figure 4.3: Periodic sawtooth signal in Problem 4.9. (a) Compute the Fourier series coefficients of yt ( ) using a direct calculation. he Fourier series coefficients are: a x() t dt For

22 j πt j πt a xte () dt ( ) te dt j πt j πt ( te ) e dt jπ jπ e jπ + ( jπ) j π ( π) j π j πt e e j π j π (b) Compute the Fourier series coefficients of yt ( ) using properties of Fourier series, your result of Problem 4.8(a) for x( t ), and the fact that yt ( ) xt ( ) x( t). We can use the following property: x( t) a. hus, the Fourier series coefficients b of yt ( ) can be obtained as follows. j ( ) j ( ) b a a + π ( π) π ( π) j j π π Problem 4. Compute and setch (magnitude and phase) the Fourier series coefficients of the following signals: (a) Signal x( t ) shown in Figure 4.4.

23 Figure 4.4: Periodic rectangular signal in Problem 4.(a). (b) x( t) sin( πt) + cos( πt) 3