Lab 3 SPECTRUM ANALYSIS OF THE PERIODIC RECTANGULAR AND TRIANGULAR SIGNALS 3.A. OBJECTIVES 3.B. THEORY

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1 Lab 3 SPECRUM ANALYSIS OF HE PERIODIC RECANGULAR AND RIANGULAR SIGNALS 3.A. OBJECIVES. he spectrum of the periodic rectangular and triangular signals.. he rejection of some harmonics in the spectrum of the periodic signals. 3. he synthesis of the rectangular and triangular periodic signals. 3.B. HEORY Let p(t) be a periodic rectangular pulse signal having the amplitude X 0, the period and the pulseduration, as shown in Figure 3.. p(t) X 0 -/ / =/f t Figure 3. he amplitude spectrum of this signal is shown in Figure 3. and is given by: A = n X0 Sa ( π nf ) where Sa function is defined by: sin( x ) Sa( x ) = x (3.) he spectral lines are constrained by a general spectral envelope (the dashed line in Figure 3.). he envelope is an imaginary line which follows the absolute value of the Sa function. he Sa envelope passes through zero at the frequencies k /, where k : / ( the first zero ), / ( the second zero ), 3/ ( the third zero ), and so on. he distance between two consecutive spectral lines is equal to f 0 (the fundamental frequency of the signal) and does not depend on the pulse-duration. If a zero of the envelope (k/) is at the frequency of the nth harmonic (nf ), then the nth harmonic will be rejected. herefore we can adjust the pulse-duration so that a certain harmonic is rejected. For the general case, to reject the n th harmonic in the k th order zero of the envelope, we must impose the condition: k k k nf = = = (3.) n f n

2 which means that the pulse-duration must be equaled to /nf 0. If the nth harmonic is rejected, then the multiple of n harmonics (nd, 3rd, 4th,...) will be rejected too. A n X 0 A 0 A A A 3 f 0 f 0 3f 0 f 3 Considering the particular case when = Figure 3., (3.) becomes: π An = X0 Sa n (3.3) he amplitudes and phases of the harmonics can be obtained by replacing n with,, 3,... in Error! Reference source not found.. In this particular case, the even harmonics are missing and the odd ones have the next parameters: A = X0 π = X 0 ϕ = 0 A A3 = X 0 = 3π 3 ϕ3 = 80 A A5 = X 0 = 5π 5 ϕ5 = 0 A A7 = X 0 = 7π 7 ϕ7 = 80 (3.4) he dc component (A 0 ) must be calculated as the time-average value of the signal p(t): A0 = p(t ) dt = X0 (3.5) Figure 3.3 shows a periodic triangular signal, centered in the origin, and having the amplitude X 0, the period and the pulse-duration.

3 x(t) X t Figura 3.3 Periodic triangular pulse he amplitude spectrum of the triangular signal is: X = π, (3.6) becomes: 0 An Sa nf Considering the particular case when = X 0 An = Sa n π 4 he dc component (A 0 ) must be calculated as the time-average value of the signal: (3.6) (3.7) ( ) A = x t dt = X (3.8) C. PROBLEMS 3.C.. Consider a periodic rectangular signal of ms period, V amplitude and the pulse-duration of 0.5ms. Determine the dc component, the amplitudes and phases of the first 0 harmonics. Sketch the amplitude spectrum and its envelope. 3.C.. Consider the same signal as at 3.B. and suppose you want to reject the 5 th harmonic in the first zero of the envelope. What must be the pulse-duration of the signal? Sketch the amplitude spectrum in this case. Which other harmonics will be rejected? 3.C.3. What must be the pulse-duration of the signal if we want to reject the 5 th harmonic in the second zero of the envelope? Sketch the amplitude spectrum. Which other harmonics will be rejected? 3.C.4 Consider a periodic triangular signal, centered in the origin, with period ms, amplitude V and pulse-duration 0.5 ms. Determine the dc component, the amplitude and phase of the first 8 harmonics. Represent the amplitude spectrum of the signal. 3

4 3.D. LAB WORK 3.D. We will first study the amplitude spectrum of a periodic rectangular signal, having the amplitude of V, the period of ms and the pulse-duration of 0.5ms. Figure 3.4. Connect the board Analog Discovery to the USB port of the computer and open the application WaveForms 05. he rectangular signal will be generated with WaveGen, and for the displaying use channel of the scope. Connect the yellow wire (W) to the orange wire (+ at the scope). Set the parameters of the signal in the WaveGen window so that it generates the rectangular signal from Figure 3.4 (the pulse width is set using the parameter Symmetry of the signal generator; it is given by the ratio k/n (in percent) in the expression (3.)). Press RUN (at both the generator and the scope) and verify if the generated signal is identical to the one from Figure 3.3. Measure the period and the amplitude of the signal using the cursors (X in the lower left side of the scope). Display the amplitude spectrum using FF from the menu View of the Scope application. Set the following: Start: 0, Stop: 30 khz (the frequency bandwidth), Units: Vpeak, op: V. Notice that the amplitude spectrum contains several spectral lines: the dc component at the 0Hz frequency; the fundamental component at khz (which is the fundamental frequency); several harmonics at frequencies multiple of the fundamental frequency. Notice that the spectrum contains only the odd harmonics ( st, 3 rd, 5 th, ); the even harmonics are missing (their amplitude is zero). Adjust the frequency axis to (0 0kHz) from the menu above the spectrum; then measure (with the cursor) and write down the amplitudes and frequencies of the dc component and of the st, 3 rd, 5 th, 7 th and 9 th harmonics (these values will be used at part 3.D.4!). 3.D.. Go back to WaveGen and modify the pulse width to 0.5 ms (modify the corresponding value for the parameter Symmetry). Press RUN and adjust the frequency bandwidth to 0 0 khz. Notice that the spectrum has now both even and odd harmonics. Which harmonics are missing? Where are the zeros of the envelope? 3.D.3. Modify now the pulse-duration in order to reject the 5 th harmonic in the first zero of the envelope. Run the simulation and display the spectrum. Which harmonics are missing now? Where are the zeros of the envelope? 4

5 Go back to the signal generator and modify the pulse width for the rejection of the 5 th harmonic in the second zero of the envelope. Which harmonics are missing now? Where are the zeros of the envelope now? 3.D.4. Now we will generate a signal that approximates the rectangular signal generated at 3.D.. We will use the harmonic form of the Fourier series. Hence we can write the rectangular signal as follows: x(t ) = A0 + An cos nf0t + n n= ( π ϕ ) he sum in Error! Reference source not found. has an infinite number of terms and we will truncate it to 9 terms by considering the signal x (t), which is an approximation of x(t): = + 9 x (t ) A0 An cos nf0t + n n= ( π ϕ ) (3.0) You will generate each term of the sum (3.0) (each cosine) with the WaveGen. he even harmonics being zero, you will have to generate only the odd harmonics (the st, 3 rd, 5 th, 7 th and 9 th harmonics), which are the harmonics that you measured at 3.D.. Generate again the rectangular signal from part 3.D., on the channel (W) of the WaveGen ; then connect output W (yellow) to channel of the scope (orange wire). In the menu of channel (W) select Custom (instead of Simple), then press New and select Math. In the left window, write the expression of the signal given by (3.0), with the values measured at 3.D. (the phases must be in radians, according to (3.4). he dc value will be introduced by setting the parameter Offset of the custom signal to the value measured at 3.D.. hen press Generate and the signal will be displayed (Custom). he approximation of the rectangular signal will be displayed on the Scope by connecting W (yellow-white wire) to channel of the scope (blue wire). What is the period of the approximated signal? Select Measurements from the View menu and display the period, the frequency and the mean value for both the rectangular and the approximated signals. In order to display the spectrum, select FF from the View menu. Notice that the first 9 spectral lines of the approximated signal are identical to those of the rectangular signal; the rest of the harmonics are zero. 3.D.5. Next, we will study the amplitude spectrum of a periodic triangular signal having the amplitude of V, the period of ms and the pulse-duration of 0.5ms. (Figure 3.5). Generate this signal on channel (W) of the wavegenerator, select Custom (instead of Simple) then, press New and select the Func window. Here you will edit the signal according to the data from able. Save and close the Custom window, and then set the parameters amplitude, offset and phase so as to generate exactly the signal from Figure 3.5. (3.9) 5

6 able. he parameters of the triangular signal Interval Signal ype Symmetry Amplitude Offset Phase 0% - 5% RampUp 00 % 00 % 0% 0 0 5% - 50% RampDown 0 % 00 % 0% % - 00% DC 0 % 0% -00 % 0 0 Press RUN and check if the wanted signal from Figure 3.5 is generated correctly. Measure the period and the amplitude of the generated signal using the cursors. Measure the average, period and the frequency of the signal using the option Measurements of the Scope, from the View menu. Display the amplitude spectrum using FF from the menu View of the Scope application. Set the following: Start: 0, Stop: 30 khz (the frequency bandwidth), Units: Vpeak, op: V. Which harmonics are missing? Where are the zeros of the envelope? Change the frequency bandwidth Start: 0, Stop: 0 khz, then using the cursors measure the dc component and the amplitude of the harmonics,, 3, 5, 6 and 7. 3.D.6. Now we will generate a signal that approximates the triangular signal generated at 3.D.5. also using the expression from (3.9). On channel (W) of the Wavegen keep the triangular signal generated at point 3.D.5, and on channel (W) select Custom (instead of Simple), then press New and select Math. In the left border of the window write the expression from (3.9), using the results from the previous measurements. his time use the cosine function directly. he approximating signal will be plotted on channel (W, yellow-white wire) of the Scope. What is the period of the approximated signal? From the View menu select Measurements and plot the values for the period, frequency and the average both of the original and the approximated signals. From the View menu select FF and plot the amplitude spectrum both of the original and the approximated signals. Observe the spectrum of the approximated signal (consisting of 6 harmonics) overlaps with good accuracy over the original signal s spectrum, in the rest the spectrum is zero. x(t) t[ms] Figura 3.5 Periodic triangular signal 6