Frequency Response Lab: Constructing Bode Plots for First and Second Order Systems, Summaries of Matlab skills

Download Frequency Response Lab: Constructing Bode Plots for First and Second Order Systems and more Summaries Matlab skills in PDF only on Docsity! ECE-205 Lab 9 Frequency Response and Experimental Bode Plot Construction Overview In this lab you will build and measure the frequency response of two circuits and then use Matlab to construct a Bode plot of each system. For many unknown (or complex) systems this is a very common method for determining the transfer function. You will also need to modify a somewhat complex Matlab function. You only need to make a few changes, but often one learns to program by modifying code that already does something close to what you want it to do. You will need to download two Matlab files from the class website. PART I: Frequency Response of a First Order System 1) Using a bread board construct the following circuit using a 1 yf capacitor and a 1 kQ resistor. These components should be available in your lab kit. We will measure the output of the circuit as the voltage across the capacitor. For this simple circuit the transfer function is of the form H(s)= ts+1 where K is the static gain (it is one for this model), and 7 is the time constant (and is equal to RC in this case). R WOE v(t) For this circuit we then expect the time constant to be around | millisecond, t =0.001. For this first order 1 1 2Qat 220.001) The cutoff frequency is the frequency at which the power in the signal has been reduced to one half of its system we then expect the corner or cutoff frequency to be at @= 27 f = L ,or f= © 159Hz. T 1 maximum value, or equivalently, the magnitude of the transfer function has decreased to E of its maximum value. For this first order circuit, the maximum value of the transfer function is at @ =0, so the cutoff 1 frequency is the frequency @, at which | H(j@,) B | H(0) |. Note that this is also the 3-dB point of the system. 2) Connect the Mobil Studio board to your circuit, using the signal generator as the input voltage. The input will initially be a 20 Hz sine wave with a 2.0 volt peak to peak voltage (an amplitude of 1.0 volt) with 0 DC offset. Set the coupling to DC coupling. Measure the input voltage as the difference between Al+ and A1- (never trust a signal generator!), and measure the output voltage as the difference between A2+ and A2-. You will need to use the cursors for this lab, so be sure to turn them on. Set the time scale so you see one or two periods of the input and output sine waves. 3) For an input x(t) = Acos(@,t) we know the steady state output wil be 9.) = Al HG@,))| cos(@,t+ ZH (ja,)) Hence if we measure the input and output amplitude at each input frequency @, , it should be pretty easy to determine | H(j@, )|at each of these frequencies. To determine the phase, we will look at the time delay between the input and output signals at each input frequency, as follows, cos(@, (t-t,)) = cos(@,t — @,t,) = cos(@,t + ZH (ja, )) Note that in this relationship the phase of the transfer function is measured in radians, ZH (j@,)=—@,t, We can then convert the phase from radians to degrees using 180 ~360 f, xt, NT ZH (ja,) (in degrees) = —27 f, t, +| Here is our (initial) procedure then: Choose a frequency for the input signal (keep the input amplitude at 1 volt, or 2 volts peak to peak) Using the cursors, measure the input amplitude (not peak to peak) Using the cursors, measure the output amplitude (not peak to peak) Using the cursors, measure the time of the peak of the input signal Using the cursors, measure the time of the next peak of the output signal. This must be the peak immediately following the peak of the input signal since we are measuring the time delay between these signals. 3) Rather than recording this and then typing it into Matlab, we will record our data directly into the Matlab program process_data_a.m. You should open this file and fill in the matrix measured as you go along. Each row in this matrix contains the data for a specific input frequency. The columns contain the data Frequency (in Hz) Amplitude of the input signal (in mV) Amplitude of the output signal (in mV) Time of the peak of the input signal (in milliseconds) Time of the subsequent output peak (in milliseconds) Put a space between each entry, and end each row (except the last row) with a semicolon (;). Note that this function modifies the contents of each column to produce the data we want to use. 4) For your initial attempt (you will add more points later), collect the appropriate data for input frequencies 20, 60, 100, 140, 180, 220, 280, 320, 520, 720, 920, and 1120 Hz. 5) Save the program process_data_a.m. This program returns an array to the calling program. In the Matlab window, type data = process_data_a; N For system A, whose frequency response is displayed in the Bode plot in Figure 1, you should be able to verify the following steady state outputs for the given steady state inputs. /f you cannot, please ask for help! x(t) = 3cos(10t) > y,, (t) # 150 cos(1 0 - 100”) x(t) = 5sin(40r +50°) — y,, (2) ® 1.6 sin(40r— 130°) f So \ YY Ie} oS / Magnitude (dB) oO T } 10° 10' 10 Frequency (rad/sec) 50 a So -100 Phase (deg) -150 nN -200 7 2 10 10 10 Frequency (rad/sec) Figure 1: Frequency response of system A For system B, whose frequency response is displayed in the Bode plot in Figure 2, you should be able to verify the following steady state outputs for the given steady state inputs. Jf you cannot, please ask for help! x(t) = 50sin(10f) > y,, ()* 0.63 sin(10r + 40°) x(t) = 100cos(200t + 35°) > y,. (t) ® 4cos(200r — 5’) Magnitude (dB) ” 10" 10° 10° 10° Frequency (rad/sec) 50 IN BR -50 3 ee oO — 8 E -100 -150 200 : 10° 10' 10° 10° 10 Frequency (rad/sec) Figure 2: Frequency response of system B