Theory & Introduction Lab #9: AC Steady State Analysis Goals for Lab #9 The main goal for lab 9 is to make the students familar with AC steady state analysis, db scale and the NI ELVIS frequency analyzer. Also important is that students understand critical damping in the frequency domain. Theory Figure 9.1 Bode Diagrams: Figure 9.2 The Bode diagram, as seen in Figure 9.1, completely characterizes a linear system. You may recall that for linear systems, superposition holds. More information on what constitutes a linear system will be covered in future engineering courses. The Bode diagram shows how the system will change the magnitude and phase of pure sinusoidal inputs. For example Figure 9.1 is the Bode plot of the 2 nd order low pass RLC circuit shown in Figure 9.2.
At very low frequencies, the magnitude of the output is very close to 0dB (unity gain), and phase shift is almost zero. This means low frequency signals pass this filter without change. At high frequencies, the magnitude drops and the phase is shifted 180 degrees which means the signal is totally attenuated. At the maximum of the magnitude curve, a resonance is clearly indicated at the resonance frequency, where the phase shift is 90 degrees. Consult Chapter 14 in your text if you wish to explore more details on this. However, our brief description in this section is sufficient to complete this lab. Decibels and the db scale: In electronic circuits, voltages and currents can be amplified or attenuated. The values encountered for such factors in circuits can vary over orders of magnitude. It is thus convenient to use a logarithmic measure for them. Let X be a voltage or current amplification or attenuation factor. A widely used logarithmic measure is: X db = 20log X The values of X db are expressed in units called decibels, or db. For example, if X=0.1, X db =20log 10 0.1=-20dB. If X>1 then the decibel value should be positive. The table below gives X db for a variety of X values. However, you should be able to calculate the decibel value of amplification or attenuation. (We also call this the gain.) X XdB 1 0dB 2 3dB 1 / 2-3dB 2 or 1/2 +6 or -6dB 10 or 0.1 +20 or -20dB 100 or 0.01 +40 or -40dB Since X db is a log function, we usually plot it on a log scale. For our purposes, this means that we equally space powers of 10 in the x-axis. We call the equally spaced steps on this scale a decade. 2nd order state variable active filter: The state variable active filter is shown in Figure 9.3; it is similar to the bi-quadratic filter we used in Lab #8. When we say filter, we mean a circuit that will allow signals of certain frequencies to either been amplified or to pass through it. The filter will reject some frequencies. To a certain extent, we can control the frequencies and the gain associated with those frequencies by selecting the filter components.
Figure 9.3 Our active filter consists of a summing node and two identical lossless integrators. From the outputs of the three op-amps, we can obtain high pass, band pass and low pass characteristics with respect to the same input signal. For simplicity, we take R R = R = R 1 = 2 3, R 4 = R 5 = R freq, C 1 = C 2 = C. The Q of the filter is determined by the ratio between RA and RB: RA + RB Q = 3RB Therefore the central frequency f = 1 0 2πR freqc, Maximum gain of the LP, HP and BP outputs are R3 R R 2 A + RB A0, LP = = 1, A 1 R 0, HP = = A0, BP = = Q.(When R1=R2=R3) R 3R 1 1 What is the maximum gain in db? Be familiar with how to switch back and forth between nominal gain and the gain in db. B
Prelab Read the purpose, principles, and procedures sections of Lab 9. Be prepared for your lab quiz which may cover this lab or any previous lab completed this semester. 1. Perform an AC steady state analysis for the RC low pass circuit shown in Figure 9.4. Be sure to print all relevant graphs to turn in to you TA. If you have questions, consult your TA. Figure 9.4 Note: The following steps will help you perform a AC steady state analysis in PSPICE. It is also good to perform the analysis by hand before starting your PSPICE simulation. a) Put a "VAC" as the input source, Set "ACMAG"=1V as shown in Figure 9.5. Figure 9.5 b) In the 1 st order RC low-pass circuit, double click the output node; label it as Vout in the pop-up window. c) In the analysis setup dialog, select "AC Sweep" as shown in Figure 9.6: Figure 9.6
d). Enter the parameters as shown in Figure 9.7: Figure 9.7 e) Start the simulation. The result window will be empty for the first time because you have not yet selected traces to display. You need to manually add the "Vout" trace and set it to the db scale: i. Click "Add Trace" on the toolbar or press "Insert" on the keyboard. See Figure 9.8. Figure 9.8 ii. In the pop-up dialog box, which is shown in Figure 9.9 on the next page, click "DB()" on the right and then V(Vout) on the left until DB(V(vout)) shows up in the "Trace Expression" on the bottom.
f) Measure the roll-off slope: Figure 9.9 Figure 9.10 Use the cursor function, whose button is circled in the Figure 9.10. Set two cursors one decade apart, say 100k and 10k, or 50k and 5k, note down the Y difference as circled. For 1 st order RC, this roll-off slope should be about 20dB/decade. Print your schematic and the graph with the slope shown and hand it in to your TA.
2. Repeat for RC high pass filter below in Figure 9.11. Be sure to print all relevant graphs to turn in to your TA. If you have questions, consult with your TA. Figure 9.11 3. Perform AC analysis for the state variable active filter at the HP, BP, and LP outputs nodes from 10Hz to 100kHz as shown in Figure 9.12. Measure the roll-off slopes from all three output signals. Place all three traces on one graph if you are able Again, submit all graphs for grading. Figure 9.12 4. Change R A and R B values yet keep R A +R B constant, (they will be replaced with a potentiometer in the lab) to observe the changes with different Q. Refer to the Q equation RA + RB Q =. For example R B =0.5k, R A =1.5k will give a Q of about 1.33. Capture AC 3RB response curves from the LP output node with Q=0.5,1 and 5. 5. Keeping R A +R B =10kOhms, calculate the R B value for critical damping Q=0.707.
Procedure 3 741 OPAMPs 5 1kΩ resistors 2 1.6kΩ resistors 2 0.1uF capacitors Task #1 Low pass RC Filter Using R=1.6kΩ and C=0.1uF, construct the circuit in Figure 9.13 on the NI ELVIS board. Be sure to use the ground on the board correctly. You will be adding to this circuit in Task #2. Generate from the HP function generator a 2Volt peak to ppeak, 500Hz sinusoid signal for the Vin. Remember to set the function generator to High Z. Monitor both Vin and Vout through CH1 and CH2 of the Tectronix oscilloscope. The two waveforms should look similar to Figure 9.14. Measure the phase difference between the two traces by using the functions associated with the "measure" button. Also find the attenuation factor between Vin p-p and Vout p-p, convert it to db scale. Record the results in Table 1. Save the graph and include it in your lab report. Gradually increase the frequency to 2kHz. The waveform should look similar to Figure 9.15. Note down the phase difference and the amplitude change. Measure the attenuation factor in db as well. Comment the change of phase and amplitude of the output signal. Save the graph and include it in your lab report. Figure 9.13 Figure 9.14
Figure 9.15 Task #2 Using the Bode Analyzer The Bode analyzer is a very useful tool provided in the NI ELVIS system. It generates sinusoid input signal at variable frequency but constant amplitude, then compares output signal with the input, finding the change in both phase and amplitude. By this way we can find the AC steady state characteristics of the circuit under test. You will use the same circuit you did in Task #1. Leave the connections to the Tectronix Oscilloscope in place. Simple add to the circuit using the following steps: a) Connect the ELVIS function generator output to ACH0+, name this node Vin; this node will provide the variable frequency input signal, whose amplitude is sensed through the ACH0 channel. b) Name the ACH1+ as Vout, this is the node where the Bode analyzer measures the output of our circuit under test. c) Connect ACH0-, ACH1- to the ground; d) Connect the Vin, Vout and ground of the RC circuit in Figure 9.13 to the Bode analyzer, with R=1.6k, C=0.1uF. Your Bode analyzer is now ready to work. e) Invoke the Bode analyzer from the ELVIS Launcher. Set the start/stop frequency from 10Hz to 20 khz, set 10 points per decade, set input amplitude 2Vpp, set the y axis ranges as default at this time.
Figure 9.16 f) Click Run button to start analysis. From the oscilloscope (by using auto scale) you can see the input signal is a sinusoid with increasing frequency. The amplitude and phase difference are then be measured by the ELVIS system and plotted on the graph. After the analysis finishes, save the log files of every trace for the lab report. You can save them as files that can be imported into Excel. Save the both the graph files and the data to be used in the lab report write-up. Task #3: State Variable Filter Build the state variable filter in Figure 9.17, connect it to the bode analyzer, with the same settings above, test its HP, BP and LP outputs individually, save log files to every trace. Figure 9.17
Task #4: Continued Analysis of the State Variable Filter: (Optional) Replace R A and R B with a 10kohm potentiometer. Adjust the circuit to critical damping. (Hint: recall the knowledge you learned in previous lab). Test the circuit with the frequency analyzer; note the shape of the AC curves around the f 0. Remove the potentiometer from the circuit, measure the resistance of the equivalent R A and R B, compare to your calculations and simulations in the pre-lab. Lab Report Requirements Be sure to include all normal requirements for the Lab Report. See your TA if you have any questions. Also include the following: 1. In Excel, import the saved log files as comma delimited text, reproduce the Bode plots in your lab report. For every roll-off slope, find two points that are 1 decade away from each other, for example 2khz and 20kHz for high frequency part, or 20 to 200 Hz for lower frequency part. Find the differences between the two marks in db. Note down the slopes in the following table. 2. Comment the difference between 1 st and 2 nd order systems. Do 2 nd order filters have a steeper or shallower slope as the gain decreases? For more information on this, consult Chapter 14 in your text. 1 st order low pass 2 nd order low pass 2 nd order high pass 2 nd order band pass lower side 2 nd order band pass higher side 3. Reproduce tables and comments in the report. Explain the theory learned from the lecture as it is reflected in the work done in the lab.
Tables and Results Table 1. Task 1: Frequency Vin p-p Vout p-p Phase (Vin->Vout) db attenuation 500Hz 2000Hz Comments to Task 1: Comments to Task 4: