Revision: April 18, E Main Suite D Pullman, WA (509) Voice and Fax

Transcription

1 Lab 9: Steady-state sinusoidal response and phasors Revision: April 18, E Main Suite D Pullman, WA (509) Voie and Fax Overview In this lab assignment, we will be onerned with the steady-state response of eletrial iruits to sinusoidal inputs. Figure 1(a) shows a blok-diagram representation of the system. The input is a osine funtion with amplitude A and phase angle. The output is a osine funtion with amplitude B and phase angle. Both the input and output waveforms have radian frequeny (reall that an important property of linear systems is that the steady-state response of a linear system to a sinusoidal input is a sinusoid with the same frequeny as the input sinusoid). The analysis of the iruit of Figure 1(a) an be simplified by representing the sinusoidal signals as phasors. The phasors provide the amplitude and phase information of the sinusoidal input and output signals. The input-output relationship governing the iruit then redues to a relationship between the output and input signal amplitudes and the output and input signal phases. The iruit an thus be represented in phasor form as an amplitude gain between the output and input signals and a phase differene between the output and input signals, as shown in Figure 1(b). Input u(t)=aos(t+) Ciruit Output y(t)=bos(t+) (a) Physial iruit Input U=AÐ B Ð ) A Output Y=BÐ (b) Phasor representation of iruit input-output relationship. Figure 1. Steady-state sinusoidal iruit analysis In this lab assignment, we will measure the gain and phase responses of several eletrial iruits and ompare these measurements with expetations based on analysis. Before beginning this lab, you should be able to: After ompleting this lab, you should be able to: Represent sinusoidal signals in phasor form (EA-???) Represent eletrial iruit steady-state sinusoidal responses in phasor form (EA-???) Measure phasor form of iruit steady-state sinusoidal response Measure input impedane of eletrial iruit Do: XXX-YYY page 1 of 9

2 This lab exerise requires: Digilent Analog Parts Kit Digilent EE board Symbol Key: Demonstrate iruit operation to teahing assistant; teahing assistant should initial lab notebook and grade sheet, indiating that iruit operation is aeptable. Analysis; inlude priniple results of analysis in laboratory report. Numerial simulation (using PSPICE or MATLAB as indiated); inlude results of MATLAB numerial analysis and/or simulation in laboratory report. Reord data in your lab notebook. I. Passive RL Ciruit General Disussion: Consider the RL iruit shown in Figure 2 below. The input to the iruit is an applied voltage and we hoose the urrent supplied by the soure to be the system output. The differential equation relating the applied voltage v (t) to the input urrent i (t) an be obtained by applying KVL around the single loop: v di ( t ) ( t ) Ri ( t ) L (1) dt If we assume that the input voltage and urrent are omplex exponentials of the form: v i ( t ) ( t ) j( t ) Ve (2) j( t ) Ie (3) We an write the iruit s input-output relation as a ratio between the urrent and the voltage: I V Ie Ve j j 1 R jl (4) Where I and V are phasors representing the magnitude and phase of the input urrent and input voltage to the iruit, respetively. This input-output relation an be written in terms of an amplitude gain and a phase shift: page 2 of 9

3 I 1 1 V R jl R L (5) 1 L tan R (6) i (t) + v (t) - + v R (t) - R L + v L (t) - Pre-lab: Figure 2. RL iruit. (a) Starting with equations (1), (2), and (3), show that the amplitude gain and phase differene between the input voltage and the input urrent are as shown in equations (5) and (6). R (b) The utoff frequeny for the iruit of Figure 2 is given to be (more on this topi in later L modules). Determine the gain as defined by equation (5) of the RL iruit for frequenies 0,, and. From equation (6), determine the phase ( ) of the RL iruit for frequenies 0,, and. (Note: your answers will, in general, be funtions of R and L.) What is the relationship between the DC gain and the gain at the utoff frequeny? () Determine the gain and phase relationships between the indutor voltage v L (t) and the indutor urrent i (t). What is the voltage-to-urrent gain of the indutor at low and high frequenies? What is the phase differene between indutor voltage and urrent at low and high frequenies? (d) We saw previously that an indutor s steady-state response to a onstant input is like a short iruit and that indutor urrents annot hange instantaneously. Do these indutor harateristis agree with your indutor gain and phase harateristis at low and high frequenies as determined in part (b) of the pre-lab? (e) Interpret your low and high frequeny gain results in part (a) of the pre-lab in terms of an indutor s steady-state response to a onstant input and an indutor s response to an instantaneous voltage hange. (e.g. replae the indutor with either an open iruit or a short iruit as appropriate for the frequeny of interest.) page 3 of 9

4 Lab Proedures: Overview: In this portion of the lab assignment, we will measure v (t), i (t), and v L (t). We will use these measurement to estimate the gain and phase differene between v (t) and i (t) and the gain and phase differene between v L (t) and i (t). These results will be ompared with our expetations based on the pre-lab analyses. We do not have the ability to diretly measure a time-varying urrent, so we will infer i (t) by measuring v (t) - v L (t) and determining i (t) by: i v ( t ) vl( t ) ( t ) (7) R All signals we will be dealing with are sinusoidal. Appendix A of this lab assignment provides tips relative to gain and phase measurement of sinusoidal signals. (a) Construt the iruit of Figure 2 with L = 1mH and R = 47 (b) Use your funtion generator to apply a sinusoidal input at v (t). Use your osillosope to display both v (t) and v L (t). Use the osillosope s math operation to display the differene between the input and indutor voltages, v (t) - v L (t). In your lab notebook, reord the amplitude of v (t), v L (t), and v (t) - v L (t). Also reord the time delay between v (t) and v L (t) and the time delay between v (t) and v (t) - v L (t). Perform this proess for the following frequenies: (low frequeny input) (high frequeny input) (orner frequeny input) Demonstrate operation of your iruit to the Teahing Assistant. Have the TA initial the appropriate page(s) of your lab notebook and the lab heklist. () Use equation (7) and your measurements from part (b) above to alulate, for eah of the frequenies listed in part (b), the amplitude of i (t), the time differene between i (t) and v (t), the time differene between i (t) and v L (t). (d) Calulate the measured gains and phase differenes between i (t) and v (t) and i (t) and v L (t) for the three frequenies listed in part (b) above. Compare your measured results with your expetations from the pre-lab. Comment on your results. page 4 of 9

5 II. Passive RC Ciruit General Disussion: In this part of the lab assignment, we will determine the input impedane of the passive RC iruit shown in Figure 3. The input impedane of a iruit is defined as the ratio of input voltage to input urrent. Thus, for the iruit of Figure 3, the input impedane is represented in phasor form as: V Z (8) I where V is the phasor representation of the iruit input voltage and I is the phasor representation of the input urrent to the iruit. The utoff frequeny for the iruit of Figure 3 is: 1 (9) RC i (t) R + v (t) - C Figure 3. Passive RC iruit. Pre-lab: (a) Determine an expression for the input impedane of the iruit of Figure 3 in terms of R, C, and. (b) If R = 100 and C = 1F, determine the utoff frequeny for the iruit. Also determine the input impedane for frequenies of: (low frequeny input) (high frequeny input) (orner frequeny input) () Chek your low and high frequeny results in part (b) relative to your expetations based on the apaitor s low and high frequeny behavior. page 5 of 9

6 Lab Proedures: Construt the iruit of Figure 3, using R = 100 and C = 1F. Measure the input impedane (magnitude and phase) of the iruit for the frequenies: Compare your measured results with your expetations based on the analysis you did in the pre-lab. Demonstrate operation of your iruit to the Teahing Assistant. Have the TA initial the appropriate page(s) of your lab notebook and the lab heklist. Hint: The proess to perform the above lab proedures is omparable to the proess performed in Part I of this lab assignment. Be sure to reord all neessary data and any alulations you perform to obtain your results in your lab notebook. page 6 of 9

7 Appendix A Measuring Gain and Phase: The gain of a system at a partiular frequeny is the ratio of the magnitude of the output voltage to the magnitude of the input voltage at that frequeny, so that: Gain = V V out in where Vout and Vin the figure below. an be measured from the sinusoidal input and output voltages as shown in Input voltage, V in Voltage V in Output voltage, V out Time V out The phase of a system at a partiular frequeny is a measure of the time shift between the output and input voltage at that frequeny, so that: Phase = T T 360 where T and T an be measured from the sinusoidal input and output voltages as shown in the figure below. page 7 of 9

8 Input voltage, V in Voltage T Output voltage, V out Time T page 8 of 9

9 Lab 9 Report Cheklist: On a title page, please inlude the following items: (5 pts) Name, title of lab, date, and lab partners(s). Brief introdution (ouple of sentenes) desribing the overall purpose of the lab. I. Passive RL Ciruit (50 pts total) 1. Gain ( I ) and phase ( Ð I ÐV ) for RL iruit at low, high, and orner frequenies. V Relationship between DC and utoff frequeny gains. (7 pts) 2. Indutor voltage-to-urrent gain and phase relationships. Indutor gain and phase relations at low and high frequenies. (8 pts) 3. Indutor physial behavior at low and high frequenies vs. expression obtained in (2) above. (2 pts) 4. RL iruit physial behavior at low and high frequenies vs. expression obtained in (1) above. (3 pts) 5. Table providing amplitudes of v (t), v L (t), and v (t) - v L (t) and time delays between v (t) and v L (t) and between v (t) - v L (t) at the three frequenies of interest. (7 pts) 6. Measured gains and phase differenes between i (t) and v (t) and i (t) and v L (t) for the three frequenies of interest. (8 pts) 7. DEMO: Have a teahing assistant initial this sheet, indiating that they have observed your system s operation (15 pts) II. Passive RC Ciruit (45 pts) 1. General expression for nput impedane (gain, phase) (7 pts) 2. Input impedane for speified R and C, at three frequenies of interest (8 pts) 3. Chek of input impedane at low, high frequenies relative to expeted apaitor behavior (5 pts) 4. Measured input impedane at three frequenies of interest. (Table inluding measured data, alulated gains and phases) (15 pts) 5. DEMO: Have a teahing assistant initial this sheet, indiating that they have observed your iruit s operation. (10 pts total) page 9 of 9