6labo ganago student lab7

Lab 7: AC Analysis

LAB EXPERIMENTS USING

NI ELVIS II AND NI MULTISIM

Alexander Ganago Jason Lee Sleight

University of Michigan

Ann Arbor

Lab 7 AC Analysis

2010 A. Ganago Introduction Page 1 of 10

Lab 7: AC Analysis

Goals for Lab 7

Learn about:

Calculations of the responses of a linear circuit to sinusoidal voltages using phasors

Conversion from time-dependent sinusoids to phasors and from phasors to time-dependent sinusoids

Measurements of time delays in the lab and conversion of the results to phase shift angles at various frequencies

The role of the capacitors and inductors impedances in phase shifts Learn how to:

Measure the phase shift between the input and output voltages and compare your lab data to pre-lab (and post-lab) calculations of the same phase shifts

Add sinusoidal output voltages in order to reproduce the sinusoidal input voltage, thus verify KVL.

Measure in the lab:

Amplitudes of sinusoidal input and output voltages in several circuits, as well as the time delays between the input and output

Calculate in the post-lab:

Phase shifts between the input and output sinusoidal voltages in several circuits, from the time delays, which you measured in the lab

Percentage differences between the amplitudes of sinusoidal input and output voltages in several circuits: (a) calculated from theory, (b) simulated with Multisim, and (c) measured in the lab

Percentage differences between the phase angles of sinusoidal output voltages with respect to the input voltages in several circuits: (a) calculated from theory, (b) simulated with Multisim, and (c) measured in the lab

Explain in the post-lab: The possible sources of error in measurements and causes of the disagreement

between theory and lab data The effects of circuit modifications on the amplitudes and phase angles of output

voltages.


Lab 7: AC Analysis

Introduction If a circuit contains sources of (co)sinusoidal voltages and currents, as well as resistors, inductors, capacitors, op amps, and other linear elements, then under certain conditions we can apply the extended form of Ohms law

= V I Z to inductors and capacitors. In other words, instead of resistance R (in the usual form of Ohms law), we can write impedance Z and use linear algebraic equations instead of differential equations. To make this great simplification of circuit analysis legitimate, we require that:

(1) all sources must be sinusoidal, at the same frequency (2) all circuit elements must be linear, and (3) the circuit should have reached its sinusoidal steady-state condition.

Condition (3) simply means that, after a voltage or current applied to the circuit was changed, you have to wait till the transient responses die out. Exactly how long do you have to wait? From what you learned about transients in Labs 5 and 6, recall that, after

the time interval equal to5 (where is the time constant of the circuit response) or 5 (where is the damping factor), the amplitude of transients falls below 1% of its initial value; therefore, after 5 or 5 , the error of using the simplification is less than 1%. The simplification we have in mind is called phasor analysis (do not mistake it for the handy weapon used by the Star Trek characters, which is called phaser). The big idea is that, under the conditions listed above, sinusoidal input at frequency , peak voltage

and phase angle max, inV in

in max, in inV (t) V cos( t ) = + produces a sinusoidal output

out max, out outV (t) V cos( t ) = + at the same frequency but with a different amplitude and phase max, outV out . We can avoid doing tedious trigonometric calculations by virtue of Eulers equation

j cos( ) j sin( )e = + , where j 1=


Lab 7: AC Analysis

so that

in inj( t ) j( ) j( t )in max, in in max, in max, inV (t) V cos( t ) V Re{ } V Re{ } Re{ }e e

+= + = = e and

out outj( t ) j( ) j( t )out max, out out max, out max, inV ( ) cos( t ) V Re{ } V Re{ } Re{ }t V e e e

+= + = = Note that the last factor is the same for the input and the output, thus we may drop jt in intermediary calculations and use the shortcut notation, which includes the magnitude and the phase angle

j( t)Re{ }e

max, inV in : in max, in inV = V Similarly, for the output voltage:

out max, out outV = V The two latter equations define the phasors for the input and output voltage: each carries the information about the amplitude and the phase of the (co)sinusoidal voltage. Note that we write equations with a cosine function for the sake of simplicity: here, the real part of complex voltage is the real voltage, which we measure in the lab. Amplitudes of sinusoidal signals are already familiar, but their phases are new to you.

Figure 7-1. Phase shift of cosine waves (1). The solid line shows the reference voltage. The dashed line shows another voltage, which leads by t.


Lab 7: AC Analysis

Figure 7-1 shows two co-sinusoidal voltages; the solid line represents the reference (or input) voltage, which reaches its maximum at time t = 0. Recall that the cosine function is maximal at zero (we consider one period from 180 to +180 degrees, or from to +); therefore, for a cosine with phase angle , we obtain: cos( t ) cos(0 ) cos(0) + = + = , Assuming that the reference voltage reaches its maximum at time = 0, as shown in Figure 7-1, we get its phase angle as 0 = . The dashed line in Figure 7-1 shows the output voltage, which reaches its maximum before the reference voltage does; in other words, the output leads by t. In the lab, you will measure t in s; for comparison with theory, you will have to express the delay between the sinusoids as a phase angle shift. The relationship is:

cos( t ) cos(0)t 0

t

t

+ = + == +

= +

If you use the frequency f is in hertz, which is more convenient in the lab, rewrite the latter equation as the following:

t2 f = +

The frequency , which is measured in radians per second, remains more convenient for theoretical calculations. The positive sign emphasizes that the output waveforms leads. Also note that the phase angle is expressed in radians. To obtain the phase shift in degrees, use the fact that 2 360 =


Lab 7: AC Analysis

Figure 7-2. Phase shift of cosine waves (2). The solid line shows the reference, or input voltage. The dashed line shows the output voltage, which lags by t. The other important case of phase shift is shown in Figure 7-2. Here the reference voltage has its peak at zero time, as before, but the other voltage reaches its maximum later, at the time t = +t. In other words, the output lags by t. Therefore the phase angle of the reference signal is zero, and the phase angle of the output signal is related to its time delay as the following:

t2 f

= =

Here the negative sign shows that the second signal lags. Again, the angle is in radians. In lab measurements and data analysis, when you are asked to calculate the phase angle from the measured time delay t, remember to take into account the signal frequency, because

2 f t = For example, time delay of 1 s, at 1 kHz corresponds to the phase angle of ( ) ( )3 1 6 3 32 f t 2 10 s 10 s 2 10 rad 360 10 degrees = 0.36 degrees = = = = which may be negligible but, at 100 kHz, the same delay corresponds to 36 degrees, which must be taken into account.


Lab 7: AC Analysis

Note that all phase angles measured in the lab are relative, with respect to the reference (input) voltage. Use cursors on the oscilloscope screen and align the first one with the peak of the input voltage (you should have learned how to find out which signal is the input). Then align the second cursor with the peak of the output signal and read out the time delay t. Make sure that the sign of t agrees with your intuitive expectations (refer to Figures 7-1 and 7-2). Do calculations to translate t into the phase angle as shown above. Now, let us take a closer look at the Ohms law

= V I Z Here, both the voltage and the current are phasors, V I

VV = V

II = I carrying information about the amplitude (or magnitude) and phase of their signals. Impedance is a complex number (in special cases, it may be a real number), Z Z Z= Z which has its magnitude Z and phase angle Z . Using the rules of complex algebra, the Ohms law equation can be rewritten as

V I

V I Z

Z =

= + The impedance of an inductor equals

L

L

L

j LZ L

Z 90

= =

= +

Z

The impedance of a capacitor equals


Lab 7: AC Analysis

C

C

C

1j C

1ZC

Z 90

= =

=

Z

The impedance of a resistor is the real number equal to the resistance:

R

R

R

RZ R

Z 0

==

=

Z

Therefore, the phase shift of the current relative to the voltage always equals +90PoP for the inductor and always equals 90PoP for the capacitor. For a resistor, there is always zero phase shift between the voltage and the current. For an entire circuit, the phase shift can be anything between 180PoP and +180PoP.

In phasor form, KCL and KVL are valid; calculations require both magnitudes and phase angles. In a series RLC circuit (see Figure 7-3), the current I through all circuit elementsthe source, the resistor, the inductor and the capacitorwill be the same, according to KCL. Since the impedances of the resistor, inductor, and capacitor include different phase shifts, the voltages across these elements will also be shifted with respect to the input voltage. In a series RLC circuit, you can obtain the current from your measurement of the resistor voltage VBRB because the phase shift between and

II RV is zero.

Figure 7-3. Series RLC circuit.


Lab 7: AC Analysis

In addition to experiments on the series RLC circuit, in this lab you will also work on a more complicated circuit (see Figure 7-4), which is similar to the phase shifter discussed in your textbook ???ERROR??? (Example 7-11, pages 335 336).

Figure 7-4. Phase-shifting RC ladder circuit. The input sinusoidal voltage comes from the function generator; the output voltage can be measured at any of the 3 nodes

, , and .

INV

O, 1V O, 2V O, 3V In your simulations and lab data, you will observe that the phase angle (referenced to the input voltage) progressively increases from to , and to . O, 1V O, 2V O, 3V If you are interested in Explorations (for extra credit), you will study three modifications of the phase-shifting circuit shown below. Each of these modifications affects the phase shifts of output voltages with respect to the input.


Lab 7: AC Analysis


Lab 7: AC Analysis

Pre-Lab:

1. Phase Shift Conversions Complete the following table:

fB0B (Hz) B0B (rad/s) tBshiftB (ms) Angle () Angle (rad)

50 5

100 45

500

1,000 0.375

1,600 90

10,000 /10

Show your work for the fB0B = 1,600 Hz case.

2010 A. Ganago Pre-Lab Page 1 of 3

Lab 7: AC Analysis

2. RLC Multisim Simulations Use the following circuit for your simulation:

R = 1 k C = 1 F L = 10 mH Use a 1 VBPPKB, 100 Hz sine wave with 0 V DC offset for the input signal. This frequency

is below the resonance 01LC

< = A. Measure VBCB

Use VBCB as the output signal. Measure and record the VBPPKB and phase shift of the output signal (clearly show the units).

B. Measure VBRB

Use VBRB as the output signal. Measure and record the VBPPKB and phase shift of the output signal (clearly show the units).

C. Measure VBLB

Use VBLB as the output signal. Measure and record the VBPPKB and phase shift of the output signal (clearly show the units).

D. Repeat parts 2.A through 2.C, this time using a 1 VBPPKB, 1.6 kHz sine wave with 0 V DC offset as the input signal. This frequency is near the

resonance 01LC

= =


Lab 7: AC Analysis

3. RC Ladder Simulations Use the following circuit for your simulation:

Use: R = 1 k C = 1 F Use a 1 VBPPKB, 200 Hz sine wave with 0 V DC offset as the input signal. Use the 4 Channel Oscilloscope tool to view the input and all three output signals simultaneously. Adjust the Oscilloscope parameters so that you can clearly view the waveforms.

Use the Cursors to fill in the following table:

Signal VBPPKB (mV) Shift (ms) Shift (rad)

VBINB 0 0

VBO, 1B

VBO, 2B

VBOUT, 3BBB

Create a printout which clearly displays the waveforms. (Pre-Lab Printout #1)


Lab 7: AC Analysis

In-Lab Work Part 1: Frequency < B0B Part 1.1 VBC Turn on the NI ELVIS II.

Build the following circuit:

Use: C = 1 F

L = 10 mH

R = 1 k

Measure VBINB on AI0 and VBCB on AI1.

From the NI ELVISmx Instrument Launcher, launch the FGEN and OSCOPE.

Power on the PB.

On the FGEN VI, create a 100 Hz, 1 VBPPKB sine wave with 0 V DC offset.

Adjust the parameters on the OSCOPE so that you can clearly view the waveforms.

Use the cursors to measure the phase shift between the input and output signals.

tBshift, 1B = ______ s

Create a printout which clearly shows the signals. (In-Lab Printout #1) Power off the PB.

2010 A. Ganago In-Lab Page 1 of 10

Lab 7: AC Analysis

Part 1.2 VBLB Continue to use the same circuit from Part 1.1.

Measure VBINB on AI0 and VBLB on AI1.

Power on the PB.




Create a printout which clearly shows the signals. (In-Lab Printout #2)


Lab 7: AC Analysis

Part 1.3 VBRB Continue to use the same circuit from Part 1.1.

Measure VBINB on AI0 and VBRB on AI1.

Power on the PB.






Lab 7: AC Analysis

Part 2: Frequency Near the Resonance 0 1LC = Part 2.1 VBC Continue to use the same circuit from Part 1.

Measure VBINB on AI0 and VBCB on AI1.

Power on the PB.

On the FGEN VI, create a 1.6 kHz, 1 VBPPKB sine wave with 0 V DC offset.






Lab 7: AC Analysis

Part 2.2 VBLB Continue to use the same circuit from Part 2.1.

Measure VBINB on AI0 and VBLB on AI1.

Power on the PB.






Lab 7: AC Analysis

Part 2.3 VBRB Continue to use the same circuit from Part 2.1.

Measure VBINB on AI0 and VBRB on AI1.

Power on the PB.






Lab 7: AC Analysis

Part 3: RC Ladder Build the following circuit:

Use: R = 1 k C = 1 F Measure VBINB on AI0 and VBOUT, 1B on AI1.

Power on the PB.

On the FGEN, create a 1 VBPPKB, 200 Hz sine wave with 0 V DC offset.

Adjust the settings on the OSCOPE so that you can clearly observe the waveforms.




Measure VBINB on AI0 and VBOUT, 2B on AI1.





Measure VBINB on AI0 and VBOUT, 3B on AI1.





Lab 7: AC Analysis

Create a printout which clearly shows the signals. (In-Lab Printout #9) Power off the PB. This is the end of the required part of this lab. If you wish to do Explorations for extra credit, turn to the next page. If you wish to finish the lab work now, please power off the NI ELVIS II and clean up your workstation.


Lab 7: AC Analysis

Explorations for Extra Credit Repeat Part 3 for each of the following modifications:


Lab 7: AC Analysis

Record the time shift of each output signal for each case. And create a printout which clearly displays the waveforms.

Modification #1:

tBshift, 10B = ______ s (In-Lab Printout #10) tBshift, 11B = ______ s (In-Lab Printout #11) tBshift, 12B = ______ s (In-Lab Printout #12) Modification #2:

tBshift, 13B = ______ s (In-Lab Printout #13) tBshift, 14B = ______ s (In-Lab Printout #14) tBshift, 15B = ______ s (In-Lab Printout #15) Modification #3:

tBshift, 16B = ______ s (In-Lab Printout #16) tBshift, 17B = ______ s (In-Lab Printout #17) tBshift, 18B = ______ s (In-Lab Printout #18)

This is the end of the lab. Please power off the NI ELVIS II and clean up your workstation.


Lab 7: AC Analysis

Post-Lab 1. RLC circuit (frequency 0 1LC < = )

A. Discuss the agreement/disagreement between your simulation (Pre-Lab 2, A) and your experimental results (In-Lab Printout #1) for the amplitude and phase shift of VBCB.

B. Discuss the agreement/disagreement between your simulation (Pre-Lab 2, B) and your experimental results (In-Lab Printout #2) for the amplitude and phase shift of VBLB.

C. Discuss the agreement/disagreement between your simulation (Pre-Lab 2, C) and your experimental results (In-Lab Printout #3) for the amplitude and phase shift of VBRB.

D. Using MATLAB, MathScript, or Excel software along with the data you collected in the lab, create a printout (Post-Lab Printout #1) which shows: each of the waveforms (VBINB, VBCB, VBLB, and VBRB) and a new waveform which shows their sum,

VBSUMB = VBCB + VBLB + VBRB.

From KVL, one expects to obtain

VBSUMB = VBCB + VBLB + VBR B= VBINB

Make a printout (Post-Lab Printout #2) that shows both VBSUMB and VBINB.

Discuss whether your results support KVL or disagree with KVL.

E. From your lab data for VBRB, and using R = 1 k, derive the equation for the sinusoidal current, I, in the circuit as a function of time.

F. From the equation for I derived in Part 1.E above, and using C = 1 F and L = 10 mH, and frequency equal to 100 Hz, derive the equation for the sinusoidal voltages VBCB and VBLB in the circuit as functions of time. Discuss their agreement/disagreement with both your lab data and your pre-lab simulations.

Continued on the next page.

2010 A. Ganago Post-Lab Page 1 of 4

Lab 7: AC Analysis

Post-Lab (continued) 2. RLC Circuit (Frequency Near the Resonance 0 1LC = )

A. Discuss the agreement/disagreement between your simulation (Pre-Lab 2, D) and your experimental results (In-Lab Printout #4) for the amplitude and phase shift of VBCB.

B. Discuss the agreement/disagreement between your simulation (Pre-Lab 2, D) and your experimental results (In-Lab Printout #5) for the amplitude and phase shift of VBLB.

C. Discuss the agreement/disagreement between your simulation (Pre-Lab 2, D) and your experimental results (In-Lab Printout #6) for the amplitude and phase shift of VBRB.

D. Using MATLAB, MathScript, or Excel software along with the data you collected in the lab, create a printout (Post-Lab Printout #3) which shows: each of the waveforms (VBINB, VBCB, VBLB, and VBRB) and a new waveform which shows their sum,

VBSUMB = VBCB + VBLB + VBRB.

From KVL, one expects to obtain

VBSUMB = VBCB + VBLB + VBR B= VBINB

Make a printout (Post-Lab Printout #4) that shows both VBSUMB and VBINB.

Discuss whether your results support KVL or disagree with KVL.

E. From your lab data for VBRB, and using R = 1 k, derive the equation for the sinusoidal current, I, in the circuit as a function of time.

F. From the equation for I derived in K above, and using C = 1 F and L = 10 mH, and frequency equal to 100 Hz, derive the equation for the sinusoidal voltages VBCB and VBLB in the circuit as functions of time. Discuss their agreement/disagreement with both your lab data and your pre-lab simulations.



Lab 7: AC Analysis

Post-Lab (continued) 3. RC Ladder Circuit

A. Discuss the agreement/disagreement between your simulation (Pre-Lab Part 3, Pre-Lab Printout #1) and your experimental results (In-Lab Printout #7) for the amplitude and phase shift of VBOUT, 1B.

B. Discuss the agreement/disagreement between your simulation (Pre-Lab Part 3, Pre-Lab Printout #1) and your experimental results (In-Lab Printout #8) for the amplitude and phase shift of VBOUT, 2B.

C. Discuss the agreement/disagreement between your simulation (Pre-Lab Part 3, Pre-Lab Printout #1) and your experimental results (In-Lab Printout #9) for the amplitude and phase shift of VBOUT, 3B.

This is the end of the required Post-Lab work.

If you did Explorations (for extra credit), continue the Post-Lab below.

4. Explorations (for extra credit): Modifications of the RC Ladder Circuit

A. In Modification #1, you added a second capacitor between the source (function generator) and the node . Discuss the effects of this additional capacitance in

the circuit on the amplitudes and phases of the output voltages , , and

relative to the unmodified circuit.

0,11V

0,11V 0,21V

0,31V

B. In Modification #1, you did not change anything between the node and the

nodes and . Explain why the output voltages and were

affected by this modification.

0,11V

0,21V 0,31V 0,21V 0,31V

C. Run a Multisim simulation of Modification #1, make a printout that shows the output voltages , , and (Post-Lab Printout #5) and compare it with

your in-lab data. 0,11V 0,21V 0,31V



Lab 7: AC Analysis

Post-Lab (continued) Explorations (for extra credit) (continued)

D. In Modification #2, you added a capacitor between the node and the node

. Discuss the effects of this additional capacitance in the circuit on the

amplitudes and phases of the output voltages , , and relative to the

Modification #1 circuit.

0,12V

0,22V

0,12V 0,22V 0,32V

E. In Modification #2, you did not change anything between the function generator and the node ; you did not change anything between the nodes and

. Explain why the output voltages and were affected by this

modification.

0,12V 0,22V

0,32V 0,12V 0,32V

F. Run a Multisim simulation of Modification #2, make a printout that shows the output voltages , , and (Post-Lab Printout #6) and compare it with


G. In Modification #3, you added a capacitor between the node and the node

. Discuss the effects of this additional capacitance in the circuit on the

amplitudes and phases of the output voltages , , and relative to the

Modification #2 circuit.

0,23V

0,33V

0,13V 0,23V 0,33V

H. In Modification #3, you did not change anything between the function generator and the node ; you did not change anything between the nodes and

. Explain why the output voltages and were affected by this

modification.

0,13V 0,13V

0,23V 0,13V 0,23V

I. Run a Multisim simulation of Modification #3, make a printout that shows the output voltages , , and (Post-Lab Printout #7) and compare it with



Student_7-1 intro_paul.docStudent_7-2_preLab_paul.docStudent_7-3_Lab_paul.docStudent_7-4_postLab_paul.doc

6labo ganago student lab7

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