Sinusoidal Response of RC & RL Circuits

Written By: Sachin Mehta

Reno, Nevada

Objective: When varying frequencies are applied to RC and RL circuits, analysis of the sinusoidal responses of the respective circuits can be accomplished somewhat easily. By using a function generator, an oscilloscope, and a few other circuit elements, we will create both an RC & RL circuit similar to the previous lab. The major difference, however, is the implementation of a sine wave as the response—instead of a step (square) wave.

Equipment used:While doing the circuit analysis, we used several devices; one of which was a multimeter. Specifics of the multimeter such as tolerance, power rating, and operation are discussed below. Two other pieces of electronics that had to be employed were the power supply (or voltage source) and the breadboard. Along with the electronic devices, resistors of different values were used (also discussed more in depth below).

a) Breadboard: This device makes building circuits easy and practical for students learning the curriculum. Instead of having to solder each joint, the student can build and test a particular circuit, and then easily disassemble the components and be on their way. The breadboard used in this experiment was a bit larger than usual, and consisted many holes, which act as contacts where wires and other electrical components, such as resistors and capacitors, can be inserted. Inside of the breadboard, metal strips connect the main rows together (five in each row) and connect the vertical columns on the side of the board together. This means that each row acts as one node.

b) Multimeter: Made by BK Precision Instruments, the 2831c model that we used during this lab allowed us to measure different currents, resistances, and voltages from our circuits. Of the four nodes on the front of the device, we used only three: the red, black, and bottom white. In theory, either of the white nodes could have been used, but the 2 Amperes node was more than capable since we barely even hit the 15 mA mark. The following ratings are manufacturer specifications for the device:

DC Volts – 1200 Volts (ac + dc peak)Ohms – 450 V dc or ac rms200 mA – 2 A --- 2 A (fuse protected)

The use of the multimeter in this lab was mainly to ensure the correct values of resistors, capacitors, and inductors were being used.

c) Wire Jumper Kit: This kit was not required to complete lab two, however the use of the wires made building the circuits more manageable. The wires were 22 AWG solid jumper wires that varied in length. The wire itself was copper, PVC insulated, and pre-stripped at ¼ inch.

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d) Resistors: This lab involved the use of many different resistors; different values for the various circuits that had to be constructed. A resistor, in the simplest definition, is an object which opposes the electrical current that is passed through it. So: the higher the Ohm value, the more a resistor will impede the current. Typically made of carbon, each resistor is color coded so identification can be made easier. The first three colored bands on the passive element are used to calculate the resistance using the following equation: R = XY * 10Z. Where X and Y and are the first two bands and Z is the third. The rightmost colored band however gives the tolerance rating of the resistor itself.

e) Inductors: The inductor is a passive element designed to store energy in the magnetic field it has. Inductors consist of a coil of wire wrapped around each other and the ones that were used in lab were of pretty big size, compared to the resistors and capacitors. An important aspect of the inductor is that it acts like a short circuit to direct current (DC) and the current that passes through an inductor cannot change instantaneously.

f) Oscilloscope: An oscilloscope is a test instrument which allows you to look at the 'shape' of electrical signals by displaying a graph of voltage against time on its screen. It is like a voltmeter with the valuable extra function of showing how the voltage varies with time. A graticule with a 1cm grid enables you to take measurements of voltage and time from the screen.

g) Function generator: A function generator is a device that can produce various patterns of voltage at a variety of frequencies and amplitudes. It is used to test the response of circuits to common input signals. The electrical leads from the device are attached to the ground and signal input terminals of the device under test.

h) In addition to the previously listed equipment a new utility was put to work in order to provide a second opinion, as well as a solid foundation of the circuits. However, the piece of equipment in question was not a machine, but software. Known as Multisim, it is a well-known program used by technicians, engineers, and scientists around the globe. Based on PSPICE, a UC Berkeley outcome, Multisim allows users to put together circuitry without the need for soldering or a permanent outcome. Mistakes can be corrected by the click of the mouse and there will be no loss in stock prices in doing so. Even students who wish to practice on the breadboard can use Multisim as there learning tool. Without it, electric circuit analysis would not be as well known, and practical in today’s ever-changing and technologically advanced society.

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Theory: The RC and RL circuits examined in this lab will consist of either a capacitor or an inductor in series with a resistor. This will make an oscillatory motion in regards to the voltage running through the circuit. By using the function generator as an input—and setting it to various frequencies—we will be able to simultaneously view the output & input waveform on the oscilloscope. We will also be able to calculate the output voltages, phase shift and resistor currents at specified frequencies, and compare the data to Multisim simulations. Since we are investigating circuits of the first-order, the mathematical equations which have to be used are not as simple as previous equations such as Ohm’s law, for example. The following are the equations needed to complete this lab experiment on the analysis of sinusoidal responses.

V out=R

√R2+X c2V ¿

or

V out=R

√R2+X L2V ¿

X c=1

2 πfC

X L=2πfL

V RMS=V out

√2

Phase Shift= ∆tt

∗360o

V= I*R

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Procedure:The first portion of this lab involved building a circuit similar to the one from the previous lab. It is an RC circuit with a 100Ω resistor in series with the capacitor equal to 0.1μF. The input is the function generator with a signal at 320 Hz at a voltage level of 10 V. It is important to remember that this voltage is the peak to peak voltage—not the peak voltage alone. Making sure to select the sine wave button on the function generator—we constructed the circuit shown in Fig. 1 on the breadboard, except we used a different resistor value.

Fig. 1: RC Circuit constructed for Part I

Table 1: Experimental Measurements for RC Circuit in Part I

Frequency Input Voltage Output Voltage (Vp-p)

Output Voltage (VRMS)

V-I Time Difference

Phase Shift (Degrees)

Resistor Current

320 Hz 10 V 352 mV 82.4 mV 800 µs 0 .5490 mA3.2 kHz 10.2 V 2.12 V 688 mV 64 µs 40 .0046 mA32 kHz 10.2 V 8.6 V 2.75 V 2.4 µs 75.5 .0183 mA

320 kHz 9.92 V 10.1 V 3.35 V .5µs 90 .0223 mA

The data in table 1 was collected by measuring the waveforms on the oscilloscope with the voltage cursor and time cursor. By comparing input and output waveforms, we were able to detect the time difference between the two waves—which in turn helped us gain a better idea about sinusoidal steady-state analysis and other circuit analysis methods. The right most-farthest column in table 1 is the current that is passing through the resistor. Usually, in the past, we have used the multimeter to measure current in the circuits we have built. But for this lab, we used a different method. Using the following equation, we were able to determine the

RMS current (IRMS): IRMS = V RMSR

Since we were able to measure the RMS voltage with the oscilloscope, and we knew the value of the resistor we used in our circuit—we could easily solve for this current. This RMS voltage is the average voltage across R and essentially, by manipulating Ohm’s Law, we could solve for current.

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Section 2.b. of Part I asked us to calculate the output voltage, time difference, phase shift, and resistor current for the ideal values. Using 0.1μF and 150 Ω as the resistance, I was able to calculate the output voltage:

V out=R

√R2+X c2V ¿ =

150

√1502+49732×10V=.301mV

V RMS=V out

√2 = 301V

√2=.213mV

Since the voltage and current are in phase—there is no time difference—meaning that there is a phase shift of 0 degrees, as well.

IRMS = V RMSR

= .213mV150Ω

= .00142 A

For3.2kHz :V out=R

√R2+Xc2V ¿ =

150

√1502+4972×10.2V=2.94V

For32kHz :V out=R

√R2+Xc2V ¿ =

150

√1502+49.72×10.2V=9.68V

For320 kHz :V out=R

√R2+Xc2V ¿ =

150

√1502+4.972×9.92V=9.91V

For the four different frequencies asked for, the ideal output voltages came out to be quite close to that of the experimental measurements. For example, the percent difference between the measured 320 kHz output voltage and the calculated output voltage is:

|9.91−10.1|9.91

×100=1.92%

With such a miniscule percent discrepancy, we can conclude that the experimental we performed, and the circuit we constructed was very close to the real thing. It was precise, accurate, and definitely intriguing. Notice how the output voltage increased steadily as the frequency was increased by a factor of 10.

On the other hand though, the RMS output voltage which was calculated for each respective frequency did not compare well to the measured values of the RMS voltages. For example, the

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VRMS measured for 320 Hz was 82.4 mV, but the calculated was .213 mV. This large discrepancy could have been due to error in the experiment, or error in the calculations done by hand. Since the peak to peak output voltages came out to be consistent, it is not likely that the equipment malfunctioned in any way. The reasonable assumption would be that the equation used to calculate the VRMS is incorrect, or a mistake was made with the voltage and time cursors when handling the oscilloscope during the lab. In order to calculate the phase for each of the trials, the following equation was utilized:

Phase = arctan ( 1RCw

¿

For 320 Hz---- phase = arctan (33.16) = 1.54 For 3.2 kHz----- phase = arctan(3.316) = 1.28 For 32 kHz------ phase = arctan(.3316) = .3202 For 320 kHz---------phase = arctan(.03316) = .0331

Using the output voltages that were measured on the oscilloscope, we were able to determine the peak voltage—and from there the RMS voltage that should have resulted from the measured output voltage.

For example, for 3.2 kHz, the peak to peak output voltage was 2.12 V. Dividing this by 2 gives us 1.06 Volts as the peak voltage. And finally multiplying by .707 gives a theoretical RMS voltage of .749 V. Instead the cursors we used on the oscilloscope measured the RMS voltage at .688 V.

- The percent difference between these two values is:

|.749−.688|.749

×100=8.14%

The time delay could be deduced using the fact that the current leads voltage by anywhere from 0 to 90 degrees in an RC circuit. Simply put, V0 leads Vin. This time difference can be analyzed by studying the input and output waveforms on the oscilloscope reading. During the lab, we did our best to estimate the correct time difference, but it all depended on how close you looked at the reading.

4) For this section, we used Multisim to perform a simple AC sweep of the circuit—and then created a Bode plot. A Bode plot is a plot of both the voltage and phase and can help a great deal in circuit analysis—especially when sinusoidal responses are being considered. Making sure to set the frequency limits from 1 Hz to 500 kHz, and specifying the measurement probe to read voltage across the two probes, a simulation was put together. It can be seen in Fig. 3.

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Fig. 2: Multisim simulation of RC circuit

Fig. 3: AC Sweep of RC circuit using MultiSim

When determining if this would be a high pass or a low-pass filter, it was necessary for me to look up these terms in the textbook. I found that a high pass filter ends up being a high and steady voltage, whereas a low-pass filter ends up at a lower voltage and approaches zero in fact. Looking at the AC sweep, I determined this RC circuit to have a high-pass filter.

When determining the half-power point, we had to find the spot at which the resistance value equals the reactance. In addition, the output would be equal to 2-.5 times the voltage input. This means that the half power point would occur at the following voltage:

10 * 2-.5 = 7.07 VMeaning that the frequency that the half-power point would occur at would be approximately 20 kHz.

Part II of the lab involved the implementation of an RL circuit, similar to the one made for the last lab (step response). The inductor and resistor were placed in series—and constructed on a breadboard. The beginning frequency we used was 800 Hz at a 10 V peak to peak sine wave. From there, we again kept increasing the frequency, but kept the input voltage the same. This would ensure that we could compare the figures, and see how a change in the frequency can drastically manipulate the circuit we are working with. The circuit constructed for this section is

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shown in Fig. 4, but instead of the 100 Ω resistor that is shown—we had to use 150 Ω for all the calculations, since the function generator had a resistance in and of itself.

Fig. 4: RL circuit constructed for part II

Setting the function generator to 800 Hz and finding the 10 V peak to peak input gave us our first experimental measurement for part II. Refer to table 2 for a complete list of data.

Table 2: Experimental Measurements for RL circuit in Part II

Frequency Input Voltage Output Voltage

Output Voltage (RMS)

V-I Time Difference

Phase Shift (Degrees)

Resistor Current

800 Hz 10 V 9.8 V 3.31 V 0 0 .022 A8 kHz 10 V 8.4 V 2.68 V 10µs 40 .018 A

80 kHz 10 V 1.68 V 55 mV 2.6µs 75.5 .366 mA500 kHz 10 V 260 mV 56.1 mV 440 ns 90 .374 mA

Part 2.b. again involved calculating data by hand. Using the following equation allowed a quick and easy way to determine theoretical values:

V out=R

√R2+X L2V ¿=

150

√1502+5.03210=9.99V

- For 8 kHz: 150

√1502+50.3210=9.48V

- For 80 kHz :150

√1502+503210=2.86V

- For 500 kHz: 150

√1502+3141210=.477V

The values calculated above vary quite greatly from the measured values in table 2. This could have been a result of inaccurate inductors, resistors, or equipment. Calibration of equipment had probably not properly been done for a long time. Such simple things as such could affect data deeply.

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Calculations of other properties of the RL network, such as RMS voltage, time difference, and phase are done in the following section. However, with the output voltage already being so different from the calculated, it is likely that any other thing we calculate for comparison purposes is not going to be a good reference. Regardless, the RMS voltage is calculated as follows:

Vp = V p−p

2 where VRMS = Vp * .707

Trial 1) VRMS = 3.53 V

Trial 2) VRMS = 3.35 V

Trial 3) VRMS = 1.01 V

Trial 4) VRMS = .169 V

Utilizing the percent discrepancy equation for trial 1:

|3.53−3.31|3.53

×100=6.23%

We can see that with a 6.23%, the measured values from the lab experiment don’t quite compare to the theoretical values, which were calculated by hand. There is an assortment of reasons for such behavior, such as wrong inductor values being used on the breadboard, incorrect use of the oscilloscope, or even just plain human error. It is not uncommon to see mistakes made with new equipment like the function generator and oscilloscope.

For the fourth trial (500 kHz)—Fig. 5 shows a screenshot I took of the oscilloscope output. As the picture shows, there is a time difference of 440 ns. Using the cursors, we were able to determine this time on the oscilloscope.

Fig. 5: Oscilloscope output of trial 4—with a frequency of 500 kHz

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4) Using Multisim to create a Bode plot was necessary for this section, just as it was for the previous section. The AC sweep is depicted in Fig. 5 below. You can see that the curve for both the phase and magnitude (voltage) decrease as the frequency increases. Just as the experimental data suggests—we have a low-pass filter in this bode plot.

Fig. 5: AC sweep of RL circuit in Part II

When determining if this was a low or high pass filter, I needed to refer back to the text to examine the definitions more closely. After studying, I found that this plot shows a low-pass filter since the voltage and phase start out at a higher degree and fade & approach zero.

To find the half-power point for this plot, we find where the input voltage is multiplied by .707, and compare this voltage to the data we have in table 2. In doing so, the half-power point would occur at approximately 15 – 20 kHz.

Conclusion:The majority of this lab was spent studying the theory behind sinusoidal waves and the responses of RC and RL circuits. With new material being introduced, such as complex numbers, frequencies of all different ranges, and reactance; it was one of the more difficult labs of the semester. In addition to this new material, we had to work with Multisim in a new fashion and create never before seen or heard of “Bode Plots” to match with our data. Overall, with circuit analysis becoming more and more important as time passes—and the use of technology expanding ever so quickly—learning how to manipulate circuits in Multisim will make future assignments much easier.

Overall, the data that was collected matched up well with the theory of what should happen as frequency of a sine wave is increased. We saw the correct response of the RC and RL circuits

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that we built on the breadboards. Viewing these sine waves and the integration of these waveforms into the study of RC & RL circuits was a major learning outcome of this lab experiment. Determining how various frequencies affect the output voltage, RMS voltage, phase shift, and even the current through the different elements was interesting.

However, when doing calculations by hand near the end of the lab experiment—we noticed that some figures were ambiguous and did not correlate well with the data that we had collected. There could be plenty of reasons for something like this to happen, but the most common and most likely is human error: error on our part. Either when collecting the data and working with the equipment, or when manipulating these new and fresh equations—mistakes could have been made in either place or even both. If we had more time to cover this material, things would have gone a lot smoother and a thorough understanding of the material would be much greater.

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