Resonant Response of RLC Circuits

Written By: Sachin Mehta

Reno, Nevada

Objective: The two most common applications of resonance in AC circuitry involves series RLC circuits and parallel RLC circuits. An RLC circuit is, of course, a circuit with a resistor, inductor and capacitor—and when implemented together helps to design communications networks and filter designs. An RLC circuit involves more complicated equations—those of second order differentials—while the circuits from the prior two lab experiments were of first order. The resonance in an AC circuit implies a characteristic & exclusive frequency that is determined by the resistor, inductor, and capacitor values. For this lab experiment, we will establish the resonance frequency of series and parallel RLC networks—comparing theoretical calculations to those measured during the lab experiment.

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.

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.

Theory:

Resonance circuits are crucial in the design of many electronic components and circuitry. In order to calculate a resonance frequency of either a series or parallel RLC circuit—we can use Eq. (1)

(1)

Note that this specific frequency depends on the product of the inductance and capacitance— not on their individual values alone, and is independent of the resistance value. In a series RLC circuit, this special frequency occurs when the reactance of both the inductor and capacitor are opposite and equal in magnitude & “cancel” each other out. Eq. (2) provides a glimpse into this property.

ωL - 1ωC

= 0 (2)

The ‘ω’ in the above equation represent the quantity 2πf. If we input frequencies, it is essentially possible to guess and check what the resonance frequency would be. Of course, when working with quantities on the degree of micro and milli, it would be extremely difficult to figure out the resonance frequency.

Since the reactance of both the capacitor and inductor cancel out, that part of the circuit is essentially a short circuit. Therefore, the only opposition to the current flow from the AC voltage is the resistor itself—making Z = R. Put complexly (in terms of complex numbers), the total impedance has no imaginary parts (capacitive reactance or inductive reactance)—it only has a “real” part. The reactance of an inductor is shown in Eq. (3). On the other hand, Eq. (4) shows the capacitive reactance. In addition, at the resonance frequency, the voltage drop across the inductor and capacitor is also equal and opposite (canceling out). The current flow

ends up being at its maximum value at the resonance frequency. And since the impedance is equal to the resistance—we can manipulate Ohm’s Law to depict the current, shown in Eq. (5)

XL = ωL (3)

XC = 1ωC

(4)I=VZ

(5)

%Difference=|expected−measured|

expected×100 (6)

RLC circuits in a series network are used in electronics a great deal. Fig. 1 shows a series RLC circuit.

Resonance occurs in any circuit with at least one capacitor and one inductor. When the imaginary part of the transfer function is zero, resonance results. At resonance, the voltage and current are in phase. The impedance is purely resistive at this point making the LC series combination act like a short circuit –and the entire voltage ends up being only across the resistor. In a parallel network, the LC combination acts like an open circuit.

Procedure:

Using Eq. (1), we were able to calculate the resonance frequency of the circuit depicted in Fig. 1. A simulation circuit constructed using Multisim shows the exact same circuit with a function generator and oscilloscope attached in Fig. 2.

Fig. 2: Multisim simulation of RLC series circuit for Part I.

The calculation of the resonance frequency is as follows:

f res=1

2 π √LC = 1

2π √¿¿¿

The resonance of both of these elements was also calculated using Eqs. (3) & (4) respectively. They were done by hand, and are as follows:

Inductive Reactance: First, we had to convert the frequency (Hz) to radians/sec.(2π)(18815 Hz) = 118218 rad

s

The next step was to input these in Eq. (3):XL = ωL = (118218 rad

s) (10.37 * 10-3) = 1225.9 Ω

Capacitive Reactance: Again, we had to convert frequency to radians/sec.(2π)(18815 Hz) = 118218 rad

s

The next step was to input these in Eq. (4):XC = 1

ωC = 1

(118218rads )(0.01∗10−6) = 845.9 Ω

Keep in mind, that the reactance calculated previously do have imaginary portions included. These imaginary parts are represented by the letter ‘j’ in the study of electrical engineering. In addition, the capacitive reactance is written as a negative reactance. So the correct form of XC should be –j845.9 Ω. The start of this lab experiment was to setup the RLC circuit shown in Fig. 1. In order to do this, we utilized the breadboard, a 1kΩ resistor, 10mH inductor, and a 0.01μF capacitor. We put these three passive elements in series and then applied the function generator as the input and the oscilloscope as the output. By doing this, we could apply a 1 V peak to peak input voltage and a 5 kHz sine wave to the circuit.

The next thing to do was monitor the output voltage on the oscilloscope screen and record the data. In addition, we record the specific input and output frequencies shown on the oscilloscope. This data is shown in Table 1. By plotting the output voltages and the frequency, we could also see the behavior of the circuit as different frequencies were applied. Fig. 3 shows a plot of the output voltage versus the various frequencies for the series RLC circuit that we constructed and analyzed. As the plot shows, the voltage reaches a minimum at around 17 kHz. The code for this plot is shown in Appendix A.

Section 4 of Part I asked to analyze this frequency a bit closer and try to find the exact frequency that produces this minimum output voltage. By examining the oscilloscope closely and manipulation the function generator setting to different frequencies around the vicinity of 17 kHz, we were able to get a closer match.

Series RLC Circuit Data-Part I

Frequency (kHz) Input Voltage (Vp-p) Output Voltage (Vp-p) Input Frequency Output Frequency

5 1.04 80 5.05 5.026 1.02 71.2 6 6.077 1.01 65.6 7.01 7.018 1 60 7.91 7.979 1.02 53.6 9.06 9.02

10 1.01 46.4 10.02 10.0111 1.04 40 11.06 11.0912 1 32.4 12.02 12.0313 1.03 26 13.02 13.0214 1.02 20 14.03 14.0415 1.01 13 15.02 15.02

16 1 7.2 16.14 16.1417 1 3.12 17.12 17.1218 1.04 5.12 18.02 18.0219 1.04 10.4 19.05 19.0520 1.02 15.2 20.02 20.0521 1.02 20.4 20.83 20.822 1.04 24.8 22.12 22.1223 1.03 28 23.2 23.0424 1.01 32 24.2 24.0725 1.02 35.6 25.06 25.06

The frequency that produced the minimum output voltage was determined to be 16.73 kHz. Before building this RLC circuit, we were asked to determine the resonance frequency using Eq. (1). Comparing these two frequencies using Eq. (6) was able to give us a percent difference that can help us see how our constructed circuit compared.

%Difference=|18815Hz−16730Hz|

18815Hz×100 = 11.1%

When analyzing this percent discrepancy, there are many factors that we need to look at. So many different variables apply that could have caused the results to go awry. Keep in mind, that when determining our theoretical resonance frequency, we used values of resistance, inductance, and capacitance that we measured. Before we built the circuit on the breadboard, we took each element (resistor, capacitor, and inductor) and found the actual values. The inductor was in fact 10.37 mH instead of 10 mH. The capacitor was not 0.01 μF, but happened to be .0069 μF.

Fig. 3: Series RLC Circuit plot of output voltage vs. frequency

When measuring the phase, we set the frequency to the calculated resonance of 18.8 kHz. This was done to ensure we were studying the resonance of the actual correct values. In doing so, we found a phase, of input to output, of 0.

A screenshot of the oscilloscope reading from the lab is shown next. It shows the reading somewhat close to the resonance frequency.

Fig. 4: Oscilloscope output taken from the lab experiment.

A Multisim output of RLC current vs. frequency is shown next in Fig. 5.

Fig. 5: AC Analysis of Circuit 1—RLC current vs. frequency.

As the plot shows, the current magnitude of the circuit is at a maximum at the resonance frequency. This behavior is called a band-pass response.

In order to provide a plot of LC impedance vs. frequency, we had to use the relationship in Eq. (5). By manipulating that equation, we could determine the impedance of the

circuit to be Z = VI

. Going into the AC analysis menu of Multisim—we could simulate an

expression to be on the Y-axis and plot it versus frequency. The resulting output is shown in Fig. 6.

Fig. 6: A plot of the LC impedance vs. frequency for Circuit 1.

We can see from our results that the output voltage is at a minimum at the resonance frequency. This occurs because the current in the circuit is at a maximum at this frequency. At resonance, the current and voltage are in phase. At frequencies below the resonance frequency, the current leads the voltage—which is sort of like a RC circuit. On the other hand, the current lags the voltage (like an RL circuit) when the frequency is above the resonance frequency.

Part II of the lab involved construction and analysis of an RLC circuit in a parallel network. Using the same resistor, capacitor, and inductor values as in Part I, it was not necessary to calculate the resonance frequency for this circuit. In addition, the inductive reactance and capacitive reactance were also the same exact values. If you wanted to work out the solutions again—you would use Eq. (1), (3), and (4) from above.

To reiterate, the resonance frequency for the RLC parallel circuit was 18.8 kHz. The reactance of the inductor was j1225.9 Ω The reactance of the capacitor was –j845.9 Ω

Fig. 7 depicts the circuit we built on the breadboard—and also shows the relative locations of where we placed the function generator and oscilloscope leads.

Fig. 7: Parallel RLC circuit built for Part II.

After attaching the function generator to the input—we set it to create a 1 V peak to peak voltage and a 5 kHz sine wave. Channel 1 of the oscilloscope was attached to the input as well, and channel 2 remained on the output. Just as we had done for Part I of the lab, we evaluated the output voltage, input frequency, and output frequency for different frequencies ranging from 5 – 25 kHz. This data is depicted in Table 2.

Parallel RLC Circuit Data-Part II

Frequency (kHz) Input Voltage (Vp-p) Output Voltage (Vp-p) Input Frequency Output Frequency

5 1.04 16.4 5 5.016 1.02 20.4 5.99 5.997 1.01 25.2 7.02 7.028 1 30 8 89 1.02 35.2 8.96 8.96

10 1.01 41.6 10.02 10.0211 1.04 49.6 11.04 11.0412 1 56.8 11.9 11.913 1.03 66.4 12.9 12.9414 1.02 75.2 14.05 14.0515 1.01 86 15.02 15.0216 1 90 15.97 15.9717 1 90 17.07 17.0718 1.04 88 17.92 17.9219 1.04 84 18.89 18.8920 1.02 74.4 19.91 19.9121 1.02 68.6 21.05 21.0522 1.04 64 21.93 21.9323 1.03 60 22.78 22.7824 1.01 55.2 24.18 24.1825 1.02 51.2 25.07 25.07

Table 2: Output Voltages/Input & Output Frequencies for Parallel RLC Circuit

As Table 2 shows, the maximum output voltage occurred at a frequency between the interval of 16 and 18 kHz.

Fig. 8 shows a plot of the output voltage versus the frequencies for Circuit 2. The code for this Matlab plot is provided in Appendix B. A minor problem that I had when developing this plot was the numbering on the axes. I did a little research and found how to number the axes the way I want—but for some reason the Y-axis did not come out as planned. Regardless, the plot still shows that the maximum output voltage occurred near the resonance frequency.

Fig. 8: Output voltage vs. Frequencies for Parallel RLC Circuit in Part II

After we determined the range that the resonance frequency occurred in, we spent a little time to find the exact frequency at which we got the highest voltage. This occurred at 17.12 kHz.

A picture of the oscilloscope readings—and the sine waves from the lab experiment itself is shown in Fig. 6.

Comparing this experimental resonance frequency to the calculated resonance frequency of 18.8 kHz occurred as follows:

%Difference=|18815Hz−17120Hz|

18815Hz×100 = 9.00%

This percent discrepancy was close to that of Part I. The imprecise resistors, capacitor, and inductor that we used for the circuit could have caused some of this error. In addition, the calibration of the oscilloscope and the function generator probably had a hand in it as well. Notice that in, both, Table 1 and 2—the input and output frequencies do not match up. Neither does the input voltage. We set the function generator to exactly 1 V—however we got readings that ranged from 1 V – 1.4 V.

A simulation of this parallel RLC circuit is depicted in Fig. 9.

Fig. 9: Simulation of Circuit 2 using Multisim

Fig. 10 actually shows the oscilloscope output from Multisim for the circuit depicted in Fig. 7. The red line represents the input voltage—while the black represents the output voltage. Notice how they are close to overlapping—but their difference represents the lag that occurs when a sinusoidal voltage is applied.

Fig. 10: Oscilloscope output from Multisim for Circuit 2 of the lab.

A picture taken during the lab experiment, itself, of the oscilloscope output and its readings are shown next in Fig. 11.

Fig. 11: Oscilloscope reading from the actual lab experiment for Circuit 2.

Using Multsim, we were able to analyze the current at various frequencies by doing an AC sweep of the circuit. The output of the circuit LC current vs. frequency is depicted in Fig. 12.

Fig. 12: AC analysis of Circuit 2—showing LC current vs. Frequency.

As the plot shows, the current is at a minimum at the resonance frequency. This is due to the fact that the inductor and capacitor act as an open circuit at resonance—making all the current go through the resistor.

A plot of the LC impedance vs. frequency was produced using Multisim, as well. Just as in Part I of the lab, we had to manipulate the expressions to get the output in the AC analysis tool. Since there is no way to measure the impedance directly using the probes, the graph took a little time to make—as Multisim is a very indepth software with many functions. Finding the correct tools provided the output in Fig. 13.

Fig. 13: LC impedance vs. Frequency for RLC parallel Circuit.

From the data in Table 2, it is clear that the output voltage became its maximum value at the resonance frequency. Since the circuit current is constant for any value of impedance, the voltage has the same shape as the total impedance. The total impedance vs. frequency is shown above and we can see that the maximum magnitude occurs at the resonance frequency. With maximum impedance, of course, the admittance of the circuit will be at its minimum (since admittance is the reciprocal of impedance). And with a very low admittance, the current in the circuit will also be very low—as we can see in the plot of LC current vs. Frequency.

Resonance circuits are very useful in the construction of filters. Their transfer functions are very highly frequency selective—so certain frequencies will pass while others will not. This allows people to select only a desired radio station, or a certain television channel. Resonance occurs in many different systems—and in all areas of engineering, physics, and chemistry. Their use in communications networks is very abundant, as well.

Conclusion:

RLC circuits have a commonplace usage among electronics of all kinds—and being able to study and analyze them is essential in electrical engineering. Using tools such as a function generator and oscilloscope can give us a better idea of how the circuits work, themselves, and how a resonance frequency is used. Finding a specific frequency is how we tune radios, channels, and many other items. In addition, the implementation of Multisim to study these circuits has given us a better sense of the relationships of RLC current and how it changes with frequency.

This lab shows the proper way to find resonant frequency of a RLC circuit. The output wave obviously changes when the frequency is varied above and below resonance. Our values were

very close to those calculated. There were many possibilities of error of course. Not using the exact capacitor values or inductor values—and reading the oscilloscope output could have resulted in the data in Tables 1 & 2 being incorrect. The oscilloscope was one of the most complicated pieces of equipment we used over the semester, and with so many different functions—it was likely that we read the output wrong. With a percent discrepancy of 11.1 % between the calculated resonance frequency and the measured for circuit 1, we came close to the correct answer. Regardless, this lab provided a lot of information that helps in the analysis of RLC circuit. Knowing how they work is very important since their use in electronics is huge. The application of filters, circuits that are designed to pass signals with only desired frequencies & reject others has been around for a long time, and their use will only become greater as time goes on.

Appendix A:

Matlab Code—Part I

Plot of Output Voltage vs. Frequency for Series RLC

Appendix B:

Matlab Code---Part II

Plot of Output Voltage vs. Frequency for Parallel RLC