Frequency Responses in RLC Circuits

Dalton Eaton and Uladzimir KasacheuskiIndiana University - Purdue University Indianapolis

(Dated: February 11, 2017)

The bandpass filter and the notch filter, the ’inverse’ of the bandpass filter, are circuits forvalued for their utility in attenuating specific frequencies of a signal. In this paper we create thetwo filters, in the forms of resistor-inductor-capacitor (RLC) circuits, and examine both circuitsfrom experimental and theoretical views. The filters are shown to accurately attenuated selectivefrequencies based on chosen component values. The results demonstrate that these filters are highlypredictable: the theoretical and experimental values match very well even in non-ideal circumstances.

I. INTRODUCTION AND BACKGROUND

Filters are commonly used circuits for the selectiveattenuation of specific frequencies. A band-pass filter isdesigned to attenuate all frequencies except those lyingwithin a certain band. A notch filter, or band-rejectfilter, is the ’inverse’ of the band pass filter; all frequen-cies are passed in a notch filter except for those lyingwith a certain range, the ’notch’. An RLC circuit is asimple yet effective method for the construction of bothof these common and powerful filters.

A. RLC Filter Theory

The RLC circuit of a band-pass filter is shown in figure1. This filter attenuates all signals except for those lyingwithin a region of frequencies center around a specificfrequency known as the resonance frequency,fo. Reso-nance occurs when the impedance, a function of appliedfrequency, of the inductor and the capacitor cancel oneanother and the total impedance of the circuit is entirelydue to the resistor. The equations for the inductive reac-tance and capacitive reactance are given in equations 1and 2 respectively, where j is the imaginary number, ω isthe angular frequency (2πfrequency), L is the inductance,and C is the capacitance.

FIG. 1. Bandpass Filter

XL = jωL (1)

XC = −1/(jωC) (2)

Equations 1 and 2 determine the impedance of the in-ductor and capacitor, the analogue of resistance for thetransient components of the two. At a specific value of ωdesignated by ωo, the two terms will be equal and oppo-site. Using Ohm’s law, the voltage found on the capaci-tor and inductor will sum to zero and thus Vsource will beequivalent to the voltage found across the resistor. Thenotch filter works on the same principles except the out-put voltage will be taken across the inductor capacitorcomponent pair.

Q = ωoL/R =fo

f2 − f1(3)

a(w) = Vo/Vs =R√

R2 + (ωL− 1/ωC)2(4)

The bandwidth of a filter is often described by its Q-Factor, defined in equation 3, where a larger Q-Factorimplies a narrower bandwidth and a smaller the oppo-site. The Q-Factor is both a function of the circuit com-ponents, the inductor and resistor, and the angular res-onance frequency, ωo. At the same time, the Q-Factorcan additionally be found by dividing the resonance fre-quency, o by the frequencies above and below resonancewhere the gain, equation 4, is half.

II. EXPERIMENTS

Experimentation in this project involved defining thevalues of the components we were working with followedby the frequency response analysis of both the bandpassand notch filter. The component values were used forcalculating theoretical values and the frequency responseanalysis defined our experimental results - used for de-termining the resonance frequency and bandwidth.

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A. Measurement of Component Values

In order to calculate theoretical values for our circuitswe needed to define the actual values of the componentsof our RLC circuits.

The measurements of the resistor and the capacitorwere very straight forward due to the availability ofa modern multi-meter capable of explicitly measuringboth. Measuring the inductance of the chosen induc-tor was more complicated and required the constructionof the circuit shown in Figure 2. Equation 5 is the rear-rangement of Ohm’s law where f is the applied frequency,Vsource is the applied voltage in root-mean-square(rms)and i is the current measured from the test circuit, whichenables us to calculate the value of the inductor. TableII A shows the measured values for the components usedin the construction of both filters.

L = Vsource/2πfi (5)

FIG. 2. Inductance Measurement Test Circuit

Measured Component ValuesResistor(R) 344ΩInductor(L) 10.3mH

Capacitor(C) 3.7nF

B. Bandpass Filter Frequency Response Analysis

The circuit shown in figure 1 was constructed usingthe previously measured component values. The appliedvoltage was swept through a wide array of frequenciesand the values of Vout and Vsource were measured at eachapplied frequency. The results of this experiment areplotted in figure 3.The phase shift induced by the inductor and capacitorwere then separately monitored, again sweeping througha similar range of frequencies and noting manually thephase change.

Our next point of interest dealt with how Vo wouldrespond to varying types of input wave function, such astriangle and square. The results were surprising and areshown in figures 4 - 6

FIG. 3. Measured Frequency Response Curve

A more advanced method of measuring phase shift wasthen additionally implemented by by using the built inmath function of the oscilloscope, Ch1-Ch2. This abilityyielded a notable but predicted result. Simultaneously,this opportunity was used to measure the voltage acrossthe inductor and capacitor separately. These values wereused as one more method of obtaining a Q-Factor valueexperimentally, as shown in equation 6.

Q = Vs/VL = Vs/VC (6)

FIG. 4. Triangle Wave Response at 10kHz

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FIG. 5. Square Wave Response at 1kHz

FIG. 6. Square Wave Response at 50kHz

C. Notch Filter Frequency Response Analysis

A notch filter, the RLC circuit for whom is shown infigure 7, was then constructed and a similar frequency re-sponse analysis as for the bandpass filter was applied. Inthis case, resonance frequency will correspond to the fre-quency at which attenuation occurs and the result shouldbe straight forward to find. Sweeping through the fre-quencies generated a full and agreeable analysis like be-fore. The results are shown in figure 8.

FIG. 7. Notch Filter

FIG. 8. Measured Notch Filter Frequency Response Curve

III. DISCUSSION

A. Band-pass Filter

Using Kirchoff’s Voltage Law (KVL) on the circuitshown in Figure 1 yields equation 7. Where ω is theangular frequency (2πf) and j is the imaginary number.

Vsource = i(jωL− 1/jωC +R) (7)

Using Ohm’s law on the resistor yields equation 8.Equations 7 and 8 can be combined to find the trans-fer function of the circuit as a function of the appliedfrequency. This relation is shown in equation 9. Theplot of the magnitude of the theoretical transfer func-tion is shown in figure 9. The plot of the magnitude ofthe transfer function experimentally measured is shownin figure 3. The experimentally measured fo is found tobe at approximately 25 KHz.

Vout = iR (8)

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VoutVsource

= T (ω) =R

(jωL− 1/jωC +R)(9)

fo = 1/(2π√LC) (10)

Calculating fo with equation 10, referencing the actualvalues for the RLC components as specified in table II Ayields a resonance frequency of 25.8Khz. This matchesclosely with the experimentally determined value of res-onance. Error in the measured values can be attributedto the internal resistance of the inductor and generalinstrumentation errors. More accuracy would likely beachieved by further increasing the resolution of appliedfrequencies.

The phase shift due to the inductor and capacitor weremonitored at a wide sweep of applied frequencies. Thephase shift was well behaved, following the theoreticallypredicted values. At frequencies below fo the voltagelead Vs, due to the capacitor, and at frequencies abovefo it lagged, due to the inductor. At fo, as expected, Voand Vs were in phase.

Using the math function of the oscilloscope as refer-enced before we were able to show that the phase differ-ence between the current and voltage is completely fre-quency independent, exactly what theory predicts. Thephase shift remained a constant ± π/2 out of phase withthe source. This result is exactly as expected.

The Q-Factor for the bandpass filter was calculatedboth theoretically as well as experimentally. Theoreti-cally, using equation 3 and the values of table II A, wefound the Q-Factor to be 4.72. Experimentally, however,we found a Q-Factor of 3 by using equation 6. Sources oferror may include the internal resistance of the circuit.

FIG. 9. Theoretical Frequency Response Curve

B. Notch Filter

The method of analysis for the Notch filter will besimilar to the method used for the analysis of the Band-pass filter. Vsource will remain as equation 7. VoutNotch

is derived using Ohm’s law on the component pair ofinductor and capacitor and is show in equation 11.

VoutNotch = ij(ωL− 1/(ωC)) (11)

The transfer function is then formed from the ratio ofequation 11 and 7 as shown in equation 12.

FIG. 10. Theoretical Notch Filter Frequency Response Curve

VsourceVoutNotch

= Tnotch(ω) =j(ωL− 1/(ωC))

(jωL− 1/jωC +R)(12)

It is evident from inspection that the frequency re-sponse yielded from the transfer function shown in figure10 matches well with the experimentally found frequencyresponse shown in figure 8. As for the theoretical andexperimental fo, due to retaining the same RLC compo-nents between the filters the theoretical and experimen-tal values will remain constant and related as describedabove for the bandpass filter.

IV. CONCLUSION

Two different filter architectures were built using a se-ries RLC circuit and tested at a wide sweep of frequen-cies. It was shown that even with a simple circuit con-struction utilizing less than ideal components that atten-uation frequencies can be selected with relatively accu-rate precision. In the future, we would like to investigatemethods of narrowing the bandwidth and otherwise con-trolling the Q-Factor.