T149/T150 Alex Trinh/Matthew Jackson

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Resonance

Introduction Resonance is the tendency of a system to oscillate at very large amplitudes at some frequency. It occurs when the

frequency of the driving force, ω, is close to the natural frequency of the system. The large amplitude occurs because the

system is oscillating under the most favourable conditions. This frequency is known as the resonance frequency, ω0. In an

electric circuit, the size of the amplitude depends on the impedance of the circuit, the lower the impedance, the larger the

amplitude. This report details the method used in exploring resonance within electrical circuits and the effect of non-ideal

components such as parasitic capacitance.

Theory

Consider a series LCR circuit, shown in Fig. 1, which is driven by an external oscillator with voltage amplitude 𝑉0 and

angular frequency ω.

The impedance of this circuit, 𝑍, is given by

𝑍 = 𝑅 + 𝑅𝐿 + 𝑗 𝜔𝐿 −1

𝜔𝐶

where 𝑅 is the resistance of the resistor and 𝑅𝐿 is the effective loss resistance of the inductor 𝐿 due to the resistance of the

wires and power loss due to the core of the inductor. Using Ohm’s Law, the current of the circuit is given by, 𝑖 = 𝑣/𝑍, so

the current through the resistor is given by

𝑣𝑅 =𝑅𝑣

𝑅 + 𝑅𝐿 + 𝑗(𝜔𝐿 −1𝜔𝐶)

The amplitude of 𝑣𝑅 is given by

𝑉𝑅 =𝑅𝑉0

(𝑅 + 𝑅𝐿)2 + (𝜔𝐿 −1𝜔𝐶)2

(1)

Fig.1 Series LCR Circuit

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The phase difference ∅ between 𝑣𝑅 and 𝑣 is given by

tan ∅ =𝜔𝐿 − 1/𝜔𝐶

𝑅 + 𝑅𝐿 (2)

Setting ∅ = 0, we can see that this is when the circuit is at resonance and the impedance of the inductor and capacitor are

equal and opposite, shown by

𝜔𝐿 −1

𝜔𝐶= 0 (3)

Perfect cancellation can never be achieved because the voltage through the inductor and capacitor is only approximately

equal

Using eqn (3) we can show that the resonance frequency 𝜔0 is given by

𝜔0 =1

𝐿𝐶 (4)

At low frequencies the high reactance of the capacitor dominates in eqn (1) so we expect 𝑉𝑅 to be

𝑉𝑅 ≈ 𝜔𝐶𝑅𝑉0 (5)

and at high frequencies above resonance the high reactance of the inductor dominates in eqn (1) so we expect 𝑉𝑅 to be

𝑉𝑅 ≈𝑅𝑉0

𝜔𝐿 (6)

Experimental Details The LCR circuit consisted of a 220 Ω resistor, 0.01 μF capacitor and a 100 mH inductor. A signal generator with voltage

𝑉0 and frequency 𝜔 was used to drive the circuit. These components were connected as shown in Fig.2

A Digital Storage Oscilloscope (DSO) was used to measure the voltage of the signal generator and the voltage across the

resistor. The values on the components were only labeled values so the actual values required measurement using the GR

Digibridge. Using eqn (4) the resonance frequency of the circuit was calculated.

Fig. 2 Experimental setup

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Using the calculated resonance frequency to save time and using the DSO, the resonance frequency was found by

adjusting the frequency of the signal generator until the phase difference between the voltages equaled zero.

The ratio 𝑉𝑅

𝑉0 was then measured over a range of frequencies spanning from 100 Hz to 1 MHz. At lower frequencies, the

voltage was increased to reduce noise being detected by the DSO but at frequencies near resonance the voltage was kept

at ~1𝑉𝑝𝑝 to reduce core saturation and to prevent any annoying whistles from the inductor. Changing the voltage

shouldn’t affect the results as only the ratio was calculated.

Results Measurements of the components were assumed to have an error of ±2% based on the manufacturer notes of the

Digibridge. The theoretical resonance frequency was calculated to be (5.1 ± 0.1) kHz and the measured frequency was

determined to be (5.0 ± 0.1) kHz. The resonance peak was also measured to be 0.92 ± 0.006 and eqn (1) was used to

calculate the theoretical value which was found to be 0.97 ± 0.01. The other values for 𝑉𝑅

𝑉0 were plotted against frequency

on a log scale for both axes shown in Fig. 3.

The two bold lines on the left and right of the resonance peak represent the theoretical responses of the circuit and were

determined using eqn (5) for low frequencies spanning from 100 Hz to 1 kHz and eqn (6) for higher frequencies above

resonance with frequencies spanning from 20 kHz to 1 MHz.

Discussion The results gathered varied from being exceptionally close to theoretical values to being far from expected. Looking at the

measured resonance frequency, the discrepancy between the experimental and theoretical values is 0.1 ± 0.2 so there is a

significant overlap between the values. The discrepancy between the measured and theoretical values for 𝑉𝑅

𝑉0 is 0.05 ± 0.02

which is a very large discrepancy. This may be caused by the signal generator. There were very large fluctuations in peak-

to-peak voltages, even when the voltage of the signal generator was maximized, causing the values to be approximated,

Fig. 3 VR/V0 vs Frequency Log-Scale Graph

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indicating that the DSO was detecting a lot of noise. Another explanation may be faulty probes as they tend to be quite

fragile. They may be connected to the components loosely so the current couldn’t be measured correctly. A different

probe or signal generator may need to be used in future to determine our measurements.

In Fig. 3 there is also a very noticeable discrepancy at 60 kHz where the current in the circuit begins to rise again.

Theoretically, the current should go to zero at very high frequencies due to the high reactance of the inductor, seen in

eqn (6). We decided to explore this problem by constructing a parallel LCR circuit with the inductor and capacitor

connected in parallel and replacing the 220 Ω resistor with a 100 kΩ resistor, as seen in Fig. 4. Again we looked for when

the circuit was at resonance. Disconnecting the capacitor, creating an RL circuit, and looking at frequencies where the

discrepancy occurred we saw that the RL circuit displayed the same behaviour as the parallel LCR circuit at resonance. It

appears that the inductor contains its own capacitance. The windings of the inductor will have some capacitance if there is

a potential difference between them and the capacitor is seen as connected in parallel with the inductor. This behaviour is

negligible at low frequencies because there is very little change in potential difference but at larger frequencies this is

more evident. This is known as parasitic capacitance and explains the discrepancy at 60 kHz. The capacitance was

calculated using eqn (4) to be (103 ± 3) pF

Conclusion In conclusion, we investigated the concept of resonance and parasitic capacitance. Resonance is the tendency of a system

to oscillate at very large amplitudes at a certain frequency. We determined the resonance frequency of our circuit

experimentally and theoretically and found them to be (5 ± 0.1) kHz and (5.1 ± 0.1) kHz respectively. We found a

discrepancy while measuring the resonance frequency as the current began to increase again and determined that the

discrepancy was due to parasitic capacitance of the windings of the inductor and the inductor was found to have a

capacitance of (103 ± 3) pF

References 1. Clifton Laboratories, Self-resonant Frequency of an Inductor

Available: http://www.cliftonlaboratories.com/self-resonant_frequency_of_inductors.htm

2. Intermediate Physics Experimental Notes

3. Serway, R. A. And Jewett, J. W, 2010, Physics for Scientists and Engineers with Modern Physics. 8th Edition,

Brooks/Cole CENGAGE Learning p. 453

Fig. 4 Parallel LCR Circuit