rlc circuit at resonance experiment

23.12 RLC Series AC Circuits

Learning objectives.

By the end of this section, you will be able to:

• Calculate the impedance, phase angle, resonant frequency, power, power factor, voltage, and/or current in a RLC series circuit.
• Draw the circuit diagram for an RLC series circuit.
• Explain the significance of the resonant frequency.

When alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other’s effect. Figure 23.46 shows an RLC series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an RLC circuit is the frequency dependence of X L X L and X C X C , and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important “resonance” features that are the basis of many applications, such as radio tuners.

The combined effect of resistance R R , inductive reactance X L X L , and capacitive reactance X C X C is defined to be impedance , an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an RLC circuit are related by an AC version of Ohm’s law:

Here I 0 I 0 is the peak current, V 0 V 0 the peak source voltage, and Z Z is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for Z Z in terms of R R , X L X L , and X C X C , we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled V R V R , V L V L , and V C V C in Figure 23.46 .

Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in R R , L L , and C C are equal and in phase. But we know from the preceding section that the voltage across the inductor V L V L leads the current by one-fourth of a cycle, the voltage across the capacitor V C V C follows the current by one-fourth of a cycle, and the voltage across the resistor V R V R is exactly in phase with the current. Figure 23.47 shows these relationships in one graph, as well as showing the total voltage around the circuit V = V R + V L + V C V = V R + V L + V C , where all four voltages are the instantaneous values. According to Kirchhoff’s loop rule, the total voltage around the circuit V V is also the voltage of the source.

You can see from Figure 23.47 that while V R V R is in phase with the current, V L V L leads by 90º 90º , and V C V C follows by 90º 90º . Thus V L V L and V C V C are 180º 180º out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage V 0 V 0 of the source does not equal the sum of the peak voltages across R R , L L , and C C . The actual relationship is

where V 0 R V 0 R , V 0 L V 0 L , and V 0 C V 0 C are the peak voltages across R R , L L , and C C , respectively. Now, using Ohm’s law and definitions from Reactance, Inductive and Capacitive , we substitute V 0 = I 0 Z V 0 = I 0 Z into the above, as well as V 0 R = I 0 R V 0 R = I 0 R , V 0 L = I 0 X L V 0 L = I 0 X L , and V 0 C = I 0 X C V 0 C = I 0 X C , yielding

I 0 I 0 cancels to yield an expression for Z Z :

which is the impedance of an RLC series AC circuit. For circuits without a resistor, take R = 0 R = 0 ; for those without an inductor, take X L = 0 X L = 0 ; and for those without a capacitor, take X C = 0 X C = 0 .

Example 23.12

Calculating impedance and current.

An RLC series circuit has a 40.0 Ω 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF 5.00 μF capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for L L and C C are the same as in Example 23.10 and Example 23.11 . (b) If the voltage source has V rms = 120 V V rms = 120 V , what is I rms I rms at each frequency?

For each frequency, we use Z = R 2 + ( X L − X C ) 2 Z = R 2 + ( X L − X C ) 2 to find the impedance and then Ohm’s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.

Solution for (a)

At 60.0 Hz, the values of the reactances were found in Example 23.10 to be X L = 1 . 13 Ω X L = 1 . 13 Ω and in Example 23.11 to be X C = 531 Ω X C = 531 Ω . Entering these and the given 40.0 Ω 40.0 Ω for resistance into Z = R 2 + ( X L − X C ) 2 Z = R 2 + ( X L − X C ) 2 yields

Similarly, at 10.0 kHz, X L = 188 Ω X L = 188 Ω and X C = 3 . 18 Ω X C = 3 . 18 Ω , so that

Discussion for (a)

In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that X L X L dominates at high frequency and X C X C dominates at low frequency.

Solution for (b)

The current I rms I rms can be found using the AC version of Ohm’s law in Equation I rms = V rms / Z I rms = V rms / Z :

I rms = V rms Z = 120 V 531 Ω = 0 . 226 A I rms = V rms Z = 120 V 531 Ω = 0 . 226 A at 60.0 Hz

Finally, at 10.0 kHz, we find

I rms = V rms Z = 120 V 190 Ω = 0 . 633 A I rms = V rms Z = 120 V 190 Ω = 0 . 633 A at 10.0 kHz

The current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in Example 23.11 . The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in Example 23.10 . The inductor dominates at high frequency.

Resonance in RLC Series AC Circuits

How does an RLC circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law, I rms = V rms / Z I rms = V rms / Z , and the expression for impedance Z Z from Z = R 2 + ( X L − X C ) 2 Z = R 2 + ( X L − X C ) 2 gives

The reactances vary with frequency, with X L X L large at high frequencies and X C X C large at low frequencies, as we have seen in three previous examples. At some intermediate frequency f 0 f 0 , the reactances will be equal and cancel, giving Z = R Z = R —this is a minimum value for impedance, and a maximum value for I rms I rms results. We can get an expression for f 0 f 0 by taking

Substituting the definitions of X L X L and X C X C ,

Solving this expression for f 0 f 0 yields

where f 0 f 0 is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At f 0 f 0 , the effects of the inductor and capacitor cancel, so that Z = R Z = R , and I rms I rms is a maximum.

Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation—in this case, forced by the voltage source—at the natural frequency of the system. The receiver in a radio is an RLC circuit that oscillates best at its f 0 f 0 . A variable capacitor is often used to adjust f 0 f 0 to receive a desired frequency and to reject others. Figure 23.48 is a graph of current as a function of frequency, illustrating a resonant peak in I rms I rms at f 0 f 0 . The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.

Example 23.13

Calculating resonant frequency and current.

For the same RLC series circuit having a 40.0 Ω 40.0 Ω resistor, a 3.00 mH inductor, and a 5.00 μF 5.00 μF capacitor: (a) Find the resonant frequency. (b) Calculate I rms I rms at resonance if V rms V rms is 120 V.

The resonant frequency is found by using the expression in f 0 = 1 2π LC f 0 = 1 2π LC . The current at that frequency is the same as if the resistor alone were in the circuit.

Entering the given values for L L and C C into the expression given for f 0 f 0 in f 0 = 1 2π LC f 0 = 1 2π LC yields

We see that the resonant frequency is between 60.0 Hz and 10.0 kHz, the two frequencies chosen in earlier examples. This was to be expected, since the capacitor dominated at the low frequency and the inductor dominated at the high frequency. Their effects are the same at this intermediate frequency.

The current is given by Ohm’s law. At resonance, the two reactances are equal and cancel, so that the impedance equals the resistance alone. Thus,

Discussion for (b)

At resonance, the current is greater than at the higher and lower frequencies considered for the same circuit in the preceding example.

Power in RLC Series AC Circuits

If current varies with frequency in an RLC circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in Figure 23.47 , voltage and current are out of phase in an RLC circuit. There is a phase angle ϕ ϕ between the source voltage V V and the current I I , which can be found from

For example, at the resonant frequency or in a purely resistive circuit Z = R Z = R , so that cos ϕ = 1 cos ϕ = 1 . This implies that ϕ = 0 º ϕ = 0 º and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because I rms I rms is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the average power is

Thus cos ϕ cos ϕ is called the power factor , which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency, cos ϕ = 1 cos ϕ = 1 .

Example 23.14

Calculating the power factor and power.

For the same RLC series circuit having a 40.0 Ω 40.0 Ω resistor, a 3.00 mH inductor, a 5.00 μF 5.00 μF capacitor, and a voltage source with a V rms V rms of 120 V: (a) Calculate the power factor and phase angle for f = 60 . 0 Hz f = 60 . 0 Hz . (b) What is the average power at 50.0 Hz? (c) Find the average power at the circuit’s resonant frequency.

Strategy and Solution for (a)

The power factor at 60.0 Hz is found from

We know Z = 531 Ω Z = 531 Ω from Example 23.12 , so that

This small value indicates the voltage and current are significantly out of phase. In fact, the phase angle is

The phase angle is close to 90º 90º , consistent with the fact that the capacitor dominates the circuit at this low frequency (a pure RC circuit has its voltage and current 90º 90º out of phase).

Strategy and Solution for (b)

The average power at 60.0 Hz is

I rms I rms was found to be 0.226 A in Example 23.12 . Entering the known values gives

Strategy and Solution for (c)

At the resonant frequency, we know cos ϕ = 1 cos ϕ = 1 , and I rms I rms was found to be 6.00 A in Example 23.13 . Thus,

P ave = ( 3 . 00 A ) ( 120 V ) ( 1 ) = 360 W P ave = ( 3 . 00 A ) ( 120 V ) ( 1 ) = 360 W at resonance (1.30 kHz)

Both the current and the power factor are greater at resonance, producing significantly greater power than at higher and lower frequencies.

Power delivered to an RLC series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next chapter on electromagnetic radiation, but it can also be suppressed as is the case in this chapter. The circuit is analogous to the wheel of a car driven over a corrugated road as shown in Figure 23.49 . The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels’ motion is a maximum if the bumps in the road are hit at the resonant frequency.

A pure LC circuit with negligible resistance oscillates at f 0 f 0 , the same resonant frequency as an RLC circuit. It can serve as a frequency standard or clock circuit—for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time. Figure 23.50 shows the analogy between an LC circuit and a mass on a spring.

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Resonance in Series RLC Circuit

Consider a series RLC circuit where a resistor , inductor and capacitor are connected in series across a voltage supply. This series RLC circuit resonates at a specific frequency known as the resonant frequency. In this circuit containing inductor and capacitor, the energy is stored in two different ways.

Variation in Inductive Reactance and Capacitive Reactance with Frequency

Variation of inductive reactance vs frequency.

We know that inductive reactance X L = 2πfL means inductive reactance is directly proportional to frequency (X L and prop ƒ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit acts as a short circuit; but when frequency increases; inductive reactance also increases. At infinite frequency, inductive reactance becomes infinity and circuit behaves as open circuit. It means that, when frequency increases inductive reactance also increases and when frequency decreases, inductive reactance also decreases. So, if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in the figure above.

Variation of Capacitive Reactance Vs Frequency

The formula for capacitive reactance X C = 1 / 2πfC shows that frequency and capacitive reactance are inversely proportional. At zero frequency (DC), capacitive reactance is infinite, and the circuit acts as an open circuit. As frequency increases, capacitive reactance decreases and becomes zero at infinite frequency, making the circuit act as a short circuit. The graph of capacitive reactance versus frequency forms a hyperbolic curve.

Inductive Reactance and Capacitive Reactance Vs Frequency

Variation of impedance vs frequency, resonant current, power factor at resonance, application of series rlc resonant circuit, leave a comment cancel reply.

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RLC circuit resonance experiment

• Thread starter Shreya
• Start date Aug 31, 2023
• Aug 31, 2023

• New research challenges conventional wisdom on wet surface adhesion
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• what circuit ?
• What C value ?
• ##\mu,\ n, \ A, \ l## ?
• what is ##x,\ y## ?

Shreya said: I have edited my question.

BvU said: μ, n, A, l ? what is x, y ?

Shreya said: The capacitor I used was rated 4.7##\mu##F.

Wait a minute, now that i think about it, the moving coil galvanometer must have had some inductance, right? That must be the reason of my higher value of calculated inductance  

A PF Mountain

The formula for the solenoid works for solenoid, which is supposed to be an ideal model with infinite length and only one layer of wire wrapped around the core. If not infinite, the ratio diameter to length should be much smaller than 1. Your coil does not seem to satisfy any of these conditions. So, why would you expect to follow that formula?  

Thanks @nasu . That makes sense.  

Shreya said: Wait a minute, now that i think about it, the moving coil galvanometer must have had some inductance, right? That must be the reason of my higher value of calculated inductance

BvU said: (But where does the 3.84 cm come from ? The picture suggests a lower value !)

BvU said: This thread is a clear statement that one should always evaluate lab results immediately . Maybe not with all the statistics and details, but at least order-of-magnitude and a preliminary result.

A PF Universe

Related to rlc circuit resonance experiment, what is the purpose of an rlc circuit resonance experiment.

The purpose of an RLC circuit resonance experiment is to study the behavior of a circuit consisting of a resistor (R), inductor (L), and capacitor (C) when subjected to varying frequencies of an AC signal. The experiment aims to identify the resonant frequency at which the circuit's impedance is minimized and the current is maximized, demonstrating the principles of resonance in electrical circuits.

How do you calculate the resonant frequency of an RLC circuit?

The resonant frequency (f 0 ) of an RLC circuit is calculated using the formula: f 0 = 1 / (2π√(LC)), where L is the inductance in henrys (H) and C is the capacitance in farads (F). At this frequency, the inductive reactance and capacitive reactance cancel each other out, resulting in the circuit behaving purely resistive.

What equipment is needed to perform an RLC circuit resonance experiment?

To perform an RLC circuit resonance experiment, you will need a signal generator to provide a range of AC frequencies, an oscilloscope to measure voltage and current, an RLC circuit board or individual components (resistor, inductor, capacitor), connecting wires, and a multimeter to measure resistance, inductance, and capacitance.

What observations are made during the RLC circuit resonance experiment?

During the RLC circuit resonance experiment, you observe the voltage across and current through the circuit components as the frequency of the AC signal is varied. At resonance, you should notice a peak in the current and a corresponding dip in the voltage across the resistor. The impedance of the circuit reaches a minimum at the resonant frequency, and the phase difference between voltage and current is zero.

How does the quality factor (Q factor) relate to the resonance of an RLC circuit?

The quality factor (Q factor) of an RLC circuit is a measure of how underdamped the circuit is and indicates the sharpness of the resonance peak. It is defined as Q = f 0 / Δf, where f 0 is the resonant frequency and Δf is the bandwidth over which the power is greater than half its peak value. A higher Q factor means a narrower and sharper resonance peak, indicating lower energy losses in the circuit.

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