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A pure LC circuit with negligible resistance oscillates at f 0 size 12{f rSub { size 8{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. [link] shows the analogy between an LC circuit and a mass on a spring.

The figure describes four stages of an L C oscillation circuit compared to a mass oscillating on a spring. Part a of the figure shows a mass attached to a horizontal spring. The spring is attached to a fixed support on the left. The mass is at rest as shown by velocity v equals zero. The energy of the spring is shown as potential energy. This is compared with a circuit containing a capacitor C and inductor L connected together. The energy is shown as stored in the electric field E of the capacitor between the plates. One plate is shown to have a negative polarity and other plate is shown to have a positive polarity. Part b of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The mass is shown to move horizontal toward the fixed support with velocity v. The energy here is stored as the kinetic energy of the spring. This is compared with a circuit containing a capacitor C and inductor L connected together. A current is shown in the circuit and energy is stored as magnetic field B in the inductor. Part c of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The spring is shown as not stretched and the energy is shown as potential energy of the spring. The mass is show to have displaced toward left. This is compared with a circuit containing a capacitor C and inductor L connected together. The energy is shown as stored in the electric field E of the capacitor between the plates. One plate is shown to have a negative polarity and other plate is shown to have a positive polarity. But the polarities are reverse of the first case in part a. Part d of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The mass is shown to move toward right with velocity v. the energy of the spring is kinetic energy. This is compared with a circuit containing a capacitor C and inductor L connected together. A current is shown in the circuit opposite to that in part b and energy is stored as magnetic field B in the inductor.
An LC circuit is analogous to a mass oscillating on a spring with no friction and no driving force. Energy moves back and forth between the inductor and capacitor, just as it moves from kinetic to potential in the mass-spring system.

Phet explorations: circuit construction kit (ac+dc), virtual lab

Build circuits with capacitors, inductors, resistors and AC or DC voltage sources, and inspect them using lab instruments such as voltmeters and ammeters.

Circuit Construction Kit (AC+DC), Virtual Lab

Section summary

  • The AC analogy to resistance is impedance Z , the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
    I 0 = V 0 Z or I rms = V rms Z , size 12{I rSub { size 8{0} } = { {V rSub { size 8{0} } } over {Z} } " or "I rSub { size 8{ ital "rms"} } = { {V rSub { size 8{ ital "rms"} } } over {Z} } ,} {}
    where I 0 size 12{I rSub { size 8{0} } } {} is the peak current and V 0 size 12{V rSub { size 8{0} } } {} is the peak source voltage.
  • Impedance has units of ohms and is given by Z = R 2 + ( X L X C ) 2 size 12{Z= sqrt {R rSup { size 8{2} } + \( X rSub { size 8{L} } - X rSub { size 8{C} } \) rSup { size 8{2} } } } {} .
  • The resonant frequency f 0 size 12{f rSub { size 8{0} } } {} , at which X L = X C size 12{X rSub { size 8{L} } =X rSub { size 8{C} } } {} , is
    f 0 = 1 LC . size 12{f rSub { size 8{0} } = { {1} over {2π sqrt { ital "LC"} } } } {}
  • In an AC circuit, there is a phase angle ϕ size 12{ϕ} {} between source voltage V size 12{V} {} and the current I size 12{I} {} , which can be found from
    cos ϕ = R Z , size 12{"cos"ϕ= { {R} over {Z} } } {}
  • ϕ = size 12{ϕ=0 rSup { size 8{ circ } } } {} for a purely resistive circuit or an RLC circuit at resonance.
  • The average power delivered to an RLC circuit is affected by the phase angle and is given by
    P ave = I rms V rms cos ϕ , size 12{P rSub { size 8{"ave"} } =I rSub { size 8{"rms"} } V rSub { size 8{"rms"} } "cos"ϕ} {}
    cos ϕ size 12{"cos"ϕ} {} is called the power factor, which ranges from 0 to 1.

Conceptual questions

Does the resonant frequency of an AC circuit depend on the peak voltage of the AC source? Explain why or why not.

Suppose you have a motor with a power factor significantly less than 1. Explain why it would be better to improve the power factor as a method of improving the motor’s output, rather than to increase the voltage input.

Problems&Exercises

An RL circuit consists of a 40.0 Ω resistor and a 3.00 mH inductor. (a) Find its impedance Z at 60.0 Hz and 10.0 kHz. (b) Compare these values of Z with those found in [link] in which there was also a capacitor.

(a) 40.02 Ω at 60.0 Hz, 193 Ω at 10.0 kHz

(b) At 60 Hz, with a capacitor, Z=531 Ω , over 13 times as high as without the capacitor. The capacitor makes a large difference at low frequencies. At 10 kHz, with a capacitor Z=190 Ω , about the same as without the capacitor. The capacitor has a smaller effect at high frequencies.

An RC circuit consists of a 40.0 Ω resistor and a 5.00 μF capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of Z with those found in [link] , in which there was also an inductor.

An LC circuit consists of a 3 . 00 mH size 12{3 "." "00" μH} {} inductor and a 5 . 00 μF size 12{5 "." "00" μF} {} capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of Z size 12{Z} {} with those found in [link] in which there was also a resistor.

(a) 529 Ω at 60.0 Hz, 185 Ω at 10.0 kHz

(b) These values are close to those obtained in [link] because at low frequency the capacitor dominates and at high frequency the inductor dominates. So in both cases the resistor makes little contribution to the total impedance.

What is the resonant frequency of a 0.500 mH inductor connected to a 40.0 μF capacitor?

To receive AM radio, you want an RLC circuit that can be made to resonate at any frequency between 500 and 1650 kHz. This is accomplished with a fixed 1.00 μH inductor connected to a variable capacitor. What range of capacitance is needed?

9.30 nF to 101 nF

Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and capacitors ranging from 1.00 pF to 0.100 F. What is the range of resonant frequencies that can be achieved from combinations of a single inductor and a single capacitor?

What capacitance do you need to produce a resonant frequency of 1.00 GHz, when using an 8.00 nH inductor?

3.17 pF

What inductance do you need to produce a resonant frequency of 60.0 Hz, when using a 2.00 μF capacitor?

The lowest frequency in the FM radio band is 88.0 MHz. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2.50 pF capacitor? (b) The capacitor is variable, to allow the resonant frequency to be adjusted to as high as 108 MHz. What must the capacitance be at this frequency?

(a) 1.31 μH

(b) 1.66 pF

An RLC series circuit has a 2.50 Ω resistor, a 100 μH inductor, and an 80.0 μF capacitor.(a) Find the circuit’s impedance at 120 Hz. (b) Find the circuit’s impedance at 5.00 kHz. (c) If the voltage source has V rms = 5 . 60 V size 12{V rSub { size 8{"rms"} } =5 "." "60"`V} {} , what is I rms size 12{I rSub { size 8{"rms"} } } {} at each frequency? (d) What is the resonant frequency of the circuit? (e) What is I rms size 12{I rSub { size 8{"rms"} } } {} at resonance?

An RLC series circuit has a 1.00 kΩ resistor, a 150 μH inductor, and a 25.0 nF capacitor. (a) Find the circuit’s impedance at 500 Hz. (b) Find the circuit’s impedance at 7.50 kHz. (c) If the voltage source has V rms = 408 V size 12{V rSub { size 8{"rms"} } ="408"`V} {} , what is I rms size 12{I rSub { size 8{"rms"} } } {} at each frequency? (d) What is the resonant frequency of the circuit? (e) What is I rms size 12{I rSub { size 8{"rms"} } } {} at resonance?

(a) 12.8 kΩ

(b) 1.31 kΩ

(c) 31.9 mA at 500 Hz, 312 mA at 7.50 kHz

(d) 82.2 kHz

(e) 0.408 A

An RLC series circuit has a 2.50 Ω resistor, a 100 μH inductor, and an 80.0 μF capacitor. (a) Find the power factor at f = 120 Hz . (b) What is the phase angle at 120 Hz? (c) What is the average power at 120 Hz? (d) Find the average power at the circuit’s resonant frequency.

An RLC series circuit has a 1.00 kΩ resistor, a 150 μH inductor, and a 25.0 nF capacitor. (a) Find the power factor at f = 7.50 Hz . (b) What is the phase angle at this frequency? (c) What is the average power at this frequency? (d) Find the average power at the circuit’s resonant frequency.

(a) 0.159

(b) 80.9º

(c) 26.4 W

(d) 166 W

An RLC series circuit has a 200 Ω resistor and a 25.0 mH inductor. At 8000 Hz, the phase angle is 45.0º . (a) What is the impedance? (b) Find the circuit’s capacitance. (c) If V rms = 408 V size 12{V rSub { size 8{"rms"} } ="408"`V} {} is applied, what is the average power supplied?

Referring to [link] , find the average power at 10.0 kHz.

16.0 W

Questions & Answers

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Source:  OpenStax, College physics ii. OpenStax CNX. Nov 29, 2012 Download for free at http://legacy.cnx.org/content/col11458/1.2
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