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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.

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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.

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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.

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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.

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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.

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

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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

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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?

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

3.17 pF

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

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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

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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?

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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

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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.

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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

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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?

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

16.0 W

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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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