<< Chapter < Page Chapter >> Page >
  • 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.

Impedance

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. [link] 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 size 12{X rSub { size 8{L} } } {} and X C size 12{X rSub { size 8{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 figure describes an R LC series circuit. It shows a resistor R connected in series with an inductor L, connected to a capacitor C in series to an A C source V. The voltage of the A C source is given by V equals V zero sine two pi f t. The voltage across R is V R, across L is V L and across C is V C.
An RLC series circuit with an AC voltage source.

The combined effect of resistance R size 12{R} {} , inductive reactance X L size 12{X rSub { size 8{L} } } {} , and capacitive reactance X C size 12{X rSub { size 8{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:

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} } "." } {}

Here I 0 size 12{I rSub { size 8{0} } } {} is the peak current, V 0 size 12{V rSub { size 8{0} } } {} the peak source voltage, and 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 size 12{Z} {} in terms of R , X L size 12{X rSub { size 8{L} } } {} , and X C size 12{X rSub { size 8{C} } } {} , we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled V R size 12{V rSub { size 8{R} } } {} , V L size 12{V rSub { size 8{L} } } {} , and V C size 12{V rSub { size 8{C} } } {} in [link] .

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 size 12{R} {} , L size 12{L} {} , and C size 12{C} {} are equal and in phase. But we know from the preceding section that the voltage across the inductor V L size 12{V rSub { size 8{L} } } {} leads the current by one-fourth of a cycle, the voltage across the capacitor V C size 12{V rSub { size 8{C} } } {} follows the current by one-fourth of a cycle, and the voltage across the resistor V R size 12{V rSub { size 8{R} } } {} is exactly in phase with the current. [link] shows these relationships in one graph, as well as showing the total voltage around the circuit V = V R + V L + V C size 12{V=V rSub { size 8{R} } +V rSub { size 8{L} } +V rSub { size 8{C} } } {} , where all four voltages are the instantaneous values. According to Kirchhoff’s loop rule, the total voltage around the circuit V is also the voltage of the source.

You can see from [link] that while V R size 12{V rSub { size 8{R} } } {} is in phase with the current, V L size 12{V rSub { size 8{L} } } {} leads by 90º , and V C size 12{V rSub { size 8{C} } } {} follows by 90º . Thus V L size 12{V rSub { size 8{L} } } {} and V C size 12{V rSub { size 8{C} } } {} are 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 size 12{V rSub { size 8{0} } } {} of the source does not equal the sum of the peak voltages across R size 12{R} {} , L size 12{L} {} , and C size 12{C} {} . The actual relationship is

V 0 = V 0 R 2 + ( V 0 L V 0 C ) 2 , size 12{V rSub { size 8{0} } = sqrt {V rSub { size 8{0R} } "" lSup { size 8{2} } + \( V rSub { size 8{0L} } - V rSub { size 8{0C} } \) rSup { size 8{2} } } ,} {}

where V 0 R size 12{V rSub { size 8{0R} } } {} , V 0 L size 12{V rSub { size 8{0L} } } {} , and V 0 C size 12{V rSub { size 8{0C} } } {} are the peak voltages across R size 12{R} {} , L size 12{L} {} , and C size 12{C} {} , respectively. Now, using Ohm’s law and definitions from Reactance, Inductive and Capacitive , we substitute V 0 = I 0 Z size 12{V rSub { size 8{0} } =I rSub { size 8{0} } Z} {} into the above, as well as V 0 R = I 0 R size 12{V rSub { size 8{0R} } =I rSub { size 8{0} } R} {} , V 0 L = I 0 X L size 12{V rSub { size 8{0L} } =I rSub { size 8{0} } X rSub { size 8{L} } } {} , and V 0 C = I 0 X C size 12{V rSub { size 8{0C} } =I rSub { size 8{0} } X rSub { size 8{C} } } {} , yielding

Questions & Answers

where are mcq?
wald Reply
Please what is the meaning of mcq
Appiah
what is static on its own
Aniedi Reply
where is the position of distance
AHMED Reply
what is physics
Mbaba Reply
calculate the angular displacement for an object which completes five revolutions
Bisalom Reply
I need someone to explain how white light disperses to form the "ROYGBIV".
Dera Reply
when it pass through a glass prism through a process called dispersion of light
Mahmud
What is an atom
Sulaimon Reply
An atom is the smallest indivisible particle of an element
Dera
When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the northern hemisphere.
Collin Reply
find the change in entropy of a 2.00 kg block of gold at 1063^0C when it meets to become liquid gold at 1063^0C
precious Reply
if you are asked to make a very sensitive thermometer which of the following fluids would you choose
precious
between mercy and gasoline
precious
it good to use mercury because mercury does not wet glass and it does not evaporate easily
Desmond
0
firdaus
SFAR Sifar SIFAT -<SIFST
firdaus
how many particles are in 2 moles of chromium
Mario Reply
if so use the normal formula number of atom= number of particle/Avogadro's number
Aki
n= np/avogadtos constant. therefore n= 24/ 6.022×10²³
albert
24÷6.022×10²³
albert
@Albert is wrong
Aki
when you cross multiple it should give you Number of particles= mole*Avogadro's number X=2m*6.022*10^²³ X=1.20*10²⁴g
Aki
1.204×10^-22
Maame
please what is final velocity and initial velocity
Nonso Reply
don't know
Ekene
what do you want to become in future
Ekene
Y ar u asking pls
Nonso
I think initial velocity is the velocity that the mobile starts with at the start time (t=0s) but I don't think I heard abt final velocity
Malak
Malak where are you now I need to learn more from you
Ekene
initial velocity is the velocity an object possess at it intial position or is the starting velocity, while a final velocity is the velocity an object or body possess at it final stage or at the end of it motion
Mubarak
Bohr is kimia, of toksid, cloud, tree have cloud, tree, river but small from toksid fish or another.
firdaus Reply
Heavy, heavy kehidupan susah, kekayaan, berlambak, bergumpul. Dikenali.
firdaus Reply
Gravitional, Gravitional mean kehidup seseorang. Kehidupan bumi, kehidupan muka bumi, kehidupan dalam longitude, kehidupan dalam momentom, kehidupan dalam mongitude. Kehidupan dalam Pelajaran, mean Pelajar kolej.
firdaus
Nonconservative. Sains belajar
firdaus
hello
LFX
what are the types of kinetics
pawi Reply
what is torque pls
Sunday Reply
Practice Key Terms 4

Get the best College physics course in your pocket!





Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask