# 23.12 Rlc series ac circuits

 Page 1 / 9
• 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 ${X}_{L}$ and ${X}_{C}$ , 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 $R$ , inductive reactance ${X}_{L}$ , and capacitive reactance ${X}_{C}$ 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:

${I}_{0}=\frac{{V}_{0}}{Z}\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}{I}_{\text{rms}}=\frac{{V}_{\text{rms}}}{Z}\text{.}$

Here ${I}_{0}$ is the peak current, ${V}_{0}$ 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 $Z$ in terms of $R$ , ${X}_{L}$ , and ${X}_{C}$ , we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled ${V}_{R}$ , ${V}_{L}$ , and ${V}_{C}$ 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 $R$ , $L$ , and $C$ are equal and in phase. But we know from the preceding section that the voltage across the inductor ${V}_{L}$ leads the current by one-fourth of a cycle, the voltage across the capacitor ${V}_{C}$ follows the current by one-fourth of a cycle, and the voltage across the resistor ${V}_{R}$ is exactly in phase with the current. [link] shows these relationships in one graph, as well as showing the total voltage around the circuit $V={V}_{R}+{V}_{L}+{V}_{C}$ , 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 ${V}_{R}$ is in phase with the current, ${V}_{L}$ leads by $\text{90º}$ , and ${V}_{C}$ follows by $\text{90º}$ . Thus ${V}_{L}$ and ${V}_{C}$ are $\text{180º}$ 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 ${V}_{0}$ of the source does not equal the sum of the peak voltages across $R$ , $L$ , and $C$ . The actual relationship is

${V}_{0}=\sqrt{{V}_{0R}^{\phantom{\rule{1.25em}{0ex}}2}+\left({V}_{0L}-{V}_{0C}{\right)}^{2}},$

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

where are mcq?
Please what is the meaning of mcq
Appiah
what is static on its own
where is the position of distance
what is physics
calculate the angular displacement for an object which completes five revolutions
I need someone to explain how white light disperses to form the "ROYGBIV".
when it pass through a glass prism through a process called dispersion of light
Mahmud
What is an atom
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.
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
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
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
don't know
Ekene
what do you want to become in future
Ekene
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.
Heavy, heavy kehidupan susah, kekayaan, berlambak, bergumpul. Dikenali.
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
what is torque pls