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  • 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

what is specific heat capacity of watee
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paul
In Inelastic collision cunculate the vilocity
Anshu Reply
elucidate
Usman
explain how a body becomes electrically charged based on the presence of charged particles
Kym Reply
induction
babar
induction
DEMGUE
definitely by induction
Raymond
induction
Raymond
induction
Shah
induction
Korodhso
please why does a needle sinks in water
DEMGUE
induction
Korodhso
induction
Auwal
what are the calculations of Newton's third law of motiow
Murtala Reply
what is dark matter
apex Reply
(in some cosmological theories) non-luminous material which is postulated to exist in space and which could take either of two forms: weakly interacting particles ( cold dark matter ) or high-energy randomly moving particles created soon after the Big Bang ( hot dark matter ).
Usman
if the mass of a trolley is 0.1kg. calculate the weight of plasticine that is needed to compensate friction. (take g=10m/s and u=0.2)
Declan Reply
what is a galaxy
Maduka Reply
a galaxy is a type of phone e.g samsung galaxy there are diff types of samsung galaxy there is s5 s6 s7 s8 s9
lasisi
what isflow rate of volume
Abcd Reply
flow rate is the volume of fluid which passes per unit time;
Rev
flow rate or discharge represnts the flow passing in unit volume per unit time
bhat
When two charges q1 and q2 are 6 and 5 coulomb what is ratio of force
Mian Reply
incomplete question
lasisi
When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
bimo Reply
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nehemiah
why is it proportional
nehemiah Reply
i don't know
Adah
y
nehemiah
what are the relationship between distance and displacement
Usman Reply
They are interchangeable.
Shii
Distance is scalar, displacement is vector because it must involve a direction as well as a magnitude. distance is the measurement of where you are and where you were displacement is a measurement of the change in position
Shii
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Usman
I'm beginner in physics so I can't reason why v=u+at change to v2=u2+2as and vice versa
Usman
what is kinematics
praveen
kinematics is study of motion without considering the causes of the motion
Theo
The study of motion without considering the cause 0f it
Usman
why electrons close to the nucleus have less energy and why do electrons far from the nucleus have more energy
Theo
thank you frds
praveen
plz what is the third law of thermodynamics
Chidera Reply
third law of thermodynamics states that at 0k the particles will collalse its also known as death of universe it was framed at that time when it waa nt posible to reach 0k but it was proved wrong
bhat
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Rev Reply
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Jeff
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Practice Key Terms 4

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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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