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V–2 AC Circuits

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Main Topics

• Power in AC Circuits.

• R, L and C in AC Circuits. Impedance.

• Description using Phasors.

• Generalized Ohm’s Law.

• Serial RC, RL and RLC AC Circuits.

• Parallel RC, RL and RLC AC Circuits.

• The Concept of the Resonance.

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The Power

• The power at any instant is a product of the voltage and current:

P(t) = V(t) I(t) = V0sin(t)I0sin(t + )• The mean value of power depends on the

phase shift between the voltage and the current:

<P> = VrmsIrmscos • The quality cos is called the power factor.

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AC Circuit with R Only

• If a current I(t) = I0sint flows through a resistor R Ohm’s law is valid at any instant. The voltage on the resistor will be in-phase:

V(t) = RI0sint = V0sint

V0 = RI0

<P> = VrmsIrms = RIrms2 = Vrms

2/R• We define the impedance of the resistor :

XR = R

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AC Circuit with L Only I

• If a current I(t) = I0sint supplied by some AC power-source flows through an inductance L Kirchhoff’s law is valid in any instant:

V(t) – LdI(t)/dt =0• This gives us the voltage on the inductor:

V(t) = LI0cost = V0sin(t+/2)

V0 = LI0

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AC Circuit with L Only II

• There is a phase-shift between the voltage and the current on the inductor. The current is delayed by = /2 behind the voltage.

• The mean power now will be zero:

<P> = VrmsIrms cos = 0

• We define the impedance of the inductance:

XL = L V0 = I0 XL

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AC Circuit with L Only III

• Since the impedance, in this case the inductive reactance, is a ratio of the peak (and also rms) values of the voltage over current we can regard it as a generalization or the resistance.

• Note the dependence on ! The higher is the frequency the higher is the impedance.

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AC Circuit with C Only I

• If a current I(t) = I0sint supplied by some AC power-source flows through an capacitor C Kirchhoff’s law is valid in any instant:

V(t) – Q(t)/C =0• This is an integral equation for voltage:

V(t) = –I0/C cost = V0sin(t – /2)

V0 = I0/C

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AC Circuit with C Only II

• There is a phase-shift between the voltage and the current on the inductor. The voltage is delayed by = /2 behind the current.

• The mean power now will be again zero:

<P> = VrmsIrms cos = 0

• We define the impedance of the capacitor:

XC = 1/C V0 = I0 XC

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AC Circuit with C Only III

• Since the impedance, in this case the capacitive reactance, is a ratio of the peak (and rms) values of the voltage over current we can regard it again as a generalization or the resistance.

• Note the dependence on ! Here, the higher is the frequency the lower is the impedance.

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A Loudspeaker Cross-over

• The different frequency behavior of the impedances of an inductor and a capacitor can be used in filters and for instance to simply separate sounds in a loud-speaker.

• high-frequency speaker ‘a tweeter’ is connected is series with an capacitor.

• low-frequency speaker ‘a woofer’ is connected is series with an inductance.

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General AC Circuits I

• If there are more R, C, L elements in an AC circuit we can always, in principle, build appropriate differential or integral equations and solve them. The only problem is that these equations would be very complicated even in very simple situations.

• There are, fortunately, several ways how to get around this more elegantly.

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General AC Circuits II

• AC circuits are a two-dimensional problem.• If we supply any AC circuit by a voltage

V0sint, the time dependence of all the voltages and currents in the circuit will also oscillate with the same t but possibly different phase.

• So it is necessary and sufficient to describe any quantity by two parameters its phase and magnitude.

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General AC Circuits III

• There are two mathematical tools commonly used:• Two-dimensional vectors, so called, phasors in

a coordinate system which rotates with t so all the phasors, which also rotate, are still

• Complex numbers in Gauss plane. This is preferred since more operations (e.g. division, roots) are defined for complex numbers.

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General AC Circuits IV

• The description by both ways is similar: The magnitude of particular quality (voltage or current) is described by a magnitude of a phasor (vector) or an absolute value of a complex number and the phase is described by the angle with the positive x-axis or a real axis.

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General AC Circuits V

• The complex number approach:• Describe voltages V, currents I, impedances Z

and admittances Y = 1/Z by complex numbers.

• Then a general complex Ohm’s law is valid:

V = ZI or I=YV

• Serial combination: Zs = Z1 + Z2 + …

• Parallel combination Yp = Y1 + Y2 + …

• Kirchhoff’s laws are valid for complex I and V

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General AC Circuits VI

• The table of complex impedances and admitancess of ideal elements R, L, C,

• j is the imaginary unit j2 = -1:• R: ZR = R YR = 1/R

• L: ZL = jL YL = -j/L

• C: ZC = -j/C YC = jC

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RC in Series

• Let’s illustrate the complex number approach on a serial RC combination:

• Let I, common for both R and C, be real.

Z = ZR + ZC = R – j/C

|Z| = (ZZ*)1/2 = (R2 + 1/2 C2)1/2

tg = –1/RC < 0 … capacity like

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RL in Series

• Let’s have a R and L in series:

• Let I, common for both R and L, be real.

Z = ZR + ZC = R + jL

|Z| = (ZZ*)1/2 = (R2 + 2L2)1/2

tg = L/R > 0 … inductance like

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RC in Parallel

• Let’s have a R and L in parallel:

• Let V, common for both R and C, be real.

Y = YR + YC = 1/R + jC

|Y| = (YY*)1/2 = (1/R2 + 2C2)1/2

tg = –[C/R] < 0 … again capacity like

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RLC in Series I

• Let’s have a R, L and C in series:

• Let again I, common for all R , L, C be real.

Z = ZR + ZC + ZL = R + j(L - 1/C)

|Z| = (R2 + (L - 1/C)2)1/2

• The circuit can be either inductance-like if:

L > 1/C … > 0

• or capacitance-like:

L < 1/C … < 0

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RLC in Series II

• New effect of resonance takes place when:

L = 1/C 2 = 1/LC

• Then the imaginary parts cancel and the whole circuit behaves as a pure resistance:• Z, V have minimum, I maximum

• It can be reached by tuning L, C or f !

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RLC in Parallel I

• Let’s have a R, L and C in parallel:

• Let now V, common for all R , L, C be real.

Y = YR + YC + YL = 1/R + j(C - 1/L)

|Y| = (1/R2 + (C - 1/L)2)1/2

• The circuit can be either inductance-like if:

L > 1/C … > 0

• or capacitance-like:

L < 1/C … < 0

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RLC in Parallel II

• Again the effect of resonance takes place when the same condition is fulfilled:

L = 1/C 2 = 1/LC

• Then the imaginary parts cancel and the whole circuit behaves as a pure resistance:• Y, I have minimum, Z,V have maximum

• It can be reached by tuning L, C or f !

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Resonance

• General description of the resonance:• If we need to feed some system capable of

oscillating on its frequency 0 then we do it most effectively if the frequency our source matches 0 and moreover is in phase.

• Good mechanical example is a swing.• The principle is used in e.g. in tuning

circuits of receivers.

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Impedance Matching

• From DC circuits we already know that if we need to transfer maximum power between two circuits it is necessary that the output resistance of the first one matches the input resistance of the next one.

• In AC circuits we have to match (complex) impedances the same way.• Unmatched phase may lead to reflection!

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Homework

• Chapter 31 – 1, 2, 3, 4, 7, 12, 13, 24, 25, 40.

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Things to read and learn

• This lecture covers:

The rest of Chapter 31

• Try to understand the physical background and ideas. Physics is not just inserting numbers into formulas!

The Mean Power I• We choose the representative time interval = T:

dttttT

IV

tdttT

IV

dttItVT

P

T

T

]sincossincos[sin

sin)sin(

)()(1

200

0

00

0

The Mean Power II

• Since only the first integral in non-zero.

coscos2

}2

2sinsin]

2

2cos

2[{cos

00

00

0

00

rmsrms

TT

T

IVIV

dtt

dtt

dt

T

IVP

^

AC Circuit with C I• From definition of the current I = dQ/dt and

relation for a capacitor Vc = Q(t)/C:

)sin(cos

sin1

)(

200

0

0

0

tC

It

C

I

tdtC

IdQ

CtV

tt

• The capacitor is an integrating device.

^

LC Circuit I• We use definition of the current I = -dQ/dt

and relation of the charge and voltage on a capacitor Vc = Q(t)/C:

0)(

2

2

LC

tQ

dt

Qd

• We take into account that the capacitor is discharged by the current. This is homogeneous differential equation of the second order. We guess the solution.

LC Circuit II

• Now we get parameters by substituting into the equation:

)cos()( 0 tQtQ

• These are un-dumped oscillations.

LCtQ

LCtQ

10)(

1)(2

LC Circuit III• The current can be obtained from the

definition I = - dQ/dt:

• Its behavior in time is harmonic.

)sin(

)sin()(

0

0

tI

tQtI

^

LC Circuit IV• The voltage on the capacitor V(t) = Q(t)/C:

• is also harmonic but note, there is a phase shift between the voltage and the current.

)cos()( 0 tC

QtV

^