V=V0sin(ωt)

V=V0cos(ωt)

V=V0sin(ωt+φ)

AC Circuit TheoryAmplitude

Phase

R

Apply AC current to a resistor…

I I=I0sin(ωt)

VR=IR=I0Rsin(ωt)

C

I ( )

( )

( )20

0

0C

tsinIC

1

tcosIC

1

dttsinIC

1

C

QV

π−ωω

=

ωω

−=

ω== ∫…capacitor…

Voltage lags current by π/2I

L( ) ( )

200L tsinLItcosLIdt

dILV π+ωω=ωω==

…and inductor

Voltage leads current by π/2

Adding voltagesR1 R2 R3I0sin(ωt)

V=I0R1sin(ωt)+I0R2sin(ωt)+I0R3sin(ωt)

=I0(R1+R2+R3)sin(ωt)

R1 C1I0sin(ωt)

( ) ( )tcosC

ItsinRIV 0

10 ωω

−ω=

voltages out of phase…

( )tsinRIV 0R ω=

( )tcosC

IV 0

C ωω

−= VT

Phasors

ωtV0sin(ωt)

V0cos(ωt)

R CI0sin(ωt) ( ) ( )tcos

C

ItsinRIV 0

0T ωω

−ω=

RC phasorsR C

I0sin(ωt) ( ) ( )tcosC

ItsinRIV 0

0T ωω

−ω=

RI0

C

I0ω VT=V0sin(ωt+φ)

= + ω

ϕ =−ω

2

20 0

1V I R

C

1tan

CR

I0sin(ωt)

V0sin(ωt+φ)

( )CRtan 1 ω=ϕ− −

Phasors – mathematics( ) ( )tcos

C

ItsinRIV 0

0T ωω

−ω=

VT=V0sin(ωt+φ)

= + ω

ϕ =−ω

2

20 0

1V I R

C

1tan

CRAsin(ωt) + Bcos(ωt) = Rsin(ωt + φ)

Rsin(ωt + φ) = Rsin(ωt)cos(φ) + Rcos(ωt)sin(φ)

A=Rcos(φ) B=Rsin(φ)

A2 + B2 = R2

B/A = tan(φ)

RL phasorsR L

I0sin(ωt)

( ) ( )tcosLItsinRIV 00T ωω+ω=

RI0

0LIωVT=V0sin(ωt+φ)

( )= + ω

ωϕ =

220 0V I R L

Ltan

R

I0sin(ωt)V0sin(ωt+φ)

ω=ϕ −

L

Rtan 1

RL filterR

LVIN VOUT

( )

( )2RL

RL

22IN

OUT

1

LR

1L

V

V

ω+

ω=

ω+ω=

0V

V 0

1V

V

IN

OUT

IN

OUT

→→ω

→∞→ω

high pass filter

RI0

0OUT LIV ω=( )22

00 LRIV ω+=

Phasors: RC filterR

CVIN VOUTRI0

C

IV 0

OUT ω=

2

200IN

C

1RIVV

ω+==

( )

( )2

2

C12

IN

OUT

RC1

1

R

1

C

1

V

V

ω+=

+ω=

ω

1V

V 0

0V

V

IN

OUT

IN

OUT

→→ω

→∞→ω

low pass filter

Bode plot

ω =0

1 R or

RC L

2

1

V

V

IN

OUT0 =ω=ω

(-3dB)

DecibelsLogarithm of power ratio

IN

OUT

2IN

2OUT

10V

Vlog20

V

Vlog10dB ==

VOUT/VIN dB

10 20

1 0

0.1 –20

0.01 –40

0.001 –60

Bode plot

Complex numbers in AC circuit theory

( ) ( ) ( )θ=+= jexpzzIm jzRez

Re(z)

Im(z)

θ j2=–1

|z|cos(θ)

|z|s

in(θ

)

( ) ( )

( ) ( )

=ω=

=ω=

ω

ω

tj

tj

eVtsinVtV

eVtcosVtV

00

00

Im

Re

Complex Impedance

ω

ω

ω

= =

= =ω

= = ω

∫

R 0

0C

L 0

j t

j t

j t

V IR I R e

1 IV Idt e

C j C

dIV L j I Le

dt

The ratio of the voltage across a component to the current through it when both are expressed in complex notation

RZR =

−= =ω ωC

1 jZ

j C C

LjZL ω=

I=I0sin(ωt) → I=I0ejωt

Complex Impedance

Real part: resistance (R)

Imaginary part: reactance

ω−ω

C

1L

( ) ( ) ( )IphaseZphaseVphase

IZV

IZV

+=

⋅=⋅=

jωL

Cj

1

ω

R

Ohm’s law

Series / parallel impedances

Z1=R Z2=jωL

Impedances in series: ZTotal=Z1+Z2+Z3…

= ∑T nn

Z Z

Z1 Z2 Z3

Impedances in parallel

=

= + +

∑

…

nT n

1 2 3

1 1

Z Z

1 1 1

Z Z Z

LR C

−=ω3

jZ

C

RC low pass filterR

CVIN VOUT

ZR

ZC

CR

CIN

ZZ

ZV

+

( )( )

( )

( )ϕ

ω+=

ω+ω−=

ω−ω−

ω+=

ω+=

+=

+=

ω

ω

jeRC1

1V

RC1

RCj1V

RCj1

RCj1

RCj1

1V

RCj1

1V

RV

ZZ

ZVV

2IN

2IN

IN

IN

Cj1

Cj1

IN

CR

CINOUT

( )RCtan 1 ω−=ϕ −

RL high pass filterR

LVIN VOUTLjR

LjVV INOUT ω+

ω=

( )ϕ

ω+

ω= je

1VV

2

RL

RL

INOUT

ω=ϕ −

L

Rtan 1

CP2 June 2003

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LR C

I0ejωt

LRC series circuit

( )

IC

1LjR

IZZZ

VVVV

CLR

CLR

ω−ω+=

++=++=

2

2

C

1LRZ

ω−ω+=

−ω=ϕ ω−

R

Ltan C

11

Purely resistive at LC

1R =ω φ=0 Z=R

−ω+ω ω−

⋅

ω−ω+⋅=

R

Ltantj2

20

C1

1

eC

1LRIV

L

LRC parallel circuit

R C

I0ejωt

= + + =R L C

VI I I I

Z

ω−ω+=

L

1Cj

R

1

Z

1

= + ω − ω

+ − ω ω = + − ω ω

2 2

1Z

1 1j C

R L

1 1j C

R L

1 1C

R L

22

CL

1

R

1

1Z

ω−ω

+

=

ω−ω

=ϕ − CL

1Rtan 1

Bandpass filter

R

CVIN VOUT

L

IN

LC

LCOUT V

ZR

ZV

+=

CjLj

1

Z

1

LC

ω+ω

=

( )C

1jZ

L1LC ω−

=ω

( )

( )IN

L1

L1

OUT V

C

jR

C

j

V

ω−+

ω−=

ω

ω

( )2L12

IN

OUT

CR1

1

V

V

ω−+=

ω

1V

V

LC

1

0V

V 0 0

V

V

IN

OUT0

IN

OUT

IN

OUT

==ω=ω

→→ω→∞→ω

Bandpass filter( )2

L12

IN

OUT

CR1

1

V

V

ω−+=

ωbuild a radio filter

f0=455kHz

∆f=20kHzLC

1f2 00 =π=ω

ω∆±ω=ω 0

C=10nF

L=12.2µH

( ) ( )( ) 2

1

CR1

12

0L12

0

=ω∆±ω−+ ω∆±ω

( ) ( )( ) 1CR2

0L12

0=ω∆±ω−ω∆±ω

R=389Ω

∆ω = π∆2 f

I(t)=I0ejωt

LCR series circuit – current driven

2

2

C

1LRZ

ω−ω+=

−ω=ϕ ω−

R

Ltan C

11

tjj0 eeZIIRV ωϕ ⋅⋅⋅==

L

R

C

I(t)

L

R

C

V(t)=V0ejωt

LCR series circuit – voltage driven

2

2

C

1LRZ

ω−ω+=

−ω=ϕ ω−

R

Ltan C

11

ϕ−== j0 eZ

V

Z

VI

V(t)

ω → ∞ →

ω → →

ω → ω = 00

I 0

0 I 0

V I maximum

R

2

2

0

C

1LR

VI

ω−ω+

=

R/L=106

R/L=105

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LC circuit – power dissipation

Power is only dissipated in the resistor

proof – power dissipated inductor and capacitor –none

( ) ( )∫=T

0dttVtI

T

1P ω

π= 2T

( ) ( )( )

( )( )tcosXIV

tcosLIV

tcosIC

1V

tsinItI

0X

0L

0C

0

ω=ωω=

ωω

−=

ω=

Lor C

1X ω

ω−=

reactance

( ) ( ) ( ) 0tsinXIT2

1dttcostsinXI

T

1P

/2T

0

220

T

00C,L

=

ωω

=ωω=ωπ=

∫

L

R

CI(t)

LRC Power dissipation

L

R

C

V(t) Z=Z0ejφ

I

V(t)=

ω−ω+=

C

1LjRZ

2

20

C

1LRZ

ω−ω+=

( )R

Ltan C

1ω−ω

=ϕ

( ) ( )( ) ( )ϕ−ω=

ω=

tsinZ

VtI

tsinVtV

0

0

0

( ) ( ) ( ) ( ) ( )ϕ−ωω== tsintsinZ

VtItVtP

0

20

Instantaneous power

Power dissipation

( ) ( ) ( ) ( ) ( )ϕ−ωω== tsintsinZ

VtItVtP

0

20

( ) ( ) ( ) ( ) ( )[ ]ϕω−ϕωω= sintcoscostsintsinIV 00

( ) ( ) ( ) ( ) ( )[ ]dtsintcostsincostsinIVT

1P

T

0

200∫ ϕωω−ϕω=

( )( ) ( ) 0 dtcost2cos1T

IVP

T

0 2100 −ϕω−= ∫

T

0

00 cos2

t

T

IV

ϕ=

ϕ= cos2

IV 00 ϕ= cosIV RMSRMS

cosφ = power factor

Power factor

RMSRMSIV

P

power apparent

power averagecos ==ϕ

Resistive load Z=R cosφ=1

Reactive load Z=X cosφ=0

Bridge circuits

V(t) V

Z1 Z3

Z2Z4

To determine an unknown impedance

Bridge balanced when VA−VB=0

VA VB

( ) ( )− =+ +2 4

1 2 3 4

Z ZV t V t 0

Z Z Z Z( ) ( )

=+ +

+ = +

=

2 4

1 2 3 4

2 3 4 4 1 2

2 3 4 1

Z Z

Z Z Z Z

Z Z Z Z Z Z

Z Z Z Z

CP2 September 2003

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