1

Chapter 3 Alternating Current Circuits I

•AC Voltage and Current - Phasors

•RMS Voltage and Current

•Reactance and Impedance

•High Pass and Low Pass filters

•RLC Resonance Circuits

2

Waves and Phasors

y(t) = A cos(ωt-φ)ω=2πf where f=1/T ( !1

2, i2)

eiωt = (0,i)

ωt

x

iy

(!12,i2)

Phasor

eiωt =

z =(x,iy) = |z|eiπ/2 = ( cos(π/2), i sin(π/2)= (0,i)

z =(x,iy) = |z|eiπ/4 = ( cos(3π/4), isin(3π/4)=

3

Phase Lag in an AC Circuit

In a general AC circuit (RLC) we have to consider that the voltageand current may be out of phase due to the circuit elements.

V(t) = Vo sin(ωt) ~ Vo eiωt

I(t) = Io sin(ωt-φ) ~ Io ei(ωt-φ)

Vo eiωt

ωt Io ei(ωt-φ)

V(t) I(t) φ=π

V(t) I(t) φ=π/2

4

VRMS( )2 =1T

|V (t ) |2 dt0

T

! = V0

2

Tsin2 ("t )!dt

0

T

! =Vo2

T1+ cos(2"t )

2!dt

0

T

! =Vo2

2+Vo2

2T1

2"cos(2"t )

0

T

VRMS( )2 ==Vo2

2+Vo2

2T1

2"cos(2"T ) # 1( ) = Vo

2

2+Vo2

2T1

2"cos(2"

2$"

) # 1%&'

()* !

=0! "## $##

!!!!= !!!VRMS =Vo2

IRMS( )2 =1T

| I (t ) |2 dt0

T

! = Io2

2!!!!!!!!!!!!!!!IRMS =

Io2

PRMS( )2 =1T

| I (t )V (t ) |2 dt0

T

! =I0V0

Tsin2 ("t + + ) sin2 ("t )dt

0

T

!

Average and RMS Voltage, Current, Power

VAVG = 1TV (t ) dt

0

T

! IAVG = 1T

I (t ) dt 0

T

! PAVG = 1T

I (t )V (t ) dt 0

T

!

5

Reactance, Impedance, and Phasors

Consider the general RLC circuit with V (t) = V0ej!t , I(t) = I0e

j (!t"# ) :

V (t) = L dIdt

+ IR + 1C

I(t)dt$V (t) = I(t) i!L + R + 1

i!C%&'

()*

Reac tance orComplex Im pedance

! "### $### Ohm ' s Law Complex (AC) Form

In any AC Circuit the Re sistive, Capacitive, and Inductive elementscan be replaces by their Complex Im pedances!

ZR =!R!=! XL Re sistive Re ac tan ce!!!!!!!!!!1 = e+ i !0 # =!0ZL = i !L( ) = +i!XL Inductive Re ac tan ce!!!!!!!!+i = e+ i !+ /2 !!!!!!!# = ++ / 2

ZC = 1i

1!C

%&'

()* = "i!XC !!!!!!Capacitive Re ac tan ce!!!!!"i = e" i !+ /2 !!!!!!!# = "+ / 2

6

Magnitude and Phase

The total Re ac tan ce can be written as Z = R + i !L " 1!C

#$%

&'(

corresponding to a complex number

XR

XLZ

The magnitude (length) of Z is

| Z | = R2 + (!L " 1!C

)2

The phase angle between XR and Z

#Z = tan"1 (y / x) = tan"1!L " 1

!CR

cos(# ) = R| Z |

φ

XC

7

Simple RC Circuit - Low pass Filter (1)

ZR

ZC

What is the output voltage V and phase ! looking across the capacitor ?

VC = ZCZR + ZC

Vo ei" t =! #i /"CR # i /"C

Vo ei" t $ voltage divider eq

|VC | = VCVC* = (#i /"C)(+i /"C)

(R # i /"C)(R + i /"C)Vo = 1 /"C

R2 + 1 /"C( )2Vo!= 1 /"RC

1+ 1 /"RC( )2Vo

Phase : VC = ( #i /"CR # i /"C

)

=Z! "# $#

(R + i /"CR + i /"C

)

=1! "# $#

= 1 /" 2C 2

R2 + 1 /"C( )2

%

&'

(

)*

X! "### $###

# i R /"CR2 + 1 /"C( )2

%

&'

(

)*

Y! "### $###

+C = tan#1 YX

%&'

()* = tan#1 #R /"C

1 /" 2C 2%&'

()* = tan#1 #"RC( ) = cot#1 #1

"RC%&'

()* = tan#1 1

"RC%&'

()* # , / 2

At !!" = 1 / RC !!!+C = tan#1 1( ) # , / 2 = , / 4 # , / 2 = #, / 4

φR

1/ωC

gain = VC /V0

ZR

ZC VC

VO

R

C

V(t) VC VC

8

Simple RC Circuit - Low pass Filter (2)

Break !Frequency!occurs!when!!!!!break =1RC

!!"!!|VC | = 12Vo = 0.707 Vo

|VC | = 1

1+ !RC( )2Vo !!!! fbreak =

12#RC

break frequency! "## $##

fb

fbφ=−π/4

ln(0.707)

9

Simple RC Circuit - High Pass Filter

ZC

V(t) ZR

What is the output voltage VR and phase !R looking across the resistor?

VR =ZR

ZR + ZC

"#$

%&' Vo!e j( t = R

(R ) j /(C)!Vo!e j( t

|VR | = VRVR* = R2

(R ) j /(C)(R + j /(C)Vo = R

R2 + 1 /(C( )2Vo = (RC

1+ (RC( )2Vo

gain = VR /VO =(RC

1+ (RC( )2

Phase : VR = R2

R2 + 1 /(C( )2

"

#$%

&'+ j R /(C

R2 + 1 /(C( )2

"

#$%

&'!!!!!*!!!R = tan)1 Y

X"#$

%&'= tan)1 1

(RC"#$

%&'

At !!( = 1 / RC !!!!R = tan)1 1( ) = + / 4

θR

1/ωC

C

R

V(t) Vout

10

Simple RC Circuit - High Pass Filter (2)

|VR | = !RC

1+ !RC( )2Vo fbreak =

12"RC

break frequency! "## $##

!!!!!!|VR | = 12Vo = 0.707 Vo

fb fb

φ=π/4

ln(0.707)

11

Differentiator and Integrator Connection

R

CV(t)

VC =1C

i ! dt

int egratorRC>>T1/T >>1/RCf >>1/RCf > fbreak

! "# $#

RV(t)

VR = R dq / dtdifferentiatorRC<<T1/T <<1/RCf << 1/RCf < fbreak

! "## $##

fbreak fbreak

12

Band Pass or Notch Filter

ωHI= ωLO=

13

RLC Circuit

damping

spring

mass

14

Resonance Condition and Q-factor

ωο

ZL=ZC at ω0

Q = |ZL|/R = |ZC|/R

15

Power in AC Circuits

16

Decibel Scale

DECIBEL SCALE When measuring power gain and voltage gain in an amplifier or circuit we often use the dcibel scale. Power P db = 10 log(Pout/Pin) Voltage

V db = 20 log 10 (Vout /Vin) = 20 log 10 (gain) Vdb at the Break Frequency At the breaking frequency the gain Vout/Vin drops by a factor of 1/

2 . This is called the -3 db point. Can you jusify this rema rk?

Sound Power• Near total silence - 0 dB• A whisper - 15 dB• Normal conversation - 60 dB• A lawnmower - 90 dB• A car horn - 110 dB• A rock concert or a jet engine - 120 dB• A gunshot or firecracker - 140 dB