•ac voltage and current - phasors •rms voltage and current …cremaldi/phys321/chapter-3.pdf ·...
TRANSCRIPT
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