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
1
Waves and Phasors
Phasor
iy
eiωt = (
!1
2
,
i
2
eiωt = (0,i)
)
ωt
x
y(t) = A cos(ωt-φ)
ω=2πf where f=1/T
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)=
(
!1 i
,
)
2 2
2
Phase Lag in an AC Circuit
In a general AC circuit (RLC) we have to consider that the voltage
and 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)
V(t)
I(t)
φ=π
Vo eiωt
ωt
Io ei(ωt-φ)
V(t)
I(t)
φ=π/2
3
Average and RMS Voltage, Current, Power
VAVG =
(V )
2
RMS
=
1
1
T
T
! V (t ) dt
0
T
! | V (t ) |
T
2
dt =
0
(V )
2
RMS
I AVG =
1
T
T
! I ( t ) dt
0
T
V02
! sin (" t )!dt =
2
T
PAVG =
0
Vo 2
T
T
!
1
T
T
! I (t )V ( t ) dt
1 + cos(2" t )
2
0
0
!dt =
Vo 2
2
+
Vo 2 1
2T 2"
T
cos(2" t )
0
Vo 2 1 %
2$
Vo
(
==
+
+
( cos(2" T ) # 1) =
'& cos(2" ) # 1*) ! !!!!=!!!VRMS =
2
2T 2"
2
2T 2" !##""##$
2
Vo 2
Vo 2 1
Vo 2
=0
(I )
2
RMS
(P )
RMS
2
=
=
1
T
1
T
! | I (t ) | dt =
2
0
Io 2
2
!!!!!!!!!!!!!!!I RMS =
T
! | I (t )V (t ) |
T
0
2
dt =
I 0V0
T
Io
2
T
! sin (" t + + ) sin (" t )dt
2
2
0
4
Reactance, Impedance, and Phasors
Consider the general RLC circuit with V (t) = V0 e j! t , I(t) = I 0 e j (! t "# ) :
dI
1
V (t) = L + IR + $ I(t)dt
dt
C
1 (
%
V (t) = I(t) ' i! L + R +
*) Ohm ' s Law Complex (AC) Form
&
i
!
C
!###"###
$
Re ac tan ce or
Complex Im pedance
In any AC Circuit the Re sistive, Capacitive, and Inductive elements
can be replaces by their Complex Im pedances!
Z R =!R!=! X L
Z L = i (! L ) = +i! X L
1% 1 (
ZC = '
* = "i! XC
i & !C )
Re sistive Re ac tan ce!!!!!!!!!!1 = e+i ! 0
# =!0
Inductive Re ac tan ce!!!!!!!!+i = e+i ! + /2 !!!!!!!# = ++ / 2
!!!!!!Capacitive Re ac tan ce!!!!!"i = e"i ! + /2 !!!!!!! # = "+ / 2
5
Magnitude and Phase
1 &
#
The total Re ac tan ce can be written as Z = R + i % ! L "
(
$
!C '
corresponding to a complex number
The magnitude (length) of Z is
XL
Z
XC
φ
XR
| Z | = R 2 + (! L "
1 2
)
!C
The phase angle between X R and Z
# Z = tan "1 (y / x) = tan "1
cos(# ) =
!L "
1
!C
R
R
|Z|
6
Simple RC Circuit - Low pass Filter (1)
R
V(t)
VC
ZR
VC
ZC
C
ZR
VO
ZC
What is the output voltage V and phase ! looking across the capacitor ?
ZC
#i / " C
VC =
Vo ei" t =!
Vo ei" t $ voltage divider eq
gain = VC / V0
Z R + ZC
R # i / "C
| VC | = VCVC
*
(#i / " C)(+i / " C)
=
Vo =
(R # i / " C)(R + i / " C)
1 / "C
R + (1 / " C )
2
2
Vo!=
1 / " RC
1 + (1 / " RC )
2
VC
R
φ
Vo
1/ωC
X ###
Y ###
#"
$ !##
#"
$
=Z #$ !#"
=1 #$ !##
!#"
2
2
% 1/" C
( %
(
#i / " C
R + i / "C
R / "C
Phase : VC = (
)(
)=' 2
#
i
' 2
2*
2*
R # i / "C R + i / "C
& R + (1 / " C ) ) & R + (1 / " C ) )
%Y (
% #R / " C (
% #1 (
% 1 (
+C = tan #1 ' * = tan #1 '
= tan #1 ( #" RC ) = cot #1 '
= tan #1 '
#, /2
*
2 2 *
& X)
&1/" C )
& " RC )
& " RC *)
At !!" = 1 / RC!!!+C = tan #1 (1) # , / 2 = , / 4 # , / 2 = #, / 4
7
Simple RC Circuit - Low pass Filter (2)
1
1
!!"!!| VC | =
Vo = 0.707 Vo
RC
2
1
!!!! fbreak =
# RC
!##"2##
$
Break!Frequency!occurs!when!!!!! break =
| VC | =
1
1 + (! RC )
2
Vo
break frequency
ln(0.707)
fb
fb
φ=−π/4
8
Simple RC Circuit - High Pass Filter
C
V(t)
ZC
Vout
ZR
V(t)
R
What is the output voltage VR and phase ! R looking across the resistor ?
" ZR %
R
j( t
VR = $
Vo!e
=
!Vo!e j( t
'
(R ) j / ( C)
# Z R + ZC &
| VR | = VRVR* =
gain = VR / VO =
2
R
Vo =
(R ) j / ( C)(R + j / ( C)
R
R + (1 / ( C )
2
2
Vo =
( RC
1 + (( RC )
2
Vo
( RC
1 + (( RC )
R
θ
1/ωC
2
"
%
"
%
R2
R / (C
1 %
)1 " Y %
)1 "
Phase : VR = $ 2
+
j
!!!!!*!!
!
=
tan
=
tan
$# '&
$#
'
R
2'
$ 2
2'
X
( RC &
# R + (1 / ( C ) &
# R + (1 / ( C ) &
At !!( = 1 / RC!!!! R = tan )1 (1) = + / 4
9
Simple RC Circuit - High Pass Filter (2)
| VR | =
! RC
1 + (! RC )
2
Vo
1
fbreak =
" RC
!##"2##
$
!!!!!!| VR | =
1
Vo = 0.707 Vo
2
break frequency
ln(0.707)
fb
fb
φ=π/4
10
Differentiator and Integrator Connection
R
V(t)
V(t)
R
C
! i dt
C
!#"#$
VR = R dq / dt
VC =
!#
#"##
$
differentiator
RC<<T
1/T <<1/RC
f << 1/RC
f < fbreak
fbreak
1
int egrator
RC>>T
1/T >>1/RC
f >>1/RC
f > fbreak
fbreak
11
Band Pass or Notch Filter
ωHI=
ωLO=
12
RLC Circuit
spring
damping
mass
13
Resonance Condition and Q-factor
ZL=ZC at ω0
Q = |ZL|/R = |ZC|/R
ωο
14
Power in AC Circuits
15
Decibel Scale
DECIBEL SCALE
When measuring power gain and
scale.
voltage gain in an amplifier or circuit we often use the dcibel
Power
P db = 10 log(Pout/Pin)
Voltage
V db = 20 log 10 (Vout /Vin) = 20 log 10 (gain)
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
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?
16