chapter 28: alternating-current circuits thursday november 3shill/teaching/2049 fall11... ·...
TRANSCRIPT
Chapter 28: Alternating-Current Circuits Thursday November 3rd
• Review of energy and oscillations in LC circuits • Alternating current theory
• Defn of terms, e.g., rms values • Resistance • Capacitive reactance • Inductive reactance
• Putting it all together – LRC circuits • Voltage/phase relations • Impedance • Resonance
Reading: up to page 501 in the text book (Ch. 28)
U =UB+U
E=12LI 2 +
12Q2
C! =
1LC
Ch. 28: Electromagnetic oscillations
Q(t) =Qm
cos!t
I(t) =!!Qm
sin!t
Ch. 28: Alternating Current V(t)=V
Psin !t +!
V( ); I(t)= IP sin !t +!I( )
Sine curve starts at ωt = -π/6 or -30o
In this example: ϕV = +π/6 or 30o
Root-Mean-Square:
Vrms
=V
P
2
Irms
=I
P
2
! = 2!fAngular frequency:
V(t)=VPsin!t
I(t)=VR=VP
Rsin!t
IP VP
I(t) V(t)
AC circuits: the resistive term
For a resistor ONLY: Current and voltage in phase
! IP
=VP
/R
Irms
=Vrms
/R
Voltage (driving term):
Current response:
AC circuits: the capacitive term Current peaks ¼ cycle before voltage
V(t)
I(t)
VP IP
!t
Q(t) =CV(t) =CVPsin!t
Charge (driving term):
I(t) =dQdt
=CVP
ddt
(sin!t)
= !CVPcos!t = !CV
Psin(!t + "
2)
Current response:
AC circuits: the capacitive term Current peaks ¼ cycle before voltage
V(t)
I(t)
VP IP
!t
Q(t) =CV(t) =CVPsin!t
Charge (driving term):
I(t)= !CVPsin(!t + !
2)
Current response: Current leads voltage by 90o
IP=VP
1/ !C=VP
XC
Capacitive reactance: XC
= 1/ !C (units - !)
AC circuits: the inductive term Voltage peaks ¼ cycle before current
!tI(t)
V(t)
VP IP
VL=!LdI /dt =!V
Psin!t
Current (driving term):
I(t)=VP
Lsin!t! dt
=!VP
!Lcos!t =
VP
!Lsin(!t ! !
2)
Current response:
AC circuits: the inductive term Voltage peaks ¼ cycle before current
!tI(t)
V(t) Current (driving term):
I(t) =
VP
!Lsin(!t! "
2)
Current response:
Inductive reactance: XL= !L (units - !)
Voltage leads current by 90o
I
P=
VP
!L=
VP
XL
VL=!LdI /dt =!V
Psin!t
VP IP
Summary
v
Phasor Diagrams Resistor Capacitor Inductor
VRp= I
pR
V
Cp= I
pX
CVLp= I
pXL
V, I in phase V lags I by 90o V leads I by 90o
http://en.wikipedia.org/wiki/Phasor_(sine_waves)
V(t)= sin!t; I(t)= sin(!t !!)
Phasor Diagrams: Adding the Voltages
tan! =VLp!V
Cp
VRp
Phasor Diagrams: Adding the Voltages
tan! =
XL!X
C
R="L!1/ "C
R
V
p= I
pR2 + I
pX
L! I
pX
C( )2
! Ip=
Vp
R2 + XL!X
C( )2=Vp
Z
Z = R2 + X
L!X
C( )2
Modified Ohm’s law:
Impedance:
Phase:
Units:ohms!
"##
$
%&&
Resonance
Ip
=V
p
R2 + XL!X
C( )2
XL= X
C
XL> X
C
XL<X
C
V leads V lags
!
o=
1LC
At resonance, Z = R, and ϕ = 0 (just like a DC circuit)
Resonance XL= X
C
XL> X
C
XL<X
C
V leads V lags
P = 1
2I
pV
pcos! = I
rmsV
rmscos!
Power delivered to the circuit: