Download November 5th Chapter 33 RLC Circuits

November
5th
Chapter 33
RLC Circuits
What’s up for the rest of the term
! (A9)
C33 – RLC circuits
! (A10) C34 – Electromagnetic (EM) waves
! (A11) C35 – Optics and images with EM waves
! (A12) C36 – Interference of EM waves
C37 – Diffraction of EM waves
Summary of Forced Oscillations
Element
Resistor
Reactance/
Resistance
R
Phase of Phase Amplitude
Current angle φ Relation
In phase
0°
VR=IRR
Capacitor XC= 1/ωdC Leads vC
(ICE)
Inductor
XL=ωdL Lags vL
(ELI)
!
-90°
VC=ICXC
+90°
VL=ILXL
ELI (positively) is the ICE man
!
!
!
Voltage or emf (E) before current (I) in an inductor (L)
Phase constant φ is positive for an inductor
Current (I) before voltage or emf (E) in capacitor (C)
RLC circuits
I=
ε
m
1
R + ( ωd L −
ωd C
2
!
!
)
2
Current is largest when
Or
1
ωd L −
=0
ωd C
ωd =
1
=ω
LC
ω is also called the resonance
frequency (in the homework)
ω =ω d = 2π f d
AC circuits
!
!
!
!
RLC circuit – resistor, capacitor
and inductor in series
Apply alternating emf
ε =ε
m
sin ω d t
Elements are in series so same
current is driven through each i = I sin ω t − φ
d
From the loop rule, at any
time t, the sum of the voltages
across the elements must
= vR + vC + vL
equal the applied emf
(
ε
)
AC circuits - Equations
ε
!
= IZ
m
Define impedance, Z to be
Z = R + (X L − XC )
2
!
XL − XC
tan φ =
R
Resonant frequency –
natural freq = driving freq
ω =ω d = 2π f d
where
2
X L = ωd L
1
XC =
ωd C
AC circuit Examples
AC circuits
!
Instantaneous rate which
energy is dissipated (power)
in a resistor is
2
P=i R
!
But
i = I sin(ωd t − φ )
P = I R sin (ωd t − φ )
2
!
2
Want average (rms) rate
!
Average over complete cycle T
〈sin θ 〉 = 1 2
2
AC circuits
!
For alternating current circuits define rootmean-square or rms values for i, V and emf
I
V
I rms =
V rms =
rms =
2
2
2
ε
!
!
Ammeters, voltmeters - give rms values
The average (rms) power dissipated by resistor
in an ac circuit is
2
Prms
!
ε
2
I R  I 
=
=
 R
2  2
(Called Pave in the book)
Prms = I
2
rms
R
AC circuits
!
If ac circuit has only resistive load Z=R
(e.g. at the resonance frequency)
Prms = ε rms I rms
!
Trade-off between current and voltage
!
!
!
For general use want low voltage
Means high current but
rms
P
=I
2
rms
R
General energy transmission rule:
Transmit at the highest possible voltage
and the lowest possible current
AC circuits
!
Transformer – device used to raise (for
transmission) and lower (for use) the ac
voltage in a circuit, keeping iV constant
!
Has 2 coils (primary and secondary) wound
on same iron core with different #s of turns
AC circuits
Alternating primary current induces
alternating magnetic flux in iron core
! Same core in both coils so induced flux
also goes through the secondary coil
! Using Faraday’s law
dΦ B
dΦ B
VP = − N P
VS = − N S
dt
dt
!
VP
VS
=
NP NS
AC circuits
!
Transformation of voltage is
NS
VS = VP
NP
!
!
If NS > NP called a
step-up transformer
If NS < NP called a
step-down transformer
AC circuits
!
Conservation of energy
I PVP = I SVS
VP
NP
IS = IP
= IP
VS
NS
AC circuits
!
The current IP appears in primary circuit
due to R in secondary circuit.
I S = VS / R
I PVP = I SVS
2
VS VS
1 V
1  NS 
 V P
=
IP =
V P = 
R VP
RV
R  NP 
2
S
2
P
!
Has for of IP = VP/Req where
2
R eq
 NP 
 R
= 
 NS 