physics for scientists and engineers ii, summer semester 2009 1 lecture 19: july 8 th 2009 physics...
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Physics for Scientists and Engineers II , Summer Semester 2009
1
Lecture 19: July 8th 2009
Physics for Scientists and Engineers II
Physics for Scientists and Engineers II , Summer Semester 2009
2
RLC Series Circuits –solved using Kirchhoff’s loop
tVv sinmax
0
0 :rule loop sKirchhoff'
2
2
C
q
dt
qdL
dt
dqRΔv
C
q
dt
dILRIΔv
Cv
i
CLR
LvRv
Physics for Scientists and Engineers II , Summer Semester 2009
3
RLC Series Circuits –solved using Kirchhoff’s loop
0sin2
2
max C
q
dt
qdL
dt
dqRtΔV
tAtqtAtqtAtq cos)(sin)(cos)(
:solution Try this2
0coscossinsin
:DE intoit Plug
2max t
C
AtLAtRAtV
sinsincoscoscos
sincoscossinsin
:identities theseUse
ttt
ttt
0sinsincoscossincoscossinsin 2max
ttLAC
AttRAtV
Physics for Scientists and Engineers II , Summer Semester 2009
4
RLC Series Circuits –solved using Kirchhoff’s loop
R
XX
RC
LL
CR
ωt
LAC
ARAt
LAC
ARAVt
ttLAC
AttRAtV
CL
1
tan0cos1
sin
term)cos (from
0cossincos
0sincossin
0sinsincoscossincoscossinsin
2
2max
2max
Physics for Scientists and Engineers II , Summer Semester 2009
5
RLC Series Circuits –solved using Kirchhoff’s loop
22
222
222
222
22
22
max
maxmax
max
2max
1
tan1cos
coscos
sincos
circuit) theof impedance"" thecalled is (Zsintancos
0sin1
cos
0sincos
term)in (from
CL
CLCL
XXRZ
XXRR
XXRZ
RR
Z
RRZ
RRZI
V
A
V
CLR
A
V
LAC
ARAV
ωts
Physics for Scientists and Engineers II , Summer Semester 2009
6
RLC Series Circuits –solved using Kirchhoff’s loop
tVv sinmax
Cv
i
CLR
LvRv
R
XX
XXRZ
VZI
tII
tVv
CL
CL
1
22
maxmax
max
max
tan
where
sin
sin
Physics for Scientists and Engineers II , Summer Semester 2009
7
RLC Series Circuits - book’s method
tVv sinmax
t
t
sinIi :assume sLet'
series).in are they (because elementscircuit threeallfor phase) and (amplitude
same theiscurrent The ).determined be to(also I amplitudean has and
much) how determined be (to voltage with thephase ofout iscurrent The
sinVv
max
max
max
Cv
i
CLR
LvRv
Physics for Scientists and Engineers II , Summer Semester 2009
8
RLC Series Circuits - book’s method
tVv sinmax
current) behind 90 lags C across (voltagecos2
sinXI
current) of ahead 90 is L across (voltage cos2
sinXI
current) with phasein is Rin (voltage sin sinI
Cmax
Lmax
max
tVtv
tVtv
tVtRv
CC
LL
RR
Cv
i
CLR
LvRv
Physics for Scientists and Engineers II , Summer Semester 2009
9
Phasor Diagram
maxI
RV
LV
CVThe voltages across the components are out of phaseas shown in the phasor diagram.They need to be added as vectors to get the total voltage.
Physics for Scientists and Engineers II , Summer Semester 2009
10
Phasor Diagram
maxI
RV
LVCV
The voltages across L and C can simply be subtracedfrom each other (180 degrees out of phase).
CL VV
Physics for Scientists and Engineers II , Summer Semester 2009
11
Phasor Diagram
maxI RV
CL VV
maxV
22max
2maxmax
2max
22max
CL
CL
CLR
XXRI
XIXIRI
VVVV
R
XX
R
XX
RI
XIXI
XXRI
VZ CL
CL1CL
max
CmaxLmax
R
CL
22
max
max
tanV
VVtan
:diagramphasor thefrom angle phase thedetermine alsocan We
e)resistenac toanalogousquantity (a Zimpedance""an definecan We
Physics for Scientists and Engineers II , Summer Semester 2009
12
Power in an AC Circuit
ttVItVtIviP sinsinsinsin maxmaxmaxmax
sincoscossinsin :Use ttt
source) by the deliveredpower ousinstantane theis (This
sincossincossin maxmax2
maxmax ttVItVIP
sincossincossin
:average) time(takePower Average
maxmax2
maxmax ttVItVIPavg
2
1sin2 t 02sin
2
1cossin ttt
coscos22
cos2
1 maxmaxmaxmax rmsrmsavg VI
VIVIP
Physics for Scientists and Engineers II , Summer Semester 2009
13
Power in an AC Circuit
cosrmsrmsavg VIP “Power factor”
maxI RV
CL VV
maxV
rms
RR
V
V
V
V
2
cosmax
RIRI
IV
VVIP rms
rmsrms
rms
Rrmsrmsavg
2
2
2
2
RIRIV rmsR 2:resistor For the max
up). heats(resistor resistor in theenergy internal
toconverted is source by the deliveredpower Average:means2RIP rmsavg
Physics for Scientists and Engineers II , Summer Semester 2009
14
Implications of power factor
cosrmsrmsavg VIP “Power factor”
rmsrmsavg VIP 1)0cos(cosresistor Only
:1 Example
inductor) pure a power toany deliver not does average,on source,power AC(an
00)90cos(cosinductor Only
:2 Example
avgP
capacitor) pure a power toany deliver not does average,on source,power AC(an
00)90cos(coscapacitor Only
:3 Example
avgP
Physics for Scientists and Engineers II , Summer Semester 2009
15
Resonance in a Series RLC Circuit – the current
circuit. theoffrequency resonance thecalled is
phase.in are voltageapplied
theandcurrent and valuemaximum a hasCurrent :1
For
1at0C
1-LX
tan:Also
10
C
1-L
for maximum a has anddependent frequency is
1
0
0
0L
0
0
22
22
LC
LCRR
X
LC
I
CLR
V
XXR
V
Z
VI
C
rms
rms
CL
rmsrmsrms
Physics for Scientists and Engineers II , Summer Semester 2009
16
0
0 20 40 60 80
R=10 Ohm
R=20 Ohm
R=30 Ohm
Resonance
R=0 Ohm
Resonance in a Series RLC Circuit – the Current
)( srad
0
rmsI
VV
mFC
HL
rms 10
0.1
0.1
22 1
CLR
V
Z
VI rmsrmsrms
Physics for Scientists and Engineers II , Summer Semester 2009
17
Resonance in a Series RLC Circuit – power
0
220
2222
22
22222
22
2222
22
2222
22
22
2
22
2
2
22
when maximum a has
1
11
1
avg
rmsrms
rmsrms
rms
CL
rmsrmsrmsavg
P
LR
RV
CLLR
RV
CLR
RV
CLR
RV
CLR
RV
XXR
RVR
Z
VRIP
Physics for Scientists and Engineers II , Summer Semester 2009
18
0
0 10 20 30 40 50 60
R=10 Ohm
R=20 Ohm
R=30 Ohm
Resonance
Resonance in a Series RLC Circuit – power
0 )( srad
avgP
220
2222
22
LR
RVP rmsavg R
LQ 00 :factorQuality
Describes sharpness of resonance.Q is larger for smaller R.
maximum) halfat width (full
Physics for Scientists and Engineers II , Summer Semester 2009
19
AC Transformer – a simple design
currents.eddy reduces Lamination
them).contains and field magnetic (increases
coreiron laminatedSoft
Primary Winding (input)
1v
S
LR2v
B
1N2N
Secondary Winding (output)
Physics for Scientists and Engineers II , Summer Semester 2009
20
AC Transformer - Voltage Transformation
ngeach windigh Flux throu :
law) s(Faraday'11
B
B
dt
dNv
1v
S
LR2v
1I B
1N2N
law) s(Faraday'12 dt
dNv B
11
22 v
N
Nv
Physics for Scientists and Engineers II , Summer Semester 2009
21
Step-up versus Step-down Transformers
former"down trans-Step":
er" transformup-Step":
12
12
11
22
NN
NN
vN
Nv
Physics for Scientists and Engineers II , Summer Semester 2009
22
AC Transformer – Connecting the load
1v LR 2v
1I B
1N2N
2I
222L :Rin dissipatedPower VIP rms values
LLLeqLeq
RN
NR
VNN
VR
V
VR
R
V
R
V
VIVI2
2
12
11
2
21
22
21
22
21
2211
:rmation)in transfo losspower (noer transformIdeal
side.primary from ed when view,resistance
load of resistance equivalent:R eq
matching" impedance"
for used becan r Transforme
Physics for Scientists and Engineers II , Summer Semester 2009
23
Transmission line economics
TR resistance has lineon Transmissi
consumer at
resistance Load :R Lgenerator AC
LR
TR
circuit Equivalent
Physics for Scientists and Engineers II , Summer Semester 2009
24
Transmission line economics
LR
TR
power)(1KW cleaner vacuumrunsConsumer
consumerlostconsumerL
TconsumerLT
L
consumerLTgenerator
TL
consumerTlost
L
consumerLCconsumer
PPPR
RPRR
R
PRRIP
RR
PRIP
R
PIRIΔVIP
2
2
22