Lecture 19: July 8 th 2009
description
Transcript of Lecture 19: July 8 th 2009
![Page 1: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/1.jpg)
Physics for Scientists and Engineers II , Summer Semester 2009
1
Lecture 19: July 8th 2009
Physics for Scientists and Engineers II
![Page 2: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/2.jpg)
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
![Page 3: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/3.jpg)
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
![Page 4: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/4.jpg)
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
![Page 5: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/5.jpg)
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
![Page 6: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/6.jpg)
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
![Page 7: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/7.jpg)
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
![Page 8: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/8.jpg)
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
![Page 9: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/9.jpg)
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.
![Page 10: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/10.jpg)
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
![Page 11: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/11.jpg)
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
![Page 12: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/12.jpg)
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
![Page 13: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/13.jpg)
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
![Page 14: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/14.jpg)
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
![Page 15: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/15.jpg)
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
![Page 16: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/16.jpg)
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
![Page 17: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/17.jpg)
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
![Page 18: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/18.jpg)
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
![Page 19: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/19.jpg)
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)
![Page 20: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/20.jpg)
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
![Page 21: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/21.jpg)
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
![Page 22: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/22.jpg)
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
![Page 23: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/23.jpg)
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
![Page 24: Lecture 19: July 8 th 2009](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815858550346895dc5b193/html5/thumbnails/24.jpg)
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