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41
alternating currents & electromagnetic waves PHY232 – Spring 2007 Jon Pumplin http://www.pa.msu.edu/~pumplin/PHY232 (Ppt courtesy of Remco Zegers)
Transcript of web.pa.msu.edu€¦ · P H Y 2 3 2-P u m p l i n-a l t e r n a t i n g c u r r e n t s a n d e l e...
alte
rna
ting
cur
rent
s &
ele
ctro
ma
gne
tic
wa
ves
PHY2
32 –
Sprin
g 2
007
Jon
Pum
plin
http
://w
ww
.pa
.msu
.ed
u/~p
ump
lin/P
HY2
32(P
ptc
our
tesy
of R
em
co
Zeg
ers
)
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s2
Que
stio
n
At t
=0, t
he s
witc
h is
clo
sed
. A
fter t
hat:
a)
the
cur
rent
slo
wly
inc
rea
ses
from
I =
0 to
I =
V/R
b)
the
cur
rent
slo
wly
de
cre
ase
s fro
m I
= V
/R to
I =
0c
) th
e c
urre
nt is
a c
ons
tant
I =
V/R
LR V
I
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s3
Ans
we
r
ØA
t t=0
, the
sw
itch
is c
lose
d.
Afte
r tha
t:a
) th
e c
urre
nt s
low
ly in
cre
ase
s fro
m I=
0 to
I=V
/Rb
) th
e c
urre
nt s
low
ly d
ec
rea
ses
from
I=V
/R to
I=0
c
) th
e c
urre
nt is
a c
ons
tant
I=V
/R
The
co
il o
pp
ose
s th
e fl
ow
of c
urre
nt d
ue to
se
lf-in
duc
tanc
e, s
o th
ec
urre
nt c
ann
ot i
mm
ed
iate
ly b
ec
om
e th
e m
axi
mum
I=V
/R. I
t will
slo
wly
rise
to th
is v
alu
e (
cha
rac
teris
tic ti
me
Ta
u=
L/R)
.
LR V
I
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s4
Alte
rna
ting
cur
rent
circ
uits
ØPr
evi
ous
ly, w
e lo
ok
at D
C c
ircui
ts:
the
vo
ltag
e d
eliv
ere
d b
y th
e s
our
ce
is
co
nsta
nt, a
s o
n th
e le
ft.Ø
No
w, w
e lo
ok
at A
C c
ircui
ts, i
n w
hic
h c
ase
th
e s
our
ce
is s
inus
oid
al.
A
is
us
ed
in c
ircui
ts to
de
note
this
.
R VV
R
II
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s5
A c
ircui
t with
a re
sist
or
ØTh
e v
olta
ge
ove
r the
resi
sto
r is
the
sa
me
as
the
vo
ltag
e d
eliv
ere
d b
y th
e s
our
ce
: V
R(t)
= V
0 si
nω ωωωt=
V0 si
n(2π πππ
ft)Ø
The
cur
rent
thro
ugh
the
resi
sto
r is:
I R
(t)=
(V
0/R)
sinω ωωω
tØ
Sinc
e V
(t)
and
I(t)
ha
ve th
e s
am
e b
eha
vio
r as
a fu
nctio
n o
f tim
e, t
hey
are
sa
id to
be
‘in
pha
se’.
ØV
0is
the
ma
xim
um v
olta
ge
ØV
(t)
is th
e in
sta
nta
neo
us v
olta
ge
Øω ωωω
is th
e a
ngul
ar f
req
uenc
y; ω ωωω
=2π πππf
f: fre
que
ncy
(Hz)
ØSE
T YO
UR
CA
LCU
LATO
R TO
RA
DIA
NS
WH
ERE
NEC
ESSA
RY
I
V(t
)=V
0sin
ω ωωωt
R
IR(A)
V0=
10 V
R=2
Ohm
ω ωωω=1
rad
/s
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s6
rms
cur
rent
s/vo
ltag
es
ØTo
und
ers
tand
ene
rgy
co
nsum
ptio
n b
y th
e c
ircui
t, it
do
esn
’t m
atte
r wha
t the
sig
n o
f th
e c
urre
nt/v
olta
ge
is. W
e n
ee
d
the
ab
solu
te a
vera
ge
cur
rent
s a
nd v
olta
ge
s (r
oo
t-m
ea
n-sq
uare
va
lue
s) :
ØV
rms=
Vm
ax/
√ √√√2Ø
I rms=
I ma
x/√ √√√2
ØTh
e fo
llow
ing
ho
ld:
ØV
rms=
I rmsR
ØV
ma
x=I m
axR
IR(A) |IR|(A) |VR|(V)
Vrm
s
I rms
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s7
po
we
r co
nsum
ptio
n in
an
AC
circ
uit
ØW
e a
lrea
dy
kno
w fo
r DC
P =
V I
= V
2 /R
= I2
R
ØFo
r AC
circ
uits
with
a s
ing
le
resi
sto
r:P(
t) =
V(t
) *
I(t)
= V
0 I 0
(sin
ω ωωωt)
2
ØA
vera
ge
po
we
r co
nsum
ptio
n:P a
ve=
Vrm
s* I r
ms=
V2 rm
s/R
= I2 rm
s R
whe
re
Vrm
s=
Vm
ax/
√ √√√2)
I rms=
I ma
x/√ √√√2
|IR|(A) |VR|(V)
Vrm
s
I rms
P(W)
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s8
vec
tor r
ep
rese
nta
tion
The
vo
ltag
e o
r cur
rent
as
a fu
nctio
n o
f tim
e c
an
be
de
scrib
ed
by
the
pro
jec
tion
of a
ve
cto
r ro
tatin
g w
ith
co
nsta
nt a
ngul
ar v
elo
city
on
one
of t
he a
xes
(x o
r y).
θ θθθ=ω ωωωt
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s9
AC
circ
uit w
ith a
sin
gle
ca
pa
cito
r
I
V(t
)=V
0sin
ω ωωωt
C
Vc
= V
0sin
ω ωωωt
Qc
= C
Vc=
C V
0 si
nω ωωωt
I c=
∆ ∆∆∆Qc/∆ ∆∆∆
t = ω ωωω
C V
0 c
osω ωωω
tSo
, the
cur
rent
pe
aks
ahe
ad
of t
he v
olta
ge
:Th
ere
is a
diff
ere
nce
in p
hase
of π πππ
/2 (
900 )
.
I(A)
Why
? W
hen
the
re is
no
t muc
h c
harg
e o
n th
e c
ap
ac
itor i
t re
ad
ily a
cc
ep
ts m
ore
a
nd c
urre
nt e
asi
ly fl
ow
s. H
ow
eve
r, th
e E
-fie
ld a
nd p
ote
ntia
l be
twe
en
the
pla
tes
inc
rea
se a
nd c
ons
eq
uent
ly it
be
co
me
s m
ore
diff
icul
t fo
r cur
rent
to fl
ow
and
the
cur
rent
de
cre
ase
s. If
the
po
tent
ial o
ver C
is m
axi
mum
, the
cur
rent
is z
ero
.
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s10
Ca
pa
citi
ve c
ircui
t -c
ont
inue
dI(A)
No
te:
I ma
x= ω ωωω
C V
0
For a
resi
sto
r we
ha
ve I
= V
0/R
so ‘1
/ω ωωωC
’is
sim
ilar t
o ‘R
’
And
we
writ
e:
I=V
/Xc
with
Xc=
1/ω ωωω
C t
hec
ap
ac
itive
rea
cta
nce
Uni
ts o
f Xc
are
Ohm
s. T
he c
ap
ac
itive
rea
cta
nce
ac
ts li
ke a
resi
sta
nce
in th
is c
ircui
t.
I
V(t
) =
V0 si
nω ωωω
t
C
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s11
Pow
er c
ons
ump
tion
in a
ca
pa
citi
ve c
ircui
t
The
re is
no
po
we
r co
nsum
ptio
n in
a p
ure
ly c
ap
ac
itive
circ
uit:
Ene
rgy
( 1/2
C V
2 ) g
ets
sto
red
whe
n th
e (
ab
solu
te)
volta
ge
ove
r the
ca
pa
cito
r is
inc
rea
sing
, and
rele
ase
d w
hen
it is
de
cre
asi
ng.
P ave
= 0
for a
pur
ely
ca
pa
citi
ve c
ircui
t
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s12
AC
circ
uit w
ith a
sin
gle
ind
ucto
r
I
V(t
) =
V0
sin
ω ωωωt
L
VL=
V0 si
nω ωωωt=
L ∆ ∆∆∆
I/∆ ∆∆∆t
I= -
(V0/
(ω ωωωL)
) c
osω ωωω
t(n
o p
roo
f he
re: y
ou
nee
d c
alc
ulus
…)
the
cur
rent
pe
aks
late
r in
time
tha
n th
e v
olta
ge
:th
ere
is a
diff
ere
nce
in p
hase
of π πππ
/2 (
900 )
I(A)
Why
? A
s th
e p
ote
ntia
l ove
r the
ind
ucto
r ris
es,
the
ma
gne
tic fl
uxp
rod
uce
s a
c
urre
nt th
at o
pp
ose
s th
e o
rigin
al c
urre
nt. T
he v
olta
ge
ac
ross
the
ind
ucto
r p
ea
ks w
hen
the
cur
rent
is ju
st b
eg
inni
ng to
rise
.
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s13
Ind
uctiv
e c
ircui
t -c
ont
inue
d
No
te:
I ma
x= V
0/(ω ωωω
L)
For a
resi
sto
r we
ha
ve I
= V
0/R
so ‘ω ωωω
L’is
sim
ilar t
o ‘R
’
And
we
writ
e:
I = V
/XL
with
XL =
ω ωωωL
the
ind
uctiv
e re
ac
tanc
eU
nits
of X
L a
re O
hms.
The
ind
uctiv
e re
ac
tanc
e a
cts
as
a re
sist
anc
ein
this
circ
uit.
IL(A)
I
V(t
) =
V0
sin
ω ωωωt
L
I(A)
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s14
Pow
er c
ons
ump
tion
in a
n in
duc
tive
circ
uit
The
re is
no
po
we
r co
nsum
ptio
n in
a p
ure
ly in
duc
tive
circ
uit:
Ene
rgy
( 1/2
L I2 )
ge
ts s
tore
d w
hen
the
(a
bso
lute
) c
urre
nt th
roug
h th
ein
duc
tor i
s in
cre
asi
ng, a
nd re
lea
sed
whe
n it
is d
ec
rea
sing
.
P ave
= 0
for a
pur
ely
ind
uctiv
e c
ircui
t
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s15
Rea
cta
nce
The
ind
uctiv
e re
ac
tanc
e (
and
ca
pa
citi
ve re
ac
tanc
e)
are
lik
e th
e re
sist
anc
e o
f a n
orm
al r
esi
sto
r, in
tha
t yo
u c
an
ca
lcul
ate
the
cur
rent
, giv
en
the
vo
ltag
e, u
sing
I =
V/X
L (o
r I =
V/X
C ).
This
wo
rks
for t
he M
axi
mum
va
lue
s, o
r fo
r the
RM
S a
vera
ge
va
lue
s.
But I
and
V a
re “
out
of p
hase
”, s
o th
e m
axi
ma
oc
cur
at
diff
ere
nt ti
me
s.
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s16
Co
mb
inin
g th
e th
ree
: the
LRC
circ
uit
ØTh
ing
s to
ke
ep
in m
ind
whe
n a
naly
zing
this
sys
tem
:Ø
1) T
he c
urre
nt in
the
sys
tem
ha
s th
e s
am
e v
alu
e
eve
ryw
here
I =
I 0 si
n(ω ωωω
t-φ φφφ)
Ø2)
The
vo
ltag
e o
ver a
ll th
ree
co
mp
one
nts
is e
qua
l to
the
so
urc
e v
olta
ge
at a
ny p
oin
t in
time
: V(t
) =
V0 si
n(ω ωωω
t)
I
V(t
)=V
0sin
ω ωωωt
LC
R
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s17
An
LRC
circ
uit
ØFo
r the
resi
sto
r: V
R =
IR a
nd V
Ra
nd I
are
in p
hase
ØFo
r the
ca
pa
cito
r: V
c=
I Xc
(“V
cla
gs
I by
900”
)
ØFo
r the
ind
ucto
r: V
L= I
XL
(“V
Lle
ad
s I b
y 90
0”)
Øa
t any
inst
ant
: VL+
Vc+V
R=V
0 si
n(ω ωωω
t).
But t
he m
axi
mum
va
lue
s o
f VL+
Vc+V
Rd
o N
OT
ad
d u
p to
V0
be
ca
use
the
y ha
ve th
eir
ma
xim
a a
t diff
ere
nt ti
me
s.
VRI VC
VL
I
V(t
)=V
0sin
ω ωωωt
LC
R
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s18
imp
ed
anc
e
ØD
efin
e X
= X
L-X
c=
rea
cta
nce
of R
LC c
ircui
t
ØD
efin
e Z
= √ √√√
[R2 +
(XL-
Xc)2
]= √ √√√
[R2 +
X2 ]
= im
pe
da
nce
of R
LC c
irØ
The
nV
tot=
I Z
loo
ks li
ke O
hms
law
!
I
V(t
)=V
0sin
ω ωωωt
LC
R
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s19
Reso
nanc
e
ØIf
the
ma
xim
um v
olta
ge
ove
r the
ca
pa
cito
r eq
uals
the
m
axi
mum
vo
ltag
e o
ver t
he in
duc
tor,
the
diff
ere
nce
in
pha
se b
etw
ee
n th
e v
olta
ge
ove
r the
who
le c
ircui
t and
the
vo
ltag
e o
ver t
he re
sist
or i
s:Ø
a)
00
Øb
)450
Øc
)900
Ød
)180
0
In t
his
ca
se, X
L
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s20
Pow
er c
ons
ump
tion
by
an
LRC
circ
uit
ØEv
en
tho
ugh
the
ca
pa
cito
r and
ind
ucto
r do
no
t co
nsum
e
ene
rgy
on
the
ave
rag
e, t
hey
affe
ct t
he p
ow
er
co
nsum
ptio
n si
nce
the
pha
se b
etw
ee
n c
urre
nt a
nd
volta
ge
is m
od
ifie
d.
ØP
= I2
rms R
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s21
Exa
mp
le
Øq
uest
ions
:Ø
wha
t is
the
ang
ula
r fre
que
ncy
of t
he s
yste
m?w
hata
re th
e in
duc
tive
and
c
ap
ac
itive
rea
cta
nce
s?Ø
wha
t is
the
imp
ed
anc
e, w
hat i
s th
e p
hase
ang
le φ φφφ
Øw
hat i
s th
e m
axi
mum
cur
rent
and
pe
ak
volta
ge
s o
ver e
ac
h e
lem
ent
Øc
om
pa
re th
e a
lge
bra
ic s
um o
f pe
ak
volta
ge
s w
ith V
0. D
oe
s th
is m
ake
se
nse
?Ø
wha
t are
the
inst
ant
ane
ous
vo
ltag
es
and
rms
volta
ge
s o
ver e
ac
h e
lem
ent
?Ø
wha
t is
po
we
r co
nsum
ed
by
ea
ch
ele
me
nt a
nd to
tal p
ow
er c
ons
ump
tion
I
V(t
)=V
0sin
ω ωωωt
LC
R
Giv
en:
R=25
0 O
hmL=
0.6
HC
=3.5
µ µµµF
f=60
Hz
V0=
150
V
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s22
ans
we
rsØ
a)
ang
ula
r fre
que
ncy
ω ωωωo
f the
sys
tem
?Ø
ω ωωω=2π πππ
f=2π πππ
60=3
77 ra
d/s
Øb
) Re
ac
tanc
es?
ØX
C=1
/ω ωωωC
=1/(
377
x 3.
5x10
-6)=
758
Ohm
ØX
L= ω ωωω
L=37
7x0.
6=22
6 O
hmØ
c)
Imp
ed
anc
e a
nd p
hase
ang
le
ØZ=
√ √√√[R2
+(X
L-X
c)2
]=√ √√√[
2502
+(22
6-75
8)2 ]
=588
Ohm
Øφ φφφ=
tan-
1 [(X
L-X
C)/
R)=t
an-
1 [(2
26-7
58)/
250]
=-64
.80
(or –
1.13
rad
)Ø
d)
Ma
xim
um c
urre
nt a
nd m
axi
mum
co
mp
one
nt v
olta
ge
s:
ØI m
ax=
Vm
ax/
Z=15
0/58
8=0.
255
AØ
VR=
I ma
xR=0
.255
x250
=63.
8 V
ØV
C=I
ma
xXC=0
.255
x758
=193
VØ
VL=
I ma
xXL=
0.25
5x26
6=57
.6 V
ØSu
m: V
R+V
C+V
L=31
4 V
. Thi
s is
larg
er t
han
the
ma
xim
um v
olta
ge
de
live
red
b
y th
e s
our
ce
(15
0 V
). T
his
ma
kes
sens
e b
ec
aus
e th
e re
leva
nt s
um is
no
t a
lge
bra
ic: e
ac
h o
f the
vo
ltag
es
are
ve
cto
rs w
ith d
iffe
rent
pha
ses.
Giv
en:
R=25
0 O
hmL=
0.6
HC
=3.5
µ µµµF
f=60
Hz
V0=
150
V
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s23
ans
we
rs
Øf)
inst
ant
ane
ous
vo
ltag
es
ove
r ea
ch
ele
me
nt (
Vto
tha
s 0
pha
se)?
Øst
art
with
the
driv
ing
vo
ltag
e V
=V0s
inω ωωω
t=V
tot
ØV
R(t)
=63.
8sin
(ω ωωωt+
1.13
) (
note
the
pha
se re
lativ
e to
Vto
t)Ø
VC(t
)=19
3sin
(ω ωωωt-
0.44
) p
hase
ang
le :
1.13
-π πππ/2
=-0.
44Ø
VL(
t)=5
7.6s
in(ω ωωω
t+2.
7) p
hase
ang
le :
1.13
+π πππ/2
=2.7
Ø
rms
volta
ge
s o
ver e
ac
h e
lem
ent
?Ø
VR,
rms=
63.8
/√ √√√2=
45.1
VØ
VC
,rms=
193/
√ √√√2=1
36 V
ØV
L,rm
s=57
.6/√ √√√
2=40
.7 V
ØI m
ax=
Vm
ax/
Z=0.
255
A
ØV
R=I m
axR
=63.
8 V
ØV
C=I
ma
xXC=1
93 V
ØV
L=I m
axX
L=57
.6 V
Øφ φφφ=
-64.
80(o
r –1.
13 ra
d)
ØV
tot=
150
V
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s24
ans
we
rs
Øg
) p
ow
er c
ons
ume
d b
y e
ac
h e
lem
ent
and
tota
l po
we
r co
nsum
ed
?Ø
P C=P
L=0
no e
nerg
y is
co
nsum
ed
by
the
ca
pa
cito
r or i
nduc
tor
ØP R
=Irm
s2 R=(
I ma
x/√ √√√2
)2 R=0
.255
2 R/2
=0.2
552 *
250/
2)=8
.13
WØ
or:
P R=V
rms2 /
R=(4
5.1)
2 /25
0=8.
13 W
(d
on’
t use
Vrm
s=V
0/√ √√√2
!!)Ø
or:
P R=V
rmsI r
msc
osφ φφφ
=(15
0/√ √√√2
)(0.
255/
√ √√√2)c
os(
-64.
80 )=8
.13
W
Øto
tal p
ow
er c
ons
ume
d=p
ow
er c
ons
ume
d b
y re
sist
or!
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s25
LRC
circ
uits
: an
ove
rvie
w
ØRe
ac
tanc
e o
f ca
pa
cito
r: X
c=
1/ω ωωωC
ØRe
ac
tanc
e o
f ind
ucto
r: X
L= ω ωωω
L
ØC
urre
nt th
roug
h c
ircui
t: sa
me
for a
ll c
om
po
nent
sØ
‘Ohm
s’ la
w fo
r LRC
circ
uit:
Vto
t=I Z
ØIm
pe
da
nce
: Z=√ √√√
[R2 +
(XL-
Xc)2 ]
Øp
hase
ang
le b
etw
ee
n c
urre
nt a
nd s
our
ce
vo
ltag
e:
tanφ φφφ
=(|V
L|-|
Vc|
)/V
R=(X
L-X
c)/
RØ
Pow
er c
ons
ume
d (
by
resi
sto
r onl
y): P
=I2 rm
sR=I
rmsV
R
P=V
rmsI r
msc
osφ φφφ
ØV
R=I m
axR
in p
hase
with
cur
rent
I, o
ut o
f pha
se b
y φ φφφ
with
Vto
t
ØV
C=I
ma
xXC
be
hind
by
900
rela
tive
to I
(and
VR)
ØV
L=I m
axX
La
hea
d o
f 900
rela
tive
to I
(and
VR)
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s26
Que
stio
n
ØTh
e s
um o
f ma
xim
um v
olta
ge
s o
ver t
he re
sist
or,
ca
pa
cito
r a
nd in
duc
tor i
n a
n LR
C c
ircui
t ca
nno
t be
hig
her t
han
the
m
axi
mum
vo
ltag
e d
eliv
ere
d b
y th
e s
our
ce
sin
ce
it
vio
late
s Ki
rchh
off’
s2n
dru
le (
sum
of v
olta
ge
ga
ins
eq
uals
th
e s
um o
f vo
ltag
e d
rop
s).
Øa
) tru
eØ
b)
fals
e
ans
we
r: fa
lse
Th
e m
axi
mum
vo
ltag
es
in e
ac
h c
om
po
nent
are
not a
chi
eve
d a
t the
sa
me
tim
e!
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s27
Reso
nanc
es
in a
n RL
C c
ircui
tØ
If w
e c
hanc
e th
e (
ang
ula
r) fr
eq
uenc
y th
e re
ac
tanc
es
will
cha
nge
si
nce
:Ø
Rea
cta
nce
of c
ap
ac
itor:
Xc=
1/ω ωωω
CØ
Rea
cta
nce
of i
nduc
tor:
XL=
ω ωωωL
ØC
ons
eq
uent
ly, t
he im
pe
da
nce
Z=√ √√√
[R2 +
(XL-
Xc)2 ]
cha
nge
sØ
Sinc
e I=
Vto
t/Z,
the
cur
rent
thro
ugh
the
circ
uit c
hang
es
ØIf
XL=
XC
(I.e
. 1/ω ωωω
C=
ω ωωωL
orω ωωω
2 =1/
LC),
Z is
min
ima
l, I i
s m
axi
mum
)Ø
ω ωωω=
√ √√√(1/
LC)
is th
e re
sona
nce
ang
ula
r fre
que
ncy
ØA
t the
reso
nanc
e fr
eq
uenc
y φ φφφ=
0
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s28
exa
mp
le
Giv
en:
R=25
0 O
hmL=
0.6
HC
=3.5
µ µµµF
f=60
Hz
V0=
150
V
Usi
ng th
e s
am
e g
ive
n p
ara
me
ters
as
the
ea
rlie
r pro
ble
m,
wha
t is
the
reso
nanc
e fr
eq
uenc
y?
ω ωωω=
√ √√√(1/
LC)=
690
rad
/sf=
ω ωωω/2
π πππ=11
0 H
z
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s29
que
stio
n
ØA
n LR
C c
ircui
t ha
s R=
50 O
hm, L
=0.5
H a
nd C
=5x1
0-3
F. A
n A
C s
our
ce
with
Vm
ax=
50V
is u
sed
. If t
he re
sist
anc
e is
re
pla
ce
d w
ith o
ne th
at h
as
R=10
0 O
hm a
nd th
e V
ma
xo
f the
so
urc
e is
inc
rea
sed
to 1
00V
, the
reso
nanc
e fr
eq
uenc
y w
ill:
Øa
) in
cre
ase
Øb
)de
cre
ase
Øc
) re
ma
in th
e s
am
e
ans
we
r c)
the
reso
nanc
e fr
eq
uenc
y o
nly
de
pe
nds
on
L a
nd C
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s30
trans
form
ers
trans
form
ers
are
use
d to
co
nve
rtvo
ltag
es
to lo
we
r/hi
ghe
r le
vels
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s31
trans
form
ers
Vp
Vs
prim
ary
circ
uit
with
Np
loo
ps
inc
oil
sec
ond
ary
c
ircui
t with
Ns
loo
ps
in c
oil
iron
co
re
If a
n A
C c
urre
nt is
ap
plie
d to
the
prim
ary
circ
uit:
Vp=-
Np∆Φ ∆Φ∆Φ∆Φ
B/∆ ∆∆∆t
The
ma
gne
tic fl
ux is
co
nta
ine
d in
the
iro
n a
nd th
e c
hang
ing
flux
ac
tsin
the
se
co
nda
ry c
oil
als
o: V
s=-N
s∆Φ ∆Φ∆Φ∆Φ
B/∆ ∆∆∆t
The
refo
re:
Vs=
(Ns/
Np)V
pif
Ns<
Np
the
n V
s<V
p
A p
erfe
ct t
rans
form
er i
s a
pur
e in
duc
tor (
no re
sist
anc
e),
so
no
po
we
r lo
ss:
P p
=PS
and
VpI p
=VsI s
; if N
s<N
pth
en
Vs<
Vp
and
I S>I
p
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s32
que
stio
n
a tr
ans
form
er i
s us
ed
to b
ring
do
wn
the
hig
h-vo
ltag
e d
eliv
ere
db
y a
po
we
rline
(10
kV)
to 1
20 V
. If t
he p
rima
ry c
oil
has
1000
0 w
ind
ing
s, a
) ho
w m
any
are
the
re in
the
se
co
nda
ry c
oil?
b
) If
the
cur
rent
in th
e p
ow
erli
neis
0.1
A, w
hat i
s th
e m
axi
mum
c
urre
nt a
t 120
V?
a)
Vs=
(Ns/
Np)V
po
r Ns=
(Vs/
Vp)N
p=
120
win
din
gs
b)
VpI p
=VsI s
so I s
=VpI p
/Vs=
8.33
A
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s33
que
stio
n
ØIs
it m
ore
ec
ono
mic
al t
o tr
ans
mit
po
we
r fro
m th
e p
ow
er
sta
tion
to h
om
es
at h
igh
volta
ge
or l
ow
vo
ltag
e?
Øa
) hi
gh
volta
ge
Øb
) lo
w v
olta
ge
ans
we
r: hi
gh
volta
ge
If th
e v
olta
ge
is h
igh,
the
cur
rent
is lo
wIf
the
cur
rent
is lo
w, t
he v
olta
ge
dro
p o
ver t
he p
ow
er
line
(w
ith re
sist
anc
e R
) is
low
, and
thus
the
po
we
r d
issi
pa
ted
in th
e li
ne (
[∆ ∆∆∆V
]2/R
=I2 R
) a
lso
low
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s34
ele
ctro
ma
gne
tic w
ave
s
ØJa
me
s M
axw
ell
form
aliz
ed
the
ba
sic
eq
uatio
ns g
ove
rnin
g
ele
ctri
city
and
ma
gne
tism
~18
70:
ØC
oul
om
b’s
law
ØM
ag
netic
forc
eØ
Am
pe
re’s
La
w (
ele
ctri
c c
urre
nts
ma
ke m
ag
netic
fie
lds)
ØFa
rad
ay’
s la
w (
ma
gne
tic fi
eld
s m
ake
ele
ctri
c c
urre
nts)
ØSi
nce
cha
ngin
g fi
eld
s e
lec
tric
fie
lds
pro
duc
e m
ag
netic
fie
lds
and
vic
e v
ers
a, h
e c
onc
lud
ed
:Ø
ele
ctri
city
and
ma
gne
tism
are
two
asp
ec
ts o
f the
sa
me
p
heno
me
non.
The
y a
re u
nifie
d u
nde
r one
se
t of l
aw
s: th
e
law
s o
f ele
ctro
ma
gne
tism
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s35
ele
ctro
ma
gne
tic w
ave
s
Ma
xwe
ll fo
und
tha
t ele
ctri
c a
nd m
ag
netic
wa
ves
trave
lto
ge
the
r thr
oug
h sp
ac
e w
ith a
ve
loc
ity o
f 1/√ √√√
(µ µµµ0ε εεε
0)v=
1/√ √√√(
µ µµµ 0ε εεε 0
)=1/
√ √√√(4π πππ
x10-7
x 8.
85x1
0-12 )
=2.9
98x1
08m
/s
whi
ch
is ju
st th
e s
pe
ed
of l
ight
(c
)
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s36
ele
ctro
ma
gne
tic w
ave
s c
an
be
use
d to
bro
ad
ca
st…
ØC
ons
ide
r the
exp
erim
ent
pe
rform
ed
by
He
rz(1
888)
I
He
rzm
ad
e a
n RL
C c
ircui
t with
L=2
.5 n
H, C
=1.0
nFTh
e re
sona
nce
fre
que
ncy
is ω ωωω
= √ √√√(
1/LC
)=6.
32x1
08ra
d/s
f= ω ωωω
/2π πππ=
100
MH
z.
Rec
all
tha
t the
wa
vele
ngth
of w
ave
s λ λλλ=
v/f=
c/f
=3x1
08 /10
0x10
6 =3.
0 m
wa
vele
ng
th: λ λλλ
=v/f
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s37
He
the
n c
ons
truc
ted
an
ant
enn
a
Øc
harg
es
and
cur
rent
s va
ry
sinu
soid
ally
in th
e p
rima
ry a
nd
sec
ond
ary
circ
uits
. The
cha
rge
s in
th
e tw
o b
ranc
hes
als
o o
scill
ate
at
the
sa
me
fre
que
ncy
f
Idip
ole
ant
enn
a
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s38
pro
duc
ing
the
ele
ctri
c fi
eld
wa
ve
ant
enn
a
++++++ ---------- ++++++----------
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s39
pro
duc
ing
the
ma
gne
tic fi
eld
wa
ve
ant
enn
a
++++++ ----------
I I
++++++----------
I I
E a
nd B
are
in p
hase
and
E=c
Bw
ithc
: sp
ee
d o
f lig
htTh
e p
ow
er/
m2 =
0.5E
ma
xBm
ax/
µ µµµ 0
The
ene
rgy
in th
e w
ave
issh
are
d b
etw
ee
n th
e
E-fie
ld a
nd th
e B
-fie
ld
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s40
que
stio
n
Ca
n a
sin
gle
wire
co
nne
cte
d to
the
+ a
nd –
po
les
of a
D
C b
atte
ry a
ct a
s a
tra
nsm
itte
r of e
lec
trom
ag
netic
wa
ves?
a)
yes
b)
no
ans
we
r: no
: the
re is
no
va
ryin
g c
urre
nt a
nd h
enc
e n
ow
ave
ca
n b
e m
ad
e.
PHY2
32 -
Pum
plin
-a
ltern
atin
g c
urre
nts
and
ele
ctro
ma
gne
tic w
ave
s41
c=f
λ λλλ
