![Page 1: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/1.jpg)
![Page 2: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/2.jpg)
![Page 3: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/3.jpg)
•
•
•
+
+
![Page 4: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/4.jpg)
•
•
for capacitance : i = Cdv
dt
for inductance : v = Ldi
dt
for resistance : i =v
R
![Page 5: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/5.jpg)
•
•
•
![Page 6: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/6.jpg)
urface of
the plates
![Page 7: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/7.jpg)
![Page 8: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/8.jpg)
•
•
•
E(t) = S (t) =
q(t)
S
Gauss
![Page 9: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/9.jpg)
•
d
0
v(t) = E(t)dl =0d E(t) d = q(t)
d
S
q(t) =S
d v(t)
C =S
d
![Page 10: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/10.jpg)
i(t) =dq(t)
dt
q(t) = C v(t)
i(t) = Cdv(t)
dt
![Page 11: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/11.jpg)
•
•
i(t) = Cdv(t)
dt
![Page 12: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/12.jpg)
•
•
i(t) = Cdv(t)
dt
![Page 13: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/13.jpg)
i(t) = Cdv(t)
dt
![Page 14: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/14.jpg)
•
•
i(t) = Cdv(t)
dt dv(t) =
1
Ci (t)dt
dv(t)v ( tO )
v ( t )
=1
Ci (t)dt
tO
t
v(t) v(tO ) =1
Ci (t)dt
tO
t
![Page 15: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/15.jpg)
•
•
•
v(t) = v(tO ) +1
Ci (t)dt
tO
t
q(tO ) =v(tO )
C
![Page 16: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/16.jpg)
•
i(t) = I 0 v(t) =I 0
Ct + v(0)
![Page 17: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/17.jpg)
•
•
dv
dt=I 0C
![Page 18: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/18.jpg)
•
•
i(t) =dq(t)
dt dq(t) = i (t)dt q(t) = i (t)dt
t
v(t) =q(t)
C v(t) =1
Ci(t)dt
t
![Page 19: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/19.jpg)
•
•
•
•
q(t) = i(t)dtt
= i (t)dttO
+ i (t)dttO
t
= q(tO ) + i (t)dttO
t
![Page 20: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/20.jpg)
•
•
•
v(t) =1
Ci(t)dt
t
=1
Ci (t)dt
tO
+ i (t)dttO
t
=q(tO )
C+
i (t)dttO
t
C= v(t0) +
1
Ci (t)dt
tO
t
![Page 21: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/21.jpg)
•
v(t) = v(t0) +1
Ci(t)dt
tO
t
![Page 22: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/22.jpg)
+
t t
v
t
v
![Page 23: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/23.jpg)
•
pE (t) =dwE (t)
dt= v(t) i (t) = v(t) C
dv(t)
dt
dwE = v(t) Cdv(t) = v(t) dq(t)
dwE = d
F •
d =
= E dq ( d) =v
d
dq d = v dq
![Page 24: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/24.jpg)
•
•
•
dwE = C v dv dwE
wE ( t )
= C v dvv ( )
v ( t )
wE (t) =C v 2 (t)
2
wE (t) =C v 2 (t)
2=q2 (t)
2 C
v =q
C
![Page 25: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/25.jpg)
•
•
wE (t) =C v 2 (t)
2=q2 (t)
2 C
![Page 26: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/26.jpg)
Mass : f = Mdv
dt Capacitor : i = C
dv
dt
![Page 27: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/27.jpg)
•
•
•
![Page 28: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/28.jpg)
v(t) =q(t)
CS
= v1 (t) + v2 (t) =q(t)
C1
+q(t)
C2
1
CS
=1
C1
+1
C2
i (t) = i1 (t) = i2 (t)
q(t) = i(t)dtt
=
= q1 (t) = i1 (t)dt =t
q2 (t) = i2 (t)dtt
![Page 29: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/29.jpg)
v(t) = v1 (t) + v2 (t)
i
C=dv
dt =
d
dt(v1 + v2) =
dv1
dt+dv2
dt=i
C1
+i
C2
1
C=
1
C1
+1
C2
1
C=d1
1A+d2
2A=1
C1+1
C2
![Page 30: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/30.jpg)
i = CP
dv
dt= i1 + i2 = C1
dv
dt+C2
dv
dt CP = C1 +C2
q(t) = i(t)dt
t = i1 (t)dt
t + i2 (t)dt
t = q1 (t) + q2 (t)
![Page 31: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/31.jpg)
v(t) =q(t)
C= v1 (t) =
q1 (t)
C1
= v2 (t) =q2 (t)
C2
Cp =1A1 + 2A2
d= 1A1
d+ 2A2
d= C1 + C2
![Page 32: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/32.jpg)
vS (t) =1
CiS (t)dt
tO
t
iS (t) = CdvSdt
![Page 33: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/33.jpg)
v(t) =1
CiS (t)dt
t
i(t) = iS (t)
v(t) = vS (t)
i(t) = CdvS (t)
dt
![Page 34: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/34.jpg)
iS (t) =0 for t 0
IO for t > 0
vS (t) =0 for t 0
VO for t > 0
![Page 35: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/35.jpg)
iS (t) =0 for t 0
IO for t > 0
v(t) =1
CiS (t)dt =
t
0 for t 0IOt
C for t > 0
![Page 36: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/36.jpg)
•
iS (t) =0 for t 0
IO for t > 0
v(t) =
0 for t 0IOt
C for t > 0
![Page 37: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/37.jpg)
vS (t) =0 for t 0
VO for t > 0
i(t) = Cdv(t)
dt =
0 for t < 0
? for t = 0
0 for t > 0
![Page 38: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/38.jpg)
•
vS (t) =0 for t 0
VO for t > 0
u(t;T ) T 0 u(t)
![Page 39: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/39.jpg)
•
du(t;T )
dt= (t;T )
![Page 40: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/40.jpg)
![Page 41: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/41.jpg)
•
for
T 0
1) (t) = 0 for t 0
2) (t)dt = u(t) du(t)
dt= (t)
t
3) (t)dt = 1+
![Page 42: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/42.jpg)
vS (t) =0 for t 0
VO for t > 0
= VO limT 0
u(t;T ) = VOu(t)
i(t) = Cdv(t)
dt = C
d
dtVO lim
T 0u(t;T )[ ] = CVO lim
T 0(t;T ) = CVO (t)
![Page 43: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/43.jpg)
•
![Page 44: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/44.jpg)
•
![Page 45: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/45.jpg)
•
![Page 46: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/46.jpg)
v1 (t) = v2 (t) v(t) for t tO +
v1 (t) =q1 (t)
C1
= v2 (t) =q2 (t)
C2
for t tO +
i1 (t) + i2 (t) = 0 for t tO +
dq1 (t)
dt+dq2 (t)
dt=d
dtq1 (t) + q2 (t)[ ] = 0 for t tO +
![Page 47: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/47.jpg)
q1 (t0 ) + q2 (t0 ) = q1 (t) + q2 (t) with t t0 +
Q1 +Q2 = C1v1 (t) +C2v2 (t) with t t0 +
Q1 +Q2 = C1v(t) +C2v(t) with t t0 +
Q1 +Q2 = (C1 +C2) v(t) with t t0 +
v(t) =Q1 +Q2
C1 +C2
=C1V1 +C2V2
C1 +C2
with t t0 +
![Page 48: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/48.jpg)
v(t) =Q1 +Q2
C1 +C2
=C1V1 +C2V2
C1 +C2
V with t t0 +
q1 (t) = C1v(t) = C1
Q1 +Q2
C1 +C2
= C1
C1V1 +C2V2
C1 +C2
with t t0 +
q2 (t) = C2v(t) = C2
Q1 +Q2
C1 +C2
= C2
C1V1 +C2V2
C1 +C2
with t t0 +
![Page 49: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/49.jpg)
v(t) =Q1 +Q2
C1 +C2
=C1V1 +C2V2
C1 +C2
V with t t0 +
v =m1v1 +m2v2m1 +m2
![Page 50: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/50.jpg)
wE (t < t0) =Q1
2
2C1
+Q2
2
2C2
wE (t > t0) =C1 (Q1 +Q2)
C1 +C2
2
1
2C1
+C2 (Q1 +Q2)
C1 +C2
2
1
2C2
=(Q1 +Q2)
C1 +C2
2
C1
2
2C1
+C2
2
2C2
=
=(Q1 +Q2)
C1 +C2
2
C1
2+C2
2
=(Q1 +Q2)
2
2(C1 +C2)
![Page 51: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/51.jpg)
wE (t < t0) wE (t > t0) =Q1
2
2C1
+Q2
2
2C2
(Q1 +Q2)2
2(C1 +C2)=
=Q1
2 (C1 +C2)C2 +Q2
2 (C1 +C2)C1 Q1
2C1C2 Q2
2C1C2 2Q1Q2C1C2
2C1C2 (C1 +C2)=
=Q1
2C2
2+Q2
2C1
2 2Q1Q2C1C2
2C1C2 (C1 +C2)=
(Q1C2 Q2C1)2
2C1C2 (C1 +C2) 0
![Page 52: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/52.jpg)
wE (t < t0) wE (t > t0) =(Q1C2 Q2C1)
2
2C1C2 (C1 +C2) 0
Q1C2 Q2C1 = 0 Q1C2 =Q2C1 Q1
C1
=Q2
C2
V1 = V2
![Page 53: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/53.jpg)
![Page 54: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/54.jpg)
•
![Page 55: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/55.jpg)
![Page 56: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/56.jpg)
![Page 57: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/57.jpg)
•
![Page 58: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/58.jpg)
•
Ampere' s Law :
B • d
l C
= μI enclosed
B • d
l C
= 0 B = 0
![Page 59: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/59.jpg)
•
B cos 0 dla
b
+ B cos 90O
b
c
dl + 0 cos 0O dlc
d
+ B cos 90O
d
a
dl = μNi
Bl = μNi B =μNi
l
Ampere' s Law :
B • d
l C
= μI enclosed
![Page 60: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/60.jpg)
•
•
•
B(t) =μNi(t)
l
(t) = B(t)S
L (t) = N (t) = N B(t) S =μN 2S
l i (t)
L =μN 2S
l
![Page 61: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/61.jpg)
•
L (t) =μ N 2 S
l i (t)
v(t) =d L (t)
dt= L
di (t)
dt
L =μN 2S
l
![Page 62: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/62.jpg)
L (t) = Li(t)
v(t) =d L (t)
dt
v(t) = Ldi (t)
dt
L
Flux Linkage : L
![Page 63: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/63.jpg)
![Page 64: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/64.jpg)
•
•
•
v(t) = Ldi (t)
dt
![Page 65: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/65.jpg)
v(t) = Ldi (t)
dt
![Page 66: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/66.jpg)
v(t) = Ldi(t)
dt v(t)dt = Ldi(t)
•
•
•
v(t)dtt0
t
= L di(t)i( t0 )
i( t )
i(t) = i(t0) +1
Lv(t)dt
t0
t
(t0) = Li(t0)
![Page 67: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/67.jpg)
•
v(t) =V0 i(t) =V0Lt + i(0)
![Page 68: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/68.jpg)
•
•
v(t) =d (t)
dtv(t)dt = d (t) (t) = v(t)dt
t
(t) = L i(t)
i(t) =1
Lv(t)dt
t
![Page 69: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/69.jpg)
•
•
•
(t) = v(t)dtt0
+ v(t)dt = (t0) +t0
t
v(t)dtt0
t
(t)
L=
(t0)
L+
v(t)dtt0
t
L i (t) = i0 (t) +
1
Lv(t)dt
t0
t
![Page 70: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/70.jpg)
•
i(t) = i0 (t) +1
Lv(t)dt
t0
t
![Page 71: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/71.jpg)
+
![Page 72: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/72.jpg)
•
wM (t) = pM (t)dt = i (t)v(t)dt = i (t)Ldi (t)
dt
ttt
= L i (t)di (t) =1
2
i ( t )
Li 2 (t)
pM (t) =dwM (t)
dt= i (t)v(t) = i (t)L
di (t)
dt
dwM (t) = i (t)Ldi (t) = i (t)d (t)
![Page 73: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/73.jpg)
•
•
•
wM (t) =Li 2 (t)
2=
2 (t)
2L
(t) = Li(t)
![Page 74: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/74.jpg)
Spring : f (t) = f (t0) + K v(t)dtt0
t
Inductor : i(t) = i (t0) +1
Lv(t)dt
t0
t
![Page 75: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/75.jpg)
• i(t) = i1 (t) + i2 (t) + i3 (t)
di
dt=d (i1 + i2 + i3)
dt
di
dt=di1dt+di2
dt+di3
dt
v(t)
Leq
=v(t)
L1
+v(t)
L2
+v(t)
L3
v(t) =d (t)
dt
![Page 76: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/76.jpg)
•
(t) = 1 (t) = 2 (t)
L i(t) = L1 i1 (t) = L2 i2 (t)
L =μN 2A
l(t) = L i (t)
![Page 77: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/77.jpg)
i(t0) = ik (t0) with k = 1,2,...,m
v(t) = v1 (t) + v2 (t) + v3 (t)
Leq
di
dt= L1
di
dt+ L2
di
dt+ L3
di
dt
v(t) =d (t)
dt
![Page 78: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/78.jpg)
•
L =μN 2A
l(t) = L i (t)
i(t) = i1 (t) = i2 (t) (t)
L= 1 (t)
L1
= 2 (t)
L2
L i(t) = L1 i (t) + L2 i (t) (t) = 1 (t) + 2 (t)
![Page 79: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/79.jpg)
iS (t) =1
LvS (t)dt
t
vS (t) = LdiS (t)
dt
![Page 80: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/80.jpg)
v(t) = vS (t)
i(t) =1
LvS (t)dt
t
![Page 81: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/81.jpg)
i (t)
i(t)
i(t) = iS (t)
v(t) = LdiS (t)
dt
![Page 82: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/82.jpg)
vS (t) =0 for t 0
VO for t > 0
i(t) =1
LvS (t)dt =
t
0 for t 0VOt
L for t > 0
![Page 83: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/83.jpg)
•
i(t) =
0 for t 0VOt
L for t > 0
![Page 84: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/84.jpg)
iS (t) =
0 for t t0
I 0 for t > t0
= I 0u(t)
v(t) = Ldi (t)
dt= L
d
dtI 0u(t)[ ] = LI 0 (t)
![Page 85: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/85.jpg)
•
1 (t0 ) = 1
2 (t0 ) = 2
i1 (t0 ) = I1 =1
L1
i2 (t0 ) = I 2 =2
L2
![Page 86: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/86.jpg)
•
i(t) = 1 + 2
L1 + L2
=L1I1 + L2I 2
L1 + L2
with t t0 +
![Page 87: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/87.jpg)
•
1 (t) = L1i (t) = L11 + 2
L1 + L2
= L1
L1I1 + L2I 2
L1 + L2
with t t0 +
2 (t) = L2i (t) = L21 + 2
L1 + L2
= L2
L1I1 + L2I 2
L1 + L2
with t t0 +
i(t) = 1 + 2
L1 + L2
=L1I1 + L2I 2
L1 + L2
I with t t0 +
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![Page 89: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/89.jpg)
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![Page 90: personal.utdallas.edukamran/EE3301/class notes/ch6.pdf · B cos0 dl a b + B cos90O b c dl+ 0 cos0O dl c d + B cos90O d a dl = μNi Bl = μNi B = μNi l Ampere's Law: B •d l C =](https://reader035.vdocuments.us/reader035/viewer/2022070714/5ed49798dc3bfa6a3e331296/html5/thumbnails/90.jpg)
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