Chapter 19: DC Circuits - Drexel University

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Chapter 19:DC Circuits
Trevor King
Drexel University
October 15,2008
DC circuits

Direct Current (DC) circuits are important

often used in digital electronic devices and almost anything
that is battery powered.
 The physics of DC currents is straightforward
 Kircho's loop rule
 Kircho's junction rule
 Understanding the individual components
 Resistors
 Capacitors
...
V
5 V
R
10

C
15 F
I
Kircho's loop rule
V
9 V
20

10

I
a
b
c
d
e
f
0
3
6
9
potential V
f
a
b
c
d
e
f
X
loop
V
i
= 0 Conserving energy
Kircho's junction rule
V
9 V
I
1
I
2
I
3
a
X
junction
I
i
= 0 Conserving charge,steady state
Elements in series
V
9 V
R
a
R
b
R
c
R
d
Two elements are in series if you cannot make a loop crossing one
of them without also crossing the other.
Elements in parallel
V
9 V
R
a
R
b
R
c
R
d
Two elements are in parallel if you can make a loop crossing both
of them which crosses no other elements.
Resistors in series
R
a
,V
a
I
R
b
,V
b
a
b
V
T
,R
T
By Kircho's junction rule,I
a
= I
b
(because there are no
intervening junctions).
V
T
= V
a
+V
b
= IR
a
+IR
b
= I (R
a
+R
b
)
V
T
= IR
T
So the eective resistance is give by
R
T
= R
a
+R
b
Resistors in parallel
R
a
,V
a
R
b
,V
b
a
b
V
T
,R
T
By Kircho's loop rule,V
a
V
b
= 0 so V
a
= V
b
= V
T
.
By Kircho's junction rule,I
T
= I
a
+I
b
V
T
= I
a
R
a
= I
b
R
b
I
b
= I
a
R
a
R
b
I
T
= I
a
+I
b
= I
a
(1 +R
a
=R
b
)
V
T
= I
a
R
a
=
I
T
1 +R
a
=R
b
R
a
= I
T
(1=R
a
+1=R
b
)
1
= I
T
R
T
So the eective resistance is give by R
T
= (1=R
a
+1=R
b
)
1
Capacitors in series
C
a
,V
a
C
b
,V
b
a
b
V
T
,C
T
By Kircho's junction rule,Q
a
= Q
b
= Q
T
(because there are no
intervening junctions).
V
T
= V
a
+V
b
= Q
a
=C
a
+Q
b
=C
b
= Q
T
(1=C
a
+1=C
b
)
V
T
= Q
T
=C
T
So the eective capacitance is give by
C
T
=

1
C
a
+
1
C
b

1
Capacitors in parallel
C
a
,V
a
C
b
,V
b
a
b
V
T
,C
T
By Kircho's loop rule,V
a
V
b
= 0 so V
a
= V
b
= V
T
.
By Kircho's junction rule,Q
T
= Q
a
+Q
b
Q
T
= Q
a
+Q
b
= C
a
V
T
+C
b
V
T
= (C
a
+C
b
)V
T
Q
T
= C
T
V
T
So the eective capacitance is give by C
T
= C
a
+C
b
.
RC circuits
E
R
C
I
From Kircho's loop rule
P
loop
V
i
= E IR Q=C = 0.
From the denition of current I =
dQ
dt
.
So
E Q=C = R
Q
t
dt = (E Q=C)
1
RdQ
Q = CE

1 e
t=RC

V
C
= E

1 e
t=RC