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Chapter 23
Fundamentals of Circuits: Direct
Current (DC)
1
Electrical Circuits: Batteries + Resistors, Capacitors
The battery establishes an electric field in the
connecting wires
 This field applies a force on electrons in the
wire just outside the terminals of the battery
2
Direct Current
• When the current in a circuit has a
constant magnitude and direction,
the current is called direct current
• Because the potential difference
between the terminals of a battery
is constant, the battery produces
direct current
• The battery is known as a source of
emf (electromotive force)
3
Basic symbols used for electric circuit drawing
Circuit diagram: abstract
picture of the circuit
4
Circuit diagram: abstract
picture of the circuit
5
Which of these diagrams represent the same circuit?
6
Analysis of a circuit
Full analysis of the circuit:
- Find the potential difference across each circuit component;
- Find the current through each circuit component.
There are two methods of analysis:
(1) Through equivalent resistors and capacitors (parallel
and series circuits) – easy approach
(2) Through Kirchhoff’s rules – should be used only if the
first approach cannot be applied.
7
Resistors in Series
Req
I


V
We introduce an equivalent circuit with just one equivalent
resistor so that the current through the battery is the same
as in the original circuit
Then from the Ohm’s law we can find the current in the
equivalent circuit
8
Resistors in Series
For a series combination of resistors, the currents are the
same through all resistors because the amount of charge
that passes through one resistor must also pass through the
other resistors in the same time interval
Ohm’s law:
Vc Vb  IR2
Vb Va  IR1
Vc Va  Vc Vb  Vb Va 
 IR2  IR1  I  R1  R2 
Req  R1  R2
The equivalent resistance has the same effect on the
circuit as the original combination of resistors
9
Resistors in Series
• Req = R1 + R2 + R3 + …
• The equivalent resistance of a series combination of
resistors is the algebraic sum of the individual resistances
and is always greater than any individual resistance
10
Resistors in Parallel
Req
I


V
We introduce an equivalent circuit with just one equivalent
resistor so that the current through the battery is the same
as in the original circuit
Then from the Ohm’s law we can find the current in the
equivalent circuit
11
Resistors in Parallel
 The potential difference across each resistor is the same
because each is connected directly across the battery terminals
 The current, I, that enters a point must be equal to the total
current leaving that point
I = I1 + I2
- Consequence of Conservation of Charge
Ohm’s law:
Vb Va  V  I1R1
Vb Va  V  I2R2
Conservation of Charge:
I  I1  I2
 1
V V
1  V
I

 V  

R1 R2
 R1 R2  Req
12
Resistors in Parallel
1
1
1


Req R1 R2
 Equivalent Resistance
1
1
1
1




Req R1 R2 R3
– The equivalent is always less than the smallest resistor in
the group
 In parallel, each device operates independently of the others
so that if one is switched off, the others remain on
 In parallel, all of the devices operate on the same voltage
 The current takes all the paths
– The lower resistance will have higher currents
– Even very high resistances will have some currents
13
Example
Req  R1  R2 
 8  4  12
1
1
1 1 1 1


  
Req R1 R2 6 3 2
Req  R1  R2 
 12  2  14
14
Example
Req  ? or
Main question:
R1
I ?
V
I
Req
R2
Req
R1
R2
I
I



V

V

15
Example
Req  ? or
Main question:
I ?
RR
R
in parallel Req ,1  1 1  1
R1  R1 2
in parallel
R1
R2
R1
R2
Req ,2 
R2R2
R
 2
R2  R2
2
I


V
16
Example
Req  ? or
Main question:
Req ,1 
R1
2
Req ,2 
in series
I ?
R2
2
Req  Req ,1  Req ,2 
R1  R2
2
I
V
2V

Req R1  R2
Req
Req ,1
Req ,2
I
I


V


V
17
Example
Main question:
Req  ? or
R1
I ?
I
V
Req
R2
Req
R1
R2
I


I


V
V
18
Example
Main question:
R1
Req  ? or
I ?
R3
Req
R5
I
I
R2


R4


V
V
To find
Req we need to use Kirchhoff’s rules.
19
Chapter 23
Kirchhoff’s rules
20
Kirchhoff’s rules
 There are two Kirchhoff’s rules
 To formulate the rules we need, at first, to choose the
directions of currents through all resistors. If we choose the
wrong direction, then after calculation the corresponding
current will be negative.
R1
I1
I
R3
I5
I2
R2


R5
I3
I4
R4
I
V
21
Junction Rule
 The first Kirchhoff’s rule – Junction Rule:
 The sum of the currents entering any junction must equal
the sum of the currents leaving that junction
- A statement of Conservation of Charge
I I
in
out
I1  I2  I3
In general, the number of times the
junction rule can be used is one fewer than
the number of junction points in the circuit
22
Junction Rule
 The first Kirchhoff’s rule – Junction Rule: Iin Iout
 In general, the number of times the junction rule can be used is one
fewer than the number of junction points in the circuit
 There are 4 junctions: a, b, c, d.
 We can write the Junction Rule for any three of them
R1
I1
b
I5
I2
a
I
R2
R5
c

R3

I3
(a) I  I1  I2
(b) I1  I5  I3
I4
(c) I2  I4  I5
d
R4
I
V
23
Loop Rule
 The second Kirchhoff’s rule – Loop Rule:
 The sum of the potential differences across all the
elements around any closed circuit loop must be zero
- A statement of Conservation of Energy

V  0
closed loop
Traveling around the loop from a to b
24
Loop Rule
 The second Kirchhoff’s rule – Loop Rule:

V  0
closed loop
25
Loop Rule
 The second Kirchhoff’s rule – Loop Rule:

V  0
closed loop
We need to write the Loop
Rule for 3 loops
Loop 1:
4
R1
I1
I
R3
I1R1  I5R5  I2R2  0
I3
Loop 2:
I2
I5
1
R5
R2
2
R4
3

I3R3  I5R5  I4R4  0
I4
Loop 3:
I
V  I2R2  I4R4  0

V
26
Kirchhoff’s Rules
I I
Junction Rule
in

 Loop Rule
out
V  0
closed loop
R1
I1
I
R3
I5
I2
R2


R5
I3
I  I1  I2
I1  I5  I3
I4
R4
I 2  I 4  I5
I
V
We have 6 equations and 6 unknown currents.
Req
V

I
I1R1  I5R5  I2R2  0
I3R3  I5R5  I4R4  0
V  I2R2  I4R4  0
27
Kirchhoff’s Rules
in

 Loop Rule
V2


I1
I
I  I1  I2
I1  I5  I3
I 2  I 4  I5
I I
Junction Rule
out
V  0
closed loop
I1R1  I5R5  I2R2  V2  0
R1
R3
I5
I2 R
2


R5
I3R3  I5R5  I4R4  0
I3
R4
V1  I2R2  I4R4  0
I4
I
V1
We have 6 equations and 6 unknown currents.
28
Example
R3
I3
R2
I2
R1
I


I
V
Req
I

Req  R1 

V
I
I
R2R3
R2  R3
V
Req
29
Example 1
R3
I3
R2
I2
Req
R1
I

I
R2R3
 R1 
R2  R3
V
Req
I

V1
I  I 2  I3
I2R2  I3R3
 R3 
I  I3 1

R
2 

R3
I 2  I3
R2
IR2
I3 
R2  R3
IR3
I2 
R2  R3
30
Example: solution based on Kirchhoff’s Rules
R3
I3
R2
I2
I  I 2  I3
R1
I3R3  I2R2  0
V  I2R2  IR1  0
I

I

V
R3
I 2  I3
R2
I3 
IR2
R2  R3
IR3
V 
R2  IR1  0
R2  R3
IR3
I2 
R2  R3
I
V
V

R3R2
 R1 Req
R2  R3
31
Example
V2 

I
R3
I3
R2
I2
R1

I

V1
I  I 2  I3
I3R3  I2R2  0
V1  V2  I2R2  IR1  0
32
Example
V
 2
I
R3
I3
R2
I2
R1

I

V1
I  I 2  I3
I3R3  I2R2  0
V1  V2  I2R2  IR1  0
33
Chapter 23
Electrical circuits with capacitors
34
Capacitors in Parallel
V
 
C2
V


C1
All the points have
the same potential
All the points have
the same potential


V
The capacitors 1 and 2 have the same potential difference
Then the charge of capacitor 1 is
The charge of capacitor 2 is
V
Q1  C1V
Q2  C2 V
35
Capacitors in Parallel

The total charge is
V

C2
Q2  C2 V
Q  Q1  Q2
Q  C1V  C2V  (C1  C2 )V

V

Q1  C1V
C1
This relation is equivalent to
the following one
Q  Ceq V
Ceq
 
Ceq  C1  C2




V
36
Capacitors in Parallel
 The capacitors can be replaced with
one capacitor with a capacitance of Ceq
 The equivalent capacitor must have
exactly the same external effect on the
circuit as the original capacitors
Q  Ceq V
37
Capacitors


The equivalence means that
Q  Ceq V

V

38
Capacitors in Series
V1



V2

C2
C1


V
V  V1  V2
39
Capacitors in Series
Q1  C1V
The total charge
is equal to 0
V1



C1
Q2  C2 V
Q1  Q2  Q
V2

C2
V  V1  V2 
V 


Q Q

C1 C2
Q
Ceq
1
1
1


Ceq C1 C2
V
Ceq 
C1C2
C1  C2
40
Capacitors in Series
 An equivalent capacitor can be found
that performs the same function as the
series combination
 The potential differences add up to the
battery voltage
41
Example
in parallel
Ceq  C1  C2  1  3  4
Ceq  C1  C2  6
Ceq  C1  C2  8
in series
Ceq 
C1C2
88

4
C1  C2 8  8
in parallel
in parallel
42
Chapter 23
RC circuits
43
RC circuit
• A direct current circuit may contain capacitors and resistors, the
current will vary with time
• When the circuit is completed, the capacitor starts to charge
• The capacitor continues to charge until it reaches its maximum
charge Q  CV
• Once the capacitor is fully charged, the current in the circuit is
zero
C
R
I
Q


V
44
RC circuit
 As the plates are being charged, the potential difference across the
capacitor increases
 At the instant the switch is closed, the charge on the capacitor is zero
 Once the maximum charge is reached, the current in the circuit is zero
V
R
Q
CV
0
0
t
R
C
Q


V
I
t
I
V  I  t  R 
Q t 
C
45
RC circuit
R
C
I
Q


V
V  I  t  R 
Q t 
0.632 C V
C

Q  t   C V 1 e  t / RC
I t  
CV
V  t / RC
e
R

46
RC circuit: time constant
R
C
Q


I

Q  t   C V 1 e  t / RC
I t  
V

V  t / RC
e
R
• The time constant represents
CV
the time required for the charge
to increase from zero to 63.2%
0.632 C V
of its maximum
 t  RC has unit of time
47
RC circuit
R
C
Q
I
q  t   Qet / RC
I t  
Q  t / RC
e
RC
• When a charged capacitor is
placed in the circuit, it can be
discharged
• The charge decreases
exponentially with
characteristic time t  RC
48