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Lecture 3
Circuit Laws,
Voltage & Current Dividers
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
KIRCHHOFF’S CURRENT
LAW
• The net current entering a node is zero.
• Alternatively, the sum of the currents
entering a node equals the sum of the
currents leaving a node.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
KIRCHHOFF’S VOLTAGE
LAW
The algebraic sum of the voltages equals
zero for any closed path (loop) in an
electrical circuit.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Using KVL, KCL, and Ohm’s
Law to Solve a Circuit
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
15 V
iy 
3A
5
ix  0.5ix  i y
ix  2 A
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ix  2 A
vx  10ix  20 V
Vs  vx  15
Vs  35 V
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Chapter 2
Resistive Circuits
1. Solve circuits (i.e., find currents and
voltages of interest) by combining
resistances in series and parallel.
2. Apply the voltage-division and currentdivision principles.
3. Solve circuits by the node-voltage
technique.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
4. Solve circuits by the mesh-current
technique.
5. Find Thévenin and Norton equivalents and
apply source transformations.
6. Apply the superposition principle.
7. Draw the circuit diagram and state the
principles of operation for the Wheatstone
bridge.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
v  v1  v2  v3  iR1  iR2  iR3  i( R1  R2  R3 )  iReq
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
 1
v
v
v
1
1 
v


i  i1  i2  i3 


 v 



R1 R2 R3
 R1 R2 R3  Req
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
1
1
1
1
1 1 1 1 2 3



      1
Req R2 R3 R4 6 3 2 6 6 6
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
1
Rtotal  R1  1  3
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
1
1 1 2 1 3 1
     
Req 3 6 6 6 6 2
Req  2.0
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
1.5
Req  8  2  10
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
1
1 1
2


   Req  5
Req 10 10 10
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Circuit Analysis using
Series/Parallel Equivalents
1. Begin by locating a combination of
resistances that are in series or parallel.
Often the place to start is farthest from the
source.
2. Redraw the circuit with the equivalent
resistance for the combination found in
step 1.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
3. Repeat steps 1 and 2 until the circuit is
reduced as far as possible. Often (but not
always) we end up with a single source and
a single resistance.
4. Solve for the currents and voltages in the
final equivalent circuit.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
v2 60V
i2 

 2A
R2 30 
v2 60V
i3 

 1A
R3 60 
v1  R1i1  10  3 A  30V
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Voltage Division
R1
v1  R1i 
v total
R1  R2  R3
R2
v2  R2 i 
v total
R1  R2  R3
Of the total voltage, the fraction that appears across a
given resistance in a series circuit is the ratio of the given
resistance to the total series resistance.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Application of the VoltageDivision Principle
R1
v1 
vtotal
R1  R2  R3  R4
1000

15
1000  1000  2000  6000
 1.5V
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Req 
R1 R2
RR
 v  Req itotal  1 2 itotal
R1  R2
R1  R2
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Current Division
R2
v
i1 

itotal
R1 R1  R2
R1
v
i2 

itotal
R2 R1  R2
For two resistances in parallel, the fraction of the total
current flowing in a resistance is the ratio of the other
resistance to the sum of the two resistances.
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Find i3
vs
100
is 

 1.25 A
R1  Rx 60  20
R2
30
1.25   0.417 A
i3 
is 
R2  R3
30  60
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Find i1
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Application of the CurrentDivision Principle
R2 R3
30  60
Req 

 20
R2  R3 30  60
Req
20
i1 
is 
15  10A
R1  Req
10  20
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Use the voltage division principle to
find the unknown voltages
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
v1 
R1
5
120V   120V  10V
vs 
R1  R2  R3  R4
5  10  15  30
12
R2
10
120V
120V  
v2 
vs 
 20V
R1  R2  R3  R4
5  10  15  30
6
v3 
R3
15
120V   120V  30V
vs 
R1  R2  R3  R4
5  10  15  30
4
R4
30
120V
120V  
v4 
vs 
 60V
R1  R2  R3  R4
5  10  15  30
2
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Req 
1
1 1

7 5
 2.92 
R1

3
320V 
20V  
v1 
vs 
 6.05V
R1  Req  R4
3  2.92  4
9.92

2.92
2.92 20V 
20V  
v2 
vs 
 5.89V
R1  Req  R4
3  2.92  4
9.92
Req
R4

4
4 20V 
20V  
v4 
vs 
 8.06V
R1  Req  R4
3  2.92  4
9.92
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Use the current division principle to
find the currents
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
R3
15
3  1A
i1 
it 
R1  R2  R3
10  20  15
R1  R2
30
3  2 A
i3 
it 
R1  R2  R3
10  20  15
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Req 
1
1 1

10 10
5
Req
5
3  1A
i1 
is 
Req  R1
5  10
Req
5
3  1A
i2 
is 
Req  R1
5  10
Req
5
3  1A
i3 
is 
Req  R1
5  10
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.
Resistor Cube
ELECTRICAL ENGINEERING: PRINCIPLES AND APPLICATIONS, Fourth Edition, by Allan R. Hambley, ©2008 Pearson Education, Inc.