Resistors in series and parallel
Direct Current Circuits
Series
Parallel
n
Req = R1 + R2 + R3 + ... = ∑ Ri
i =1
EF 152 Fall, 2010 Lecture 4-5
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n
1
1
1
1
1
= +
+ + ... = ∑
Req R1 R2 R3
i =1 Ri
EF 152 Fall, 2010 Lecture 4-5
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Example: Series-parallel Combination Circuit
Equivalent Resistance
Find Itotal, IR1, IR2, IR3, IR4, VR1, VR2, VR3 and VR4
EF 152 Fall, 2010 Lecture 4-5
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EF 152 Fall, 2010 Lecture 4-5
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Kirchhoff’s rules
Example:
Some circuits cannot be broken down into
series and parallel connections.
Find IR1, IR2 and IR3
R1
R2
R3
R4
=
=
=
=
1
2
3
4
Ω
Ω
Ω
Ω
The battery emf's values:
ε1 = 12 V
ε2 = 3 V
ε3 = 10 V
Junction Rule – Valid at any Junction
∑I = 0
Gustav Robert Kirchhoff
(12 March 1824 – 17
October 1887)
Step 1 - Label the currents in
each branch, indicating direction.
Loop Rule – Valid for any closed loop
Step 2 - Label the + and - ends of the
resistors. Current goes through a resistor
from the + end to the - end.
∑V = 0
EF 152 Fall, 2010 Lecture 4-5
Example:
R1
R2
R3
R4
=
=
=
=
1
2
3
4
Ω
Ω
Ω
Ω
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Find IR1, IR2 and IR3
EF 152 Fall, 2010 Lecture 3-8
Kirchhoff’s rules step-by-step
The battery emf's
values:
ε1 = 12 V
ε2 = 3 V
ε3 = 9 V
•
•
•
•
Step 3 - Apply the junction rule at either one of
the junctions.
-This will reduce the number of uinknowns
I1 + I2 + I3 = 0 or: I3 =-(I1 + I2 )
Step 4 – Choose your loops and write the
equations.
•
•
•
(It doesn't matter where on the loop you start or
which direction you go.)
Loop 1 (green):
Step 5 – Solve
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-ε1 + I1 R1 + ε2 - I2 R2 + I1 R4 = 0
•
•
(1) -12+I1+3-2I2+4I1=0
Loop 2 (Purple):
-ε2 + ε3 - I3 R3 + I2 R2 = 0
(2) -2+9-3I3+2I2 = 0
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Draw a large diagram
Label all quantities (V, R, I for each path), assume a
direction for unknown I’s
Identify unknown values
Use junction rule to reduce number of unknown currents.
Use all but one junction. Use + for in, - for out.
Choose a closed loop and assume a direction of travel
Apply ΣV=0 for the loop, be careful with signs
Repeat ΣV=0 for other loops until you have enough
equations. Be sure to chose loops so all paths are included
at least once.
Solve simultaneous linear equations.
If you get a negative answer it means the direction is
opposite the assumption.
EF 152 Fall, 2010 Lecture 4-5
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Solutions – Excel, Matlab, TI-83
Another Example
Find the current in each part of the circuit and the
equivalent resistance of the network.
Microsoft Office
Excel 2007 Workbook
EF 152 Fall, 2010 Lecture 4-5
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EF 152 Fall, 2010 Lecture 4-5
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