Series & Parallel Circuits

Series and Parallel Circuits
Series and Parallel
Circuits

--------------------------------------------------

Joseph F. Alward, PhD � �
Department of �Physics �
University of the Pacific

� Alternating Voltage Sources � � � � � � �

Power station provides alternating current (AC).


�Alternating Current� � �
� �
-------------------------------------------------------------------
Current surges first one way, then the other
through the toaster heating filament, changing
direction 120 times per second.

Clockwise current is�positive, counter-clockwise
is negative.


� Series Wiring� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

What is the equivalent
series resistance Rs?

� � � � � Rs = 9 + 6
� � �� � � � � = 15 W



� � � � �I �= 30 / 15
� � � � � ��= 2 A

� Series Wiring Example� � � � � � � � � � � � � � � � ��

� � � � � �What is the power dissipated
� � � � � � in the 6 ohm resistor?

� � � �
�I �= 12 / 9
� � = 1.33 A
----------------
�P = �I2 R �
� � = (1.33)2 (6)
� � = 10.6 W

� �Resistors in Series: �The Extension Cord�� � � �

Which resistor gets hotter?
I = V / (R + r) �
----------------------------
PR �= �I2 R
� � �
= [ V2 / (R+r)2 ] R
----------------------------
Pr � �= �I2 r � �
� � � = [
V2 /(R+r)2 ] r

� �Blowing a Fuse � � � � � � � � � �

� � � � � � � � � �I = 9 / 0.5
� � � � � � � � � � �= 18 A

What is the least R which will
prevent the fuse from blowing?

� Parallel Wiring� � � � � � � � � � � �� � � �� � � � � � � � � � � � � � � � � � � � � �

� �Two pipes in parallel can be replaced by one. �The single pipe
� �will have less resistance than the pipe with the least resistance.

� Light Bulbs in Parallel� � � � � � � � � � � � ����

Light bulbs
connected
in parallel.

Equivalent resistance of this
parallel combination found
through the relation

1 / Rp = 1 / R1 + 1 / R2


Rp is the single resistance
which is equivalent to R1
and R2 in parallel.

House Wiring is Parallel� � � � � �
�Which connecting wire, A, B, C,
�D, or E, will be the first to become
�dangerously hot �if too many
�appliances are turned on?

How can overheating be
prevented, even if all the
appliances in the house are
turned on?

� �Circuit Breakers� � � � � � � � �

� Brass expands more than steel �
� As temperature increases, the
��bimetallic strip arcs to the left,
� settles into groove, and the
� spring pulls the metal bar down.

� �Four Resistors in Parallel � � � � � � � ��� � �� �
Find the resistance between A and B:

1 / Rp = 1 / 2 + 1 / 3 + 1 / 6 + 1 / 1
� �Rp = 0.50 W


----------------------------------
VAB �= 12 V

What is the current in each resistor?

What is the sum of all currents?

�Three-Way Light � � � � � � �� �

� � �What is the resistance
� � �of each filament?
P = V2 / R
V = 120 volts for each filament
------------------------------------------
R1
= resistance of 75-W bulb:
75 = (120)2 / R1
R1 = (120)2 / 75
� � � �= 192 �W

------------------------------------------
R2
= resistance of 100-W bulb:
100 �= (120)2 / R2
R2 = (120)2 / 100
� � � � = 144 �W


Equivalent Resistance of Series and
�Parallel Combination of Resistors
Parallel combination:

1 / Rp = 1 / 6 + 1 / 3
� � � � =� 0.5

� � Rp = 1 / 0.5�
� � � � � = 2 W
---------------------

� Series and Parallel Circuits� � � � � � � � � � � � � � � � � �
What is the total resistance between Points A and B?

4 + 6 = 10
--------------------------------------
1 / Rp = 1 / 9 + 1 / 8 + 1 / 10
� �Rp = 2.98 W
-------------------------------------
�2.98 + 3 = 5.98 �
-------------------------------------
1 / Rp = 1 / 5.98 + 1 / 20

� �Rp = 4.60�W


� �A Series-Parallel Circuit Problem: �Part One� �

�What is the potential difference between
�Points A and B?
1 / Rp = �1/3 + 1/6

� �Rp = 2 W
-------------------------
�Total: �6 W
-------------------------
�I = 12 V / 6 W
� �= �2 A
-------------------------
Ohm's Law:

VAB = (2 A) (2 W�)
� � � � = 4 V


A Series-Parallel Circuit Problem: �Part Two
� � � �

�What is the current in the 5-W resistor?
I = V / R (Ohm's Law)
-------------------------------
From Part One:
VAB = 4 V
-------------------------------
R = 6 W �(lower branch)
-------------------------------
I = 4 V / 6 W
� = 2 / 3 A

This is also the current
in the 1 W resistor.
>


Series and Parallel
� Connections

�What happens when the wire
� is connected?

� Kirchhoff's Junction Rule � � � � � � � � � � � � � � � � � � � � � � � � �
� �
Junctions are places where
�wires are connected together.

Kirchhoff's Junction Rule:

Total current into a junction
is equal to the total current
out of the junction.

� �Gustav Robert Kirchhoff
� � � � � � �(1824-1887)
� Kirchhoff's Loop Rule � � � � � � � � � � � � � � � ��

� A loop is a circuit, which is any
� closed path with batteries and
� resistors.
The sum of the potential differences
encountered around a closed circuit
equals zero.

---------------------------------------------------
Traveling clockwise from A:
-12 I �- 6 - 8 I + 24 = 0
� � � � � � � �or.............

Traveling counter-clockwise from A:
-24 + 8 I +6 + 12 I = 0

�� Kirchhoff's Rules Problem Part One:��


� � � � �

���Kirchhoff's Rules Problem Part Two: � � � � � � �

� � � � � � Loop BCDB: �
� � � � � � -4 I2 + 16 + 8 (I1 - I2) � = 0 � � � � �(1)
� � � � � � -4 I2 + 16 +8 I1 - 8 I2 � �= 0 � � �� � (2)
� � � � � � �8 I1 -12 I2 � � � � � � � � � ��= -16 � � � (3)
Loop ABDA: �

-2 I1 -8 (I 1- I2) - 16 + 6 �= 0 � � � �(4)
-2 I1 -8 I 1 + 8�I2 - 10 � � �= 0 � � � �(5)
-10I1 + 8�I2 � � � � � � � � � ��= 10 �
� �(6)
----------------------------------------------
Solve Equations 3 and 6:

I1 = 3 / 21 A
I2 = 30 / 21 A

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