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E-Eng. Fundamentals
Parallel Circuits
Total Conductance
Total Conductance
Total Resistance For Equal Parallel Resistors

Two Resistances in Parallel


Three Resistances in Parallel
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E-Eng. Fundamentals
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EXAMPLE 6.7 Calculate the total resistance of the parallel network of Fig. 6.13.
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E-Eng. Fundamentals
The voltage across parallel elements is the same.
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E-Eng. Fundamentals
KIRCHHOFF’S CURRENT LAW
Kirchhoff’s current law (KCL) states that the algebraic sum of the
currents entering and leaving an area, system, or junction is zero.
Important Definitions




Node: A point at which two or more elements
have a common connection is called a node.
Path: then the set of nodes and elements that we have passed through is defined as a path.
Loop: If the node at which we started is the same as the node on which we ended, then the path is, by definition,
a closed path or a loop.
Branch: a single path in a network, composed of one simple element and the node at each end of that element is
defined as branch.
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E-Eng. Fundamentals
Example/ In the circuit of Figure there are eight circuit elements. Find vR2
(the voltage across R2) and the voltage labeled vx.
 we find vR2 by writing a KVL equation around the
loop on the left, starting at point c:
4 − 36 + vR2 = 0


vR2 = 32
apply KVL c  a  b  c :
+4 − 36 + 12 + 14 + vx = 0  vx = 6 V
Independent Voltage Sources 
Independent Current Sources 
Dependent Sources 
To distinguish between dependent and
independent sources, we introduce the diamond
symbols shown in Fig. 2.18. In Fig. 2.18a and c, K
is a dimensionless scaling constant. In Fig.2.18b, g
is a scaling factor with units of A/V;
in Fig. 2.18d, r is a scaling factor with units of
V/A. The controlling current ix and the controlling
voltage vx must be defined in the circuit.
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E-Eng. Fundamentals
At node b, since 12 A are entering and 8 A are leaving, I4 must be leaving. Therefore, 
At node c, I3 is leaving at 2 A and I4 is entering at 4 A, requiring that I6 be leaving
Applying Kirchhoff’s current law at node c, 
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Application 
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E-Eng. Fundamentals
Noting the total input current versus that leaving clearly reveals that I1
is a current of 22 mA - 17 mA = 5 mA leaving the system.
Current Divider Rule
 For two parallel elements of equal value, the current will divide equally.
 For parallel elements with different values, the smaller the resistance,
the greater the share of input current.
 For parallel elements of different values, the current will split with a
ratio equal to the inverse of their resistor values.

Current seeks the path of least resistance.
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E-Eng. Fundamentals
Open and Short Circuits
--------------------------------------------------------------------------------Example/ Determine the value of v and the power supplied by the independent current source
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E-Eng. Fundamentals
Problems in pages 204 ----- 2011 in Text Book (Boylestad)
Example/
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E-Eng. Fundamentals
Series-Parallel Networks
series-parallel networks are networks that contain both series and
parallel circuit configurations.
Example/ Find I1, I2, IA, IB and IC
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E-Eng. Fundamentals
Example / Find the indicated currents and voltages for the network of following Figure.
Solution:
Example/ For the network of Figure, determine the voltages V1
and V2 and the current I.
Applying Kirchhoff’s voltage law to the loop indicated,
we obtain
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E-Eng. Fundamentals
Example /Calculate the indicated currents and voltage of Figure.
Solution:
Redrawing the network after combining series elements yields next figure