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ELECTRICITY & MAGNETISM (Fall 2011)
LECTURE # 13
BY
MOEEN GHIYAS
TODAY’S LESSON
(Parallel Circuits – Chapter 6)
Introductory Circuit Analysis by Boylested (10th Edition)
Today’s Lesson Contents
• Parallel Elements
• Total Conductance and Total Resistance
• Parallel Circuits
• Kirchhoff’s Current Law (KCL)
• Solution to Problems
Parallel Elements
• Two elements, branches, or networks are in
parallel if they have two points in common.
Parallel Elements
• Different ways in which three parallel elements may
appear.
Parallel Elements
• In fig,
• Elements 1 and 2 are in parallel because they have
terminals a and b in common. The parallel
combination of 1 and 2 is then in series with element
3 due to the common terminal point b.
Parallel Elements
• In fig, elements 1 and 2 are in series because they
have only terminal b in common. The series
combination of 1 and 2 is then in parallel with element
3 due to the common terminals point b and c.
Total Conductance and Total Resistance
• For parallel elements,
the total conductance
is the sum of the
individual
conductances
• Note that the equation
is for 1 divided by the
total resistance rather
than total resistance.
Total Conductance and Total Resistance
• Example – Determine the total conductance and
resistance for the parallel network of Fig
• Solution:
Total Conductance and Total Resistance
• Example – Determine the effect on total conductance
and resistance of the network of fig if another resistor of
10Ω were added in parallel with the other elements
• Solution:
Note that adding additional terms Note that adding additional terms
increases the conductance level increases the conductance level
and decreases the resistance level.and decreases the resistance level.
Total Conductance and Total Resistance
• Recall for series circuits that the total resistance will
always increase as additional elements are added in
series.
• For parallel resistors, the total resistance will always
decrease as additional elements are added in parallel.
Total Conductance and Total Resistance
• Example – Determine the total resistance for the
network of Fig
• Solution:
Total Conductance and Total Resistance
• The total resistance of parallel resistors is always less
than the value of the smallest resistor.
• The wider the spread in numerical value between two
parallel resistors, the closer the total resistance will be
to the smaller resistor.
Total Conductance and Total Resistance
• For equal resistors in parallel, the equation becomes,
• For same conductance levels, we have
Total Conductance and Total Resistance
• For two parallel resistors,
• For three parallel resistors,
Total Conductance and Total Resistance
• Example – Find the total resistance of the network of
Fig
• Solution:
Total Conductance and Total Resistance
• Example – Calculate the total resistance for the network
of Fig
• Solution:
Total Conductance and Total Resistance
• Parallel elements can be interchanged without changing
the total resistance or input current.
Total Conductance and Total Resistance
• Example – Determine the values of R1, R2, and R3 in fig
if R2 = 2R1 and R3 = 2R2 and total resistance is 16 kΩ.
• Solution:
• . =
• Since
Parallel Circuits
• The voltage across parallel elements is the same.
• or
• But and
• Take the equation for the total resistance and multiply
both sides by the applied voltage, For single-source For single-source
parallel networks, the parallel networks, the
source current (Isource current (Iss) is ) is
equal to the sum of the equal to the sum of the
individual branch individual branch
currents.currents.
Parallel Circuits
• The power dissipated by the resistors and delivered
by the source can be determined from
Parallel Circuits
• Example – Given the information provided in fig:
a) Determine R3.
b) Calculate E.
c) Find Is.
d) Find I2.
e) Determine P2.
Parallel Circuits
a) Determine R3.
Solution:
Parallel Circuits
b) Calculate E.
c) Find Is.
Solution:
Parallel Circuits
d) Find I2.
e) Determine P2.
Solution:
Kirchhoff’s Current Law (KCL)
• Kirchhoff’s current law (KCL) states that
the algebraic sum of the currents
entering and leaving an area, system, or
junction is zero.
• In other words, the sum of the currents
entering an area, system, or junction
must equal the sum of the currents
leaving the area, system, or junction.
Kirchhoff’s Current Law (KCL)
• In technology the term node is commonly used to
refer to a junction of two or more branches. Therefore,
this term will be used frequently in future.
Kirchhoff’s Current Law (KCL)
• At node a:
• At node b:
• At node c:
• At node d:
Kirchhoff’s Current Law (KCL)
• Example – Determine unknown current I1.
• Solution:
• I1 is 5mA and leaving system.
Kirchhoff’s Current Law (KCL)
• Example – Determine the currents I3 and I5 of fig using
Kirchhoff’s current law (KCL).
• Solution:
• . At node a: I1 + I2 = I3
• . At node b: I3 = I4 + I5
Summary / Conclusion
• Parallel Elements
• Total Conductance and Total Resistance
• Parallel Circuits
• Kirchhoff’s Current Law (KCL)
• Solution to Problems