electricity & magnetism (fall 2011) lecture # 13 by moeen ghiyas

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.


• Example – Determine the total conductance and

resistance for the parallel network of Fig

• Solution:


• 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.


• 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.


• Example – Determine the total resistance for the

network of Fig

• Solution:


• 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.


• For equal resistors in parallel, the equation becomes,

• For same conductance levels, we have


• For two parallel resistors,

• For three parallel resistors,


• Example – Find the total resistance of the network of

Fig

• Solution:


• Example – Calculate the total resistance for the network

of Fig

• Solution:


• Parallel elements can be interchanged without changing

the total resistance or input current.


• 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.


• 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.


• At node a:

• At node b:

• At node c:

• At node d:


• Example – Determine unknown current I1.

• Solution:

• I1 is 5mA and leaving system.


• 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

electricity & magnetism (fall 2011) lecture # 13 by moeen ghiyas

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