ELECTRICITY & MAGNETISM (Fall 2011) LECTURE # 13 BY MOEEN GHIYAS

TODAY’S LESSON (Parallel Circuits – Chapter 6) Introductory Circuit Analysis by Boylested (10 th 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 increases the conductance 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 R 1, R 2, and R 3 in fig if R 2 = 2R 1 and R 3 = 2R 2 and total resistance is 16 kΩ. Solution:. = Since

Parallel Circuits The voltage across parallel elements is the same. or Butand Take the equation for the total resistance and multiply both sides by the applied voltage, For single-source parallel networks, the source current (I s ) is equal to the sum of the individual branch 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 R 3. b)Calculate E. c)Find I s. d)Find I 2. e)Determine P 2.

Parallel Circuits a)Determine R 3. Solution:

Parallel Circuits b)Calculate E. c)Find I s. Solution:

Parallel Circuits d)Find I 2. e)Determine P 2. 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 I 1. Solution: I 1 is 5mA and leaving system.

Kirchhoff’s Current Law (KCL) Example – Determine the currents I 3 and I 5 of fig using Kirchhoff’s current law (KCL). Solution:.At node a:I 1 + I 2 = I 3.At node b: I 3 = I 4 + I 5

Summary / Conclusion Parallel Elements Total Conductance and Total Resistance Parallel Circuits Kirchhoff’s Current Law (KCL) Solution to Problems