Chapter 6

Parallel Circuits

Parallel Circuits

House circuits contain parallel circuits

The parallel circuit will continue to operate even though one component may be open

Only the open or defective component will no longer continue to operate

Parallel Circuits

Elements in parallel

When they have exactly two nodes in common

Elements between nodes

Any device like resistors, light bulbs, etc.

Elements connected in parallel

Same voltage across them

Parallel Circuits

Series - Parallel Circuits

Circuits may contain a combination of series and parallel components

Being able to recognize the various connections in a network is an important step in analyzing these circuits

Series - Parallel Circuits

Parallel Circuits

To analyze a particular circuit

First identify the node

Next, label the nodes with a letter or number

Then, identify types of connections

Parallel Circuits

Kirchhoff’s Current Law (KCL)

The algebraic sum of the currents entering and leaving a node is equal to zero

Kirchhoff’s Current Law (KCL)

Currents entering the node are taken to be positive, leaving are taken to be negative

Sum of currents entering a node is equal to the sum of currents leaving the node

Kirchhoff’s Current Law (KCL)

An analogy:

When water flows in a pipe, the amount of water entering a point is the amount leaving that point

Direction of Current

If unsure of the direction of current through an element, assume a direction

Base further calculations on this assumption

Direction of Current

If this assumption is incorrect, calculations will show that the current has a negative sign

Negative sign simply indicates that the current flows in the opposite direction

Resistors in Parallel

Voltage across all parallel elements in a circuit will be the same

Resistors in Parallel

For a circuit with 3 resistors: I_T = I₁ + I₂ + I₃

Resistors in Parallel

Total resistance of resistors in parallel will always be less than resistance of smallest resistor

Equal Resistors in Parallel

For n equal resistors in parallel, each resistor has the same conductance G

G_T = nG

R_T = 1/G_T = 1/nG = R/n

Equal Resistors in Parallel

Total resistance of equal resistors in parallel is equal to the resistor value divided by the number of resistors

Two Resistors in Parallel

For only two resistors connected in parallel, the equivalent resistance may be found by the product of the two values divided by the sum

Often referred to as “product over the sum” formula

Three Resistors in Parallel

For three resistors in parallel:

Rather than memorize this long expression

Use basic equation for resistors in parallel

Voltage Sources in Parallel

Voltage sources with different potentials should never be connected in parallel

When two equal sources are connected in parallel

Each source supplies half the required current

Voltage Sources in Parallel

Jump starting automobiles

If two unequal sources are connected

Large currents can occur and cause damage

Current Divider Rule

Allows us to determine how the current flowing into a node is split between the various parallel resistors

Current Divider Rule

Current Divider Rule

For only two resistors in parallel:

Current Divider Rule

If current enters a parallel network with a number of equal resistors, current will split equally between resistors

In a parallel network, the smallest value resistor will have the largest current

Largest resistor will have the least current

Current Divider Rule

Most of the current will follow the path of least resistance

Analysis of Parallel Circuits

Voltage across all branches is the same as the source voltage

Determine current through each branch using Ohm’s Law

Find the total current using Kirchhoff’s Current Law

Analysis of Parallel Circuits

To calculate the power dissipated by each resistor, use either VI, I²R, or V²/R

Total power consumed is the sum of the individual powers

Compare with I_T²R_T

Ammeter Design

Coil of the meter can only handle a small amount of current

A shunt resistor in parallel allows most of current to bypass the coil

Ammeter Design

Voltmeter Loading Effects

A voltmeter

Meter movement in series with a current-limiting resistance

If resistance is large compared with the resistance across which the voltage is to be measured, the voltmeter will have a very small loading effect

Voltmeter Loading Effects

If this resistance is more than 10 times the resistance across which the voltage is being measured, the loading effect can generally be ignored.

However, it is usually much higher.


	House circuits contain parallel circuits
	The parallel circuit will continue to operate even though one component may be open
	Only the open or defective component will no longer continue to operate


	Elements in parallel
		When they have exactly two nodes in common
	Elements between nodes
		Any device like resistors, light bulbs, etc.
	Elements connected in parallel
		Same voltage across them


	Circuits may contain a combination of series and parallel components
	Being able to recognize the various connections in a network is an important step in analyzing these circuits


	To analyze a particular circuit
		First identify the node
		Next, label the nodes with a letter or number
		Then, identify types of connections


	The algebraic sum of the currents entering and leaving a node is equal to zero


	Currents entering the node are taken to be positive, leaving are taken to be negative
	Sum of currents entering a node is equal to the sum of currents leaving the node


	An analogy:
		When water flows in a pipe, the amount of water entering a point is the amount leaving that point


	If unsure of the direction of current through an element, assume a direction
	Base further calculations on this assumption


	If this assumption is incorrect, calculations will show that the current has a negative sign
	Negative sign simply indicates that the current flows in the opposite direction


	Voltage across all parallel elements in a circuit will be the same


	Total resistance of resistors in parallel will always be less than resistance of smallest resistor


	For n equal resistors in parallel, each resistor has the same conductance G
	G_T = nG
	R_T = 1/G_T = 1/nG = R/n


	Total resistance of equal resistors in parallel is equal to the resistor value divided by the number of resistors


	For only two resistors connected in parallel, the equivalent resistance may be found by the product of the two values divided by the sum



	Often referred to as “product over the sum” formula


	For three resistors in parallel:




	Rather than memorize this long expression
		Use basic equation for resistors in parallel


	Voltage sources with different potentials should never be connected in parallel
	When two equal sources are connected in parallel
		Each source supplies half the required current


	Jump starting automobiles
	If two unequal sources are connected
		Large currents can occur and cause damage


	Allows us to determine how the current flowing into a node is split between the various parallel resistors


	If current enters a parallel network with a number of equal resistors, current will split equally between resistors
	In a parallel network, the smallest value resistor will have the largest current
		Largest resistor will have the least current


	Most of the current will follow the path of least resistance


	Voltage across all branches is the same as the source voltage
	Determine current through each branch using Ohm’s Law
	Find the total current using Kirchhoff’s Current Law


	To calculate the power dissipated by each resistor, use either VI, I²R, or V²/R
	Total power consumed is the sum of the individual powers
	Compare with I_T²R_T


	Coil of the meter can only handle a small amount of current
	A shunt resistor in parallel allows most of current to bypass the coil


	A voltmeter
		Meter movement in series with a current-limiting resistance
	If resistance is large compared with the resistance across which the voltage is to be measured, the voltmeter will have a very small loading effect


	If this resistance is more than 10 times the resistance across which the voltage is being measured, the loading effect can generally be ignored.
	However, it is usually much higher.