Chapter 5

Series Circuits

Series Circuits

Two elements in a series

Connected at a single point

No other current-carrying connections at this point

A series circuit is constructed by connecting various elements in series

Series Circuits

Normally

Current will leave the positive terminal of a voltage source

Move through the resistors

Return to negative terminal of the source

Series Circuits

Current is similar to water flowing through a pipe

Current leaving the element must be the same as the current entering the element

Same current passes through every element of a series circuit

Series Circuits

The laws, theorems, and rules that you apply to DC circuits

Also apply to AC circuits

Kirchhoff’s Voltage Law (KVL)

The algebraic sum of the voltage that rises and drops around a closed loop is equal to zero

E_T - V₁ - V₂ - V₃ - ∙∙∙ - V_n= 0

Kirchhoff’s Voltage Law (KVL)

Another way of stating KVL is:

Summation of voltage rises is equal to the summation of voltage drops around a closed loop

V₁ + V₂ + V₃ + ∙∙∙ + V_n = E_T

Resistors in Series

Most complicated circuits can be simplified

For a series circuit

V₁ + V₂ + V₃ = E

IR₁ + IR₂ + IR₃ = E

I(R₁ + R₂ + R₃ )= E

I(R₁ + R₂ + R₃ )= IR_total (Note: I’s cancel)

Resistors in Series

Total resistance in a series circuit is the sum of all the resistor values

Power in a Series Circuit

Power dissipated by each resistor is determined by the power formulas:

P = VI = V²/R = I²R

Power in a Series Circuit

Since energy must be conserved, power delivered by voltage source is equal to total power dissipated by resistors

P_T = P₁ + P₂ + P₃ + ∙∙∙ + P_n

Voltage Sources in Series

In a circuit with more than one source in series

Sources can be replaced by a single source having a value that is the sum or difference of the individual sources

Polarities must be taken into account

Voltage Sources in Series

Resultant source

Sum of the rises in one direction minus the sum of the voltages in the opposite direction

Interchanging Series Components

Order of series components

May be changed without affecting operation of circuit

Sources may be interchanged, but their polarities can not be reversed

After circuits have been redrawn, it may become easier to visualize circuit operation

The Voltage Divider Rule

Voltage applied to a series circuit

Will be dropped across all the resistors in proportion to the magnitude of the individual resistors

The Voltage Divider Rule

Voltage dropped across each resistor may be determined by the voltage across any other resistor (or combination of resistors) by using the voltage divider rule expressed as:

The subscripts must match (x and y)

Voltage Divider Rule Application

If a single resistor is very large compared to the other series resistors, the voltage across that resistor will be the source voltage

If the resistor is very small, the voltage across it will be essentially zero

Voltage Divider Rule Application

If a resistor is more than 100 times larger than another resistor

Smaller resistor can be neglected

Circuit Ground

Ground

Point of reference or a common point in a circuit for making measurements

One type of grounding is chassis ground

In this type of grounding

Common point of circuit is often the metal chassis of the piece of equipment

Circuit Ground

Chassis ground

Often connected to Earth Ground

Earth ground

Physically connected to the earth by a metal pipe or rod

Circuit Ground

If a fault occurs within a circuit, the current is redirected to the earth

Voltages are often measured with respect to ground

Double Subscripts

For the circuit shown, we can express the voltage between any two node points (a and b) as V_ab.

If a is at a higher potential than b, then V_ab is positive

Double Subscripts

If b is at a higher potential than a, then V_ab is negative

In this case, V_abwould be negative

Double Subscripts

Double Subscripts

To determine correct polarity

Calculate all voltage drops across all components

Assign polarities

As you go around a circuit, add the gains and subtract the drops

Double Subscripts

Be sure to take the sign of the polarity on the same side of the source or element as you go around the circuit

Single Subscripts

In a circuit with a ground reference point

Voltages may be expressed with respect to that reference point

V_a

Voltage at point a with respect to ground

Ground represents a point of zero reference potential

Single Subscripts

Any voltage with a single subscript is with respect to ground

This is the same as V_a(0)

Single Subscripts

If voltages at various points in a circuit are known with respect to ground, then the voltage between points is easily determined

V_ab = V_a – V_b

Point Sources

Voltage source given with respect to ground

May be represented by a voltage at a single point (node) in the circuit

This voltage may be referred to as a point source

Point Sources

Voltages at these points represent voltages with respect to ground, even if ground is not shown

Point sources simplify representation of a circuit

Internal Resistance of Voltage Sources

Ideal sources have no internal resistance

In an ideal source

Terminal voltage does not change when the load changes

For a practical source

There is internal resistance

Internal Resistance of Voltage Sources

As the load changes

Drop across the internal resistance changes

Terminal voltage changes

Ammeter Loading Effects

An ammeter is placed in a circuit to make a measurement

Resistance will affect the circuit

Amount of loading is dependent upon the instrument and the circuit

Ammeter Loading Effects

If resistance of the meter is small compared to the resistance of the circuit, the loading effect will be small


	Two elements in a series
		Connected at a single point
		No other current-carrying connections at this point
	A series circuit is constructed by connecting various elements in series


	Normally
		Current will leave the positive terminal of a voltage source
		Move through the resistors
		Return to negative terminal of the source


	Current is similar to water flowing through a pipe
		Current leaving the element must be the same as the current entering the element
	Same current passes through every element of a series circuit


	The laws, theorems, and rules that you apply to DC circuits
		Also apply to AC circuits


	The algebraic sum of the voltage that rises and drops around a closed loop is equal to zero
	E_T - V₁ - V₂ - V₃ - ∙∙∙ - V_n= 0


	Another way of stating KVL is:
		Summation of voltage rises is equal to the summation of voltage drops around a closed loop

	V₁ + V₂ + V₃ + ∙∙∙ + V_n = E_T


	Most complicated circuits can be simplified
	For a series circuit
		V₁ + V₂ + V₃ = E
		IR₁ + IR₂ + IR₃ = E
		I(R₁ + R₂ + R₃ )= E
		I(R₁ + R₂ + R₃ )= IR_total (Note: I’s cancel)


	Total resistance in a series circuit is the sum of all the resistor values


	Power dissipated by each resistor is determined by the power formulas:

	P = VI = V²/R = I²R


	Since energy must be conserved, power delivered by voltage source is equal to total power dissipated by resistors

	P_T = P₁ + P₂ + P₃ + ∙∙∙ + P_n


	In a circuit with more than one source in series
		Sources can be replaced by a single source having a value that is the sum or difference of the individual sources
	Polarities must be taken into account


	Resultant source
		Sum of the rises in one direction minus the sum of the voltages in the opposite direction


	Order of series components
		May be changed without affecting operation of circuit
	Sources may be interchanged, but their polarities can not be reversed
	After circuits have been redrawn, it may become easier to visualize circuit operation


	Voltage applied to a series circuit
		Will be dropped across all the resistors in proportion to the magnitude of the individual resistors


	Voltage dropped across each resistor may be determined by the voltage across any other resistor (or combination of resistors) by using the voltage divider rule expressed as:




	The subscripts must match (x and y)


	If a single resistor is very large compared to the other series resistors, the voltage across that resistor will be the source voltage
	If the resistor is very small, the voltage across it will be essentially zero


	If a resistor is more than 100 times larger than another resistor
		Smaller resistor can be neglected


	Ground
		Point of reference or a common point in a circuit for making measurements
	One type of grounding is chassis ground
	In this type of grounding
		Common point of circuit is often the metal chassis of the piece of equipment


	Chassis ground
		Often connected to Earth Ground
	Earth ground
		Physically connected to the earth by a metal pipe or rod


	If a fault occurs within a circuit, the current is redirected to the earth
	Voltages are often measured with respect to ground


	For the circuit shown, we can express the voltage between any two node points (a and b) as V_ab.
	If a is at a higher potential than b, then V_ab is positive


	If b is at a higher potential than a, then V_ab is negative
	In this case, V_abwould be negative


	To determine correct polarity
		Calculate all voltage drops across all components
		Assign polarities
	As you go around a circuit, add the gains and subtract the drops


	Be sure to take the sign of the polarity on the same side of the source or element as you go around the circuit


	In a circuit with a ground reference point
		Voltages may be expressed with respect to that reference point
	V_a
		Voltage at point a with respect to ground
		Ground represents a point of zero reference potential


	Any voltage with a single subscript is with respect to ground
	This is the same as V_a(0)


	If voltages at various points in a circuit are known with respect to ground, then the voltage between points is easily determined
	V_ab = V_a – V_b


	Voltage source given with respect to ground
		May be represented by a voltage at a single point (node) in the circuit
		This voltage may be referred to as a point source


	Voltages at these points represent voltages with respect to ground, even if ground is not shown
	Point sources simplify representation of a circuit


	Ideal sources have no internal resistance
	In an ideal source
		Terminal voltage does not change when the load changes
	For a practical source
		There is internal resistance