Chapter 14

Inductive Transients

Transients

Voltages and currents during a transitional interval

Referred to as transient behavior of the circuit

Capacitive circuit

Voltages and currents undergo transitional phase

Capacitor charges and discharges

Transients

Inductive circuit

Transitional phase occurs as the magnetic field builds and collapses

Voltage Across an Inductor

Induced voltage across an inductor is proportional to rate of change of current

Voltage Across an Inductor

If inductor current could change instantaneously

Its rate of change would be infinite

Would cause infinite voltage

Continuity of Current

Infinite voltage is not possible

Inductor current cannot change instantaneously

It cannot jump from one value to another, but must be continuous at all times

Use observation when analyzing circuits

Circuit and Waveforms for
Current Build-up

Inductor Voltage

Immediately after closing the switch on an RL circuit

Current is zero

Voltage across the resistor is zero

Inductor Voltage

Voltage across resistor is zero

Voltage across inductor is source voltage

Inductor voltage will then exponentially decay to zero

Open-Circuit Equivalent

After switch is closed

Inductor has voltage across it and no current through it

Inductor with zero initial current looks like an open circuit at instant of switching

Open-Circuit Equivalent

This statement will later be applied to include inductors with nonzero initial currents

Initial Condition Circuits

Voltages and currents in circuits immediately after switching

Determined from the open-circuit equivalent

Initial Condition Circuits

By replacing inductors with opens

We get initial condition circuit

Initial condition networks

Yield voltages and currents only at switching

Circuit Current

Current in an RL circuit is an exponentially increasing function of time

Current begins at zero and rises to a maximum value

Circuit Voltages

Voltage across resistor is given by i•R

Voltage across resistor is an increasing function as well (because the inductor current is passing through it)

Circuit Voltages

By KVL,voltage across inductor is

E – v_R

Voltage across inductor is an exponentially decreasing function of time

Time Constant

t = L/R

Units are seconds

Equations may now be written as

Time Constant

The larger the inductance

The longer the transient

The larger the resistance

The shorter the transient

Time Constant

As R increases

Circuit looks more and more resistive

If R is much greater than L

Circuit looks purely resistive

Interrupting Current in an Inductive Circuit

When switch opens in an RL circuit

Energy is released in a short time

This may create a large voltage

Induced voltage is called an inductive kick

Opening of inductive circuit may cause voltage spikes of thousands of volts

Interrupting a Circuit

Switch flashovers are generally undesirable

They can be controlled with proper engineering design

These large voltages can be useful

Such as in automotive ignition systems

Interrupting a Circuit

It is not possible to completely analyze such a circuit

Resistance across the arc changes as the switch opens

Interrupting a Circuit

In circuit shown, we see changes after switch opens:

Inductor Equivalent at Switching

Current through an inductor

Same after switching as before switching

An inductance with an initial current

Looks like a current source at instant of switching

Its value is value of current at switching

De-energizing Transients

If an inductor has an initial current I₀, equation for current becomes

t ' = L/R. R equals total resistance in discharge path

De-energizing Transients

Voltage across inductor goes to zero as circuit de-energizes

De-energizing Circuits

Voltage across any resistor is product of current and that resistor

Voltage across each of resistors goes to zero

More Complex Circuits

For complex circuits

Like capacitive circuits

Necessary to determine Thévenin equivalent circuit using inductor as the load

More Complex Circuits

R_Th is used to determine time constant

t = L/R_Th

E_Th is used as source voltage


	Voltages and currents during a transitional interval
		Referred to as transient behavior of the circuit
	Capacitive circuit
		Voltages and currents undergo transitional phase
		Capacitor charges and discharges


	Inductive circuit
		Transitional phase occurs as the magnetic field builds and collapses


	Induced voltage across an inductor is proportional to rate of change of current


	If inductor current could change instantaneously
		Its rate of change would be infinite
		Would cause infinite voltage


	Infinite voltage is not possible
		Inductor current cannot change instantaneously
		It cannot jump from one value to another, but must be continuous at all times
	Use observation when analyzing circuits


	Immediately after closing the switch on an RL circuit
		Current is zero
		Voltage across the resistor is zero


	Voltage across resistor is zero
		Voltage across inductor is source voltage
	Inductor voltage will then exponentially decay to zero


	After switch is closed
		Inductor has voltage across it and no current through it
	Inductor with zero initial current looks like an open circuit at instant of switching


	This statement will later be applied to include inductors with nonzero initial currents


	Voltages and currents in circuits immediately after switching
		Determined from the open-circuit equivalent


	By replacing inductors with opens
		We get initial condition circuit
	Initial condition networks
		Yield voltages and currents only at switching


	Current in an RL circuit is an exponentially increasing function of time
	Current begins at zero and rises to a maximum value


	Voltage across resistor is given by i•R
	Voltage across resistor is an increasing function as well (because the inductor current is passing through it)


	By KVL,voltage across inductor is
	E – v_R
	Voltage across inductor is an exponentially decreasing function of time


	The larger the inductance
		The longer the transient
	The larger the resistance
		The shorter the transient


	As R increases
		Circuit looks more and more resistive
		If R is much greater than L
		Circuit looks purely resistive


	When switch opens in an RL circuit
		Energy is released in a short time
		This may create a large voltage
		Induced voltage is called an inductive kick
	Opening of inductive circuit may cause voltage spikes of thousands of volts


	Switch flashovers are generally undesirable
		They can be controlled with proper engineering design
	These large voltages can be useful
		Such as in automotive ignition systems


	It is not possible to completely analyze such a circuit
		Resistance across the arc changes as the switch opens


	Current through an inductor
		Same after switching as before switching
	An inductance with an initial current
		Looks like a current source at instant of switching
	Its value is value of current at switching


	If an inductor has an initial current I₀, equation for current becomes


	t ' = L/R. R equals total resistance in discharge path


	Voltage across inductor goes to zero as circuit de-energizes


	Voltage across any resistor is product of current and that resistor



	Voltage across each of resistors goes to zero


For complex circuits
	Like capacitive circuits
		Necessary to determine Thévenin equivalent circuit using inductor as the load


	R_Th is used to determine time constant
	t = L/R_Th
	E_Th is used as source voltage