Inductive Transients

1
Chapter 14

• Inductive Transients

2
Transients

• Voltages and currents during a transitional
  interval are called transients.
• In a capacitive circuit, voltages and currents go
  through a transitional phase while the capacitor
  charges and discharges.
• In an inductive circuit, a transitional phase
  occurs as the magnetic field builds and
  collapses.

3
Voltage across an Inductor

• The induced voltage across an inductor is
  proportional to the rate of change of current.
• If the inductor current could change
  instantaneously, its rate of change would be
  infinite. This would cause infinite voltage.

4
Continuity of Current

• Since infinite voltage is not possible, inductor
  current cannot change instantaneously.
• This means it cannot jump abruptly from one value
  to another, but must be continuous at all values
  of times.
• Use this observation when analyzing circuits.

5
Circuit and Waveforms for Current Build-up
6
Inductor Voltage

• Immediately after closing the switch on an RL
  circuit, the current is zero, so the voltage
  across the resistor is zero.
• Since the voltage across the resistor is zero,
  the voltage across the inductor is source
  voltage.
• The inductor voltage will then decay to zero.

7
Open-Circuit Equivalent

• Just after the switch is closed, the inductor has
  voltage across it and no current through it.
• An inductor with zero initial current looks like
  an open circuit at the instant of switching.
• This statement will later be applied to include
  inductors with nonzero initial currents.

8
Initial Condition Circuits

• Voltages and currents in circuits immediately
  after switching can be determined from the
  open-circuit equivalent.
• By replacing inductors with opens, we get the
  initial condition circuit.
• Initial condition networks yield voltages and
  currents only at the instant of switching.

9
Circuit Current

• Current in an RL circuit is an increasing
  function.
• The current begins at zero and rises to a maximum
  value.

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Circuit Voltages

• The voltage across the resistor is given by iR.
• The voltage across the resistor is an increasing
  function.

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Circuit Voltages

• By KVL, the voltage across the inductor is
• E - vR.
• The voltage across the inductor is a decreasing
  function.

12
Time Constant

• ? L/R
• The units are seconds.
• The equations may now be written as

13
Time Constant

• The larger the inductance, the longer the
  transient.
• The larger the resistance, the shorter the
  transient.
• As R increases, the circuit looks more and more
  resistive if R is much greater than L, the
  circuit looks purely resistive.

14
Interrupting Current in an Inductive Circuit

• When the switch opens in an RL circuit, a great
  deal of energy is released in a short time.
• This may create large voltage.
• This induced voltage is called an inductive kick.
• The breaking of current may cause voltage spikes
  of thousands of volts.

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Interrupting a Circuit

• Switch flashovers are generally undesirable, but
  they can be controlled with proper engineering
  design.
• These large voltages are sometimes useful, such
  as in automotive ignition systems.
• It is not possible to completely analyze such a
  circuit because the resistance across the arc
  changes as the switch opens.

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Interrupting a Circuit

• In the circuit shown, we can see the changes as
  after the switch opens

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Inductor Equivalent at Switching

• The current through an inductor is the same
  immediately after switching as before switching.
• An inductance with an initial current looks like
  a current source at the instant of switching.
• Its value is the value of the current at
  switching.

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De-energizing Transients

• If an inductor has an initial current I0, the
  equation for the current becomes
• ? ' L/R. R equals total resistance in the
  discharge path.

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De-energizing Transients

• The voltage across the inductor goes to zero as
  the circuit de-energizes.

20
De-energizing Circuits

• The voltage across any resistor is the product of
  the current and that resistor.
• The voltage across any of the resistors goes to
  zero.

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More Complex Circuits

• For complex circuits, it is necessary to
  determine the Thévenin equivalent of the circuit
  with respect to the inductor.
• RTh is used to determine the time constant.
• ETh is used as the source voltage.