Chapter 14: Inductors and Inductance

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Chapter 14Inductors and Inductance

• What are inductors?
• An inductor is an electronic component that will
  oppose any changes in the current in a circuit.
• An inductor is an electromagnet in the manner in
  which it is constructed, but its function is not
  to create a magnetic field, but to oppose current
  flow.
• Inductors generated a counter-emf or back-emf
  in the coil due to magnetic lines of forces that
  cut across a conductive wire.
• The term self-induction is used to describe the
  counter-emf that is generated in a coil.

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Basic Inductor
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Electromagnetic Field - DC
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Units of Inductance

• What is the unit and symbol for inductance?
• The unit of inductance is the henry (H), which is
  defined as the production of one volt of
  counter-emf in a coil when current has a change
  of one ampere per second.
• The symbol for inductance is the letter (L) and
  is defined mathematically as

Vind L Di/Dt where L inductance,
in henrys (H) Di a change in current
(deltadifference) Dt a
change in time Vind voltage inducted
in a coil
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Electromagnetic Field - AC
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Basic Inductor
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Inductor Construction
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What Determines Inductance?

• Factors that determine Inductance
• Number of turns (direct relationship)
• Area of the coil (direct relationship)
• Length of coil (inverse relationship)
• Permeability of the core used (direct
  relationship)

Metric FormulaL N2 A m
l L in henrys N number of
turnsA area, square metersl length in
metersm permeability of core
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Calculating Inductance for coils
English FormulaL (n r)2 m
9r 10 l L in microhenrys, mH n number
of turnsl length of coil in inchesm
permeability of core
When finding the inductance of a single layer air
core coil, the value for the permeability is 1.
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Inductors in Series

• Because inductors are connected end-to-end the
  number of turns, and the area increases
  proportionally with the total inductance.
• With this type of circuit, this effectively
  increases the opposition to a change in current
  offered by the coils.
• The formula is identical to the series resistance
  formula used in DC circuits.

LT L1 L2 L3 LN
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Inductors in Parallel

• Because inductors are connected in parallel, more
  paths are provided this effectively reduces the
  opposition to a change in current.
• Each branch has its own separate inductance, and
  the smallest inductance determines the total
  opposition to a change in current.
• The formula is identical to the parallel
  resistance formula used in DC circuits.

With this formula this means that the total
inductance is always less than the smallest
inductor.
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Inductive (L/R)Time Constants

• Inductors will not have any effect on current in
  a circuit that uses purely a DC power source.
• When current is either pulsating DC or AC,
  magnetic fields in the coil expands and
  collapses, generating an opposition to the flow
  in current.
• In a pulsating DC circuit the polarity never
  changes direction, and when a resistor is placed
  in a DC circuit with an inductor this delays the
  flow of current in the circuit.
• There is a time constant ( in seconds)for RL
  circuits and it is found by this formula L/R
• There is always voltage on the coil, but current
  increases exponentially with respect to time, and
  is limited by the resistance in the circuit.
• Just as it was in RC time constant circuits, it
  takes 5 time constants for the circuit to store
  the electromagnetic energy in a coil. (growth
  curve)

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Inductive (L/R)Time Constants-2

• When a coil releases its magnetic energy, it
  takes again, 5 time constants to release its
  stored magnetic energy. (decay curve)
• To have a large time constant, inductance must be
  very large compared to resistance in a DC
  circuit.

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AC Inductive circuits

• In an AC circuit, current is constantly changing
  direction, which causes the magnetic fields to
  constantly expand and collapse.
• The higher the frequency (rate of change) of an
  AC source, the time available to store the
  magnetic field is greatly reduced, and the
  current to flow in a circuit decreases.
• A decrease in current flow in an inductor is
  similar to the effect a resistor has on current
  flow in DC or AC circuits.
• In a purely inductive circuit, the flow of
  current is 90 degrees out of phase with the
  voltage that is applied to the coil (ELI).
• In summary, in AC inductive circuits, voltage
  always leads current by 90 degrees due to an
  inductors ability to oppose a change in current.
  
• The measure of the opposition to current flow in
  a circuit using an inductor without the
  dissipation of energy with an AC power source is
  called inductive reactance.

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AC Inductive circuits

• In an AC circuit, current is constantly changing
  direction, which causes the magnetic fields to
  constantly expand and collapse.
• The higher the frequency (rate of change) of an
  AC source, the time available to store the
  magnetic field is greatly reduced, and the
  current to flow in a circuit decreases.
• A decrease in current flow in an inductor is
  similar to the effect a resistor has on current
  flow in DC or AC circuits.
• In a purely inductive circuit, the flow of
  current is 90 degrees out of phase with the
  voltage that is applied to the coil (ELI).
• In summary, in AC inductive circuits, voltage
  always leads current by 90 degrees due to an
  inductors ability to oppose a change in current.
  
• The measure of the opposition to current flow in
  a circuit using an inductor without the
  dissipation of energy with an AC power source is
  called inductive reactance.

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Inductive Reactance

• The symbol is XL the unit is the ohm, and is
  found by the following formula
• XL 2pfL
• where
• f frequency in hertz
• L inductance in henrys
• XL inductive reactance in ohms

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Series AC-RL Circuits

• The vector diagram and the phase angle are
  plotted in the solution process when resolving
  the vectors of R, XL and Z.
• Since the voltage in an inductor leads current by
  90 degrees, the XL or the EL vectors are plotted
  on the positive portion of the y-axis.
• The phase angle is positive since the resultant
  vector or phase angle rotates counterclockwise
  from the x-axis (R).
• An AC source with a high frequency will create a
  large value for XL effectively causes a higher
  EL/ER ratio in the circuit, and the circuit acts
  inductively reactive.
• The power factor, the PVAR, PTRUE and PAPP are
  found effectively the same way. PVAR IT2
  XLT PTRUE IT2 RT
  PAPP IT Es

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Solving Series RL-AC circuits-1

• R1 R2 L1
  L2
• 50v 10 kW 25 kW 50.0 mH
  20.0 mH
• f 32 kHz
• Steps

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Solving Series RL-AC circuits-2

• R1 R2 L1
  L2
• 50v 10 kW 25 kW 50.0 mH
  20.0 mH
• f 32 kHz
• Steps

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Solving Series RL-AC circuits-3

• R1 R2 L1
  L2
• 50v 10 kW 25 kW 50.0 mH
  20.0 mH
• f 32 kHz
• Steps

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Solving Series RL-AC circuits-4

• R1 R2 L1
  L2
• 50v 10 kW 25 kW 50.0 mH
  20.0 mH
• f 32 kHz
• Steps

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Series AC-RL- 2

• PVAR some times is referred to as imaginary
  power, and the symbol PX is sometimes used in
  problem solving.
• The diagram vector in an AC-RL circuit looks like
  this

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Quality Factor of a Coil

• Because inductors are coils of wire that contain
  resistance, they differ from capacitors in the
  manner they operate in AC circuits.
• At lower AC frequencies, the resistance of the
  wire can greatly affect the way an inductor
  performs in a circuit. WHY?
• All inductors act like individual Series RL
  circuits, and for this reason a term was created
  to describe the quality of an inductors
  operating performance.
• Quality factor- (Q) is a ratio of the energy
  stored in an inductor by the energy dissipated by
  the inductor and has no units.
• A high Q value means that an inductor acts like
  an inductor, that is, it will pass all DC current
  signals and block high frequency AC signals.

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Q formula

• Q XL
• RLT of the coil

• Generally speaking a Q greater than 10 is an
  indication that the inductor will act inductively
  reactive in an AC circuit.
• Caution The symbol Q can be used to indicate
  other quantities in complex AC circuits, so often
  the terms are clearly identified as the Q of
  the coil.

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Parallel AC-RL Circuits

• Voltage is always constant in a parallel circuit
  therefore, XLT, RT and Z are never plotted in
  parallel AC-RL circuits.
• The vector diagram and the phase angle are
  plotted in the solution process when resolving
  the vectors of IT, ILT and IRT.
• Instead, the current flowing through inductors,
  ILT and resistors, IRT, are used in plotting the
  vectors of these circuits, along with IT
  (resultant vector).
• Since the current in a resistor leads current in
  an inductor by 90 degrees, the ILT vector is
  plotted on the negative portion of the y-axis.
• The current flowing through the resistors, IRT is
  plotted on the x-axis, and the resultant vector
  is the total current, IT flowing in the circuit.

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Parallel AC-RL-2

• The phase angle is negative since the resultant
  vector or phase angle rotates clockwise from the
  x-axis (IRT).
• An AC source with a high frequency will create a
  large value for XL and effectively causes a lower
  current to flow in the branch that contains the
  inductor.
• Since the frequency has no effect on the current
  flow through the resistor, when the frequency is
  very high, the only current to flow in the
  circuit is through the resistor.
• The power in AC-RL circuits are found using a
  different formula. Pvar
  Es2/XLT Ptrue Es2/RT
  PAPP IT Es

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Solving Parallel RL-AC circuits-1

• 80V R1 R2
  L1 L2
• f 20 kHz
• 10kW 15kW 8 mH
  4 mH
• Steps

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Solving Parallel RL-AC circuits-2

• 80V R1 R2
  L1 L2
• f 20 kHz
• 10kW 15kW 8 mH
  4 mH
• Steps

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Solving Parallel RL-AC circuits-3

• 80V R1 R2
  L1 L2
• f 20 kHz
• 10kW 15kW 8 mH
  4 mH
• Steps

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Solving Parallel RL-AC circuits-4

• 80V R1 R2
  L1 L2
• f 20 kHz
• 10kW 15kW 8 mH
  4 mH
• Steps

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Parallel AC-RL Vector