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ECE 3301 General Electrical Engineering
Presentation 23 Inductance
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Inductance An electrical current in any conductor causes a magnetic field to exist around the conductor. The magnetic field forms a closed loop around the conductor as illustrated below.
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Inductance The magnetic field obeys the right-hand rule.
If the current is in the direction of the thumb of the right hand, the magnetic field is pointed in the direction the fingers form around the conductor.
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Inductance If the conductor is formed into a coil, the magnetic field is reinforced by adjacent conductors.
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Inductance The magnetic field is said to “link” the turns of the coil.
Energy is stored in the magnetic field.
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Inductance The effect of the magnetic field is to maintain the current in the conductor. This effect is called inductance.
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Inductance An inductance in a circuit is represented by the symbol shown below.
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Inductance An inductance in a circuit is represented by the symbol shown below. Inductance is measured in henries (H).
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Inductance The current-voltage relationship for inductance is given by the equation
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Inductance The voltage across an inductance is proportional to the time-rate-of-change of the current through the inductance. The constant of proportionality is called the inductance.
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Drives a time-varying current through the inductor
The current source Drives a time-varying current through the inductor
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The voltage across the inductance
Is proportional to the time rate-of-change of the current
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The voltage across the inductance
Is proportional to the slope of this waveform
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Slope = 0 Voltage = 0
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Slope = 1 Voltage = L(1)
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Slope = 0 Voltage = 0
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Slope = -1 Voltage = L(-1)
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Slope = 0 Voltage = 0
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Drives a time-varying, sinusoidal current through the inductor
The current source Drives a time-varying, sinusoidal current through the inductor
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Period = T0 Amplitude
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The voltage across the inductance
Is proportional to the time rate-of-change of the current
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The current and voltage are 90 degrees “out of phase” with each other.
The voltage leads the current by 90 degrees.
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Drives a time-varying current through the inductor
The current source Drives a time-varying current through the inductor
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The voltage across the inductance
Is proportional to the slope of this waveform
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The instantaneous change in current
Causes an infinite voltage pulse across the inductor
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An infinite pulse cannot be achieved !
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Rule 1 of Inductance Since this infinite voltage impulse cannot be physically realized, we conclude the first rule-of-thumb about inductance. One cannot instantaneously change the current through an inductance.
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Rule 2 of Inductance Since the voltage across the inductance is proportional to the time rate-of-change of the current through the inductance, when the current is constant (DC), the voltage is zero. A voltage of zero across a circuit element is the definition of a short circuit. This leads to the second rule-of-thumb about inductances. An inductance is a short circuit to Steady State DC.
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Inductance Consider an inductance driven by a voltage source.
The inductance has an initial current of i0 amps.
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Inductance The voltage-current relationship is given by:
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Inductance Performing a bit of calculus:
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Inductance Integrating both sides:
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Inductance Completing the integration:
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Inductance Solving for i(t):
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Inductance The current through an inductance is proportional to the integral of the voltage across the inductance plus the initial current through the inductance.
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Places a time-varying voltage across the inductance
The voltage source Places a time-varying voltage across the inductance
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The current through the inductance
Is proportional to the area under the voltage waveform
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Final current Accumulated area Initial current
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Places a sinusoidal voltage across the inductance
The voltage source Places a sinusoidal voltage across the inductance
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Is proportional to the integral of the voltage
The current Is proportional to the integral of the voltage
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The current and voltage are 90 degrees “out of phase” with each other.
The voltage leads the current by 90 degrees.
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Power and Energy in an Inductance
The instantaneous power absorbed by any circuit element is given by: Using the voltage-current relationship for an inductance:
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Power and Energy in an Inductance
Leads to the instantaneous power in an inductance: By definition, power is given by:
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Power and Energy in an Inductance
Consequently : and :
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Power and Energy in an Inductance
The energy stored in an inductance may be found by integration. Assume an initial energy of zero and an initial current of zero.
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Power and Energy in an Inductance
The energy stored in an inductance is:
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Inductors in Series
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Inductors in Parallel
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Inductors in Parallel
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Inductors in Parallel
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Inductors in Parallel
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