ECE 3301 General Electrical Engineering Presentation 23 Inductance

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.

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.

Inductance If the conductor is formed into a coil, the magnetic field is reinforced by adjacent conductors.

Inductance The magnetic field is said to “link” the turns of the coil. Energy is stored in the magnetic field.

Inductance The effect of the magnetic field is to maintain the current in the conductor. This effect is called inductance.

Inductance An inductance in a circuit is represented by the symbol shown below.

Inductance An inductance in a circuit is represented by the symbol shown below. Inductance is measured in henries (H).

Inductance The current-voltage relationship for inductance is given by the equation

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.

Drives a time-varying current through the inductor The current source Drives a time-varying current through the inductor

The voltage across the inductance Is proportional to the time rate-of-change of the current

The voltage across the inductance Is proportional to the slope of this waveform

Slope = 0 Voltage = 0

Slope = 1 Voltage = L(1)

Slope = 0 Voltage = 0

Slope = -1 Voltage = L(-1)

Slope = 0 Voltage = 0

Drives a time-varying, sinusoidal current through the inductor The current source Drives a time-varying, sinusoidal current through the inductor

Period = T0 Amplitude

The voltage across the inductance Is proportional to the time rate-of-change of the current

The current and voltage are 90 degrees “out of phase” with each other. The voltage leads the current by 90 degrees.

Drives a time-varying current through the inductor The current source Drives a time-varying current through the inductor

The voltage across the inductance Is proportional to the slope of this waveform

The instantaneous change in current Causes an infinite voltage pulse across the inductor

An infinite pulse cannot be achieved !

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.

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.

Inductance Consider an inductance driven by a voltage source. The inductance has an initial current of i0 amps.

Inductance The voltage-current relationship is given by:

Inductance Performing a bit of calculus:

Inductance Integrating both sides:

Inductance Completing the integration:

Inductance Solving for i(t):

Inductance The current through an inductance is proportional to the integral of the voltage across the inductance plus the initial current through the inductance.

Places a time-varying voltage across the inductance The voltage source Places a time-varying voltage across the inductance

The current through the inductance Is proportional to the area under the voltage waveform

Final current Accumulated area Initial current

Places a sinusoidal voltage across the inductance The voltage source Places a sinusoidal voltage across the inductance

Is proportional to the integral of the voltage The current Is proportional to the integral of the voltage

The current and voltage are 90 degrees “out of phase” with each other. The voltage leads the current by 90 degrees.

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:

Power and Energy in an Inductance Leads to the instantaneous power in an inductance: By definition, power is given by:

Power and Energy in an Inductance Consequently : and :

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.

Power and Energy in an Inductance The energy stored in an inductance is:

Inductors in Series

Inductors in Parallel

Inductors in Parallel

Inductors in Parallel

Inductors in Parallel