Electronics Inductive Reactance Copyright © Texas Education Agency, 2014. All rights reserved

Electronics

Inductive Reactance

Copyright © Texas Education Agency, 2014. All rights reserved.

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Presentation Overview Terms and definitions Symbols and definitions Factors needed to compute inductive reactance, XL Formula for computing inductive reactance (sinusoidal waveforms) Current and voltage relationships in RL circuits Computing applied voltage and impedance in series RL circuits Formulas for determining

true power apparent power reactive power power factor

Formula for determining quality factor (Q) or figure of merit of an inductor

Inductive time constants Universal time constant chart


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Terms and DefinitionsA. Resistance- opposition to current flow, which results in

energy dissipation.B. Reactance- opposition to a change in current or voltage,

which does not result in energy dissipation. (NOTE: this opposition is caused by inductive and capacitive effects.)

C. Impedance- opposition to current including both resistance and reactance. (NOTE: Resistance, reactance, and impedance are all measured in ohms.)

D. Inductive reactance- the opposition to a change in current caused by inductance.

E. Power- the rate of energy consumption in a circuit (true power).

F. Reactive power- the product of reactive voltage and current in an AC circuit.


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Terms and Definitions (cont’d.)

G. Apparent power- the product of volts and amperes (or the equivalent) in an AC circuit.

H. Power factor- the ratio of the true power (watts) to apparent power (volts-amperes) in an AC circuit.

I. Phase angle- the angle that the current leads or lags the voltage in an AC circuit. (NOTE: The phase angle is expressed in degrees or radians.)

J. Angular velocity- the rate of change of cyclical motion. (NOTE: angular velocity is expressed in radians per second.)

K. Time constant- the time required for an exponential quantity to change by an amount equal to 0.632 times the total change that will occur.


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Symbols and Units

A. X - Reactance in ohms B. XL - Inductive reactance in ohmsC. f - Frequency in hertzD. R - Resistance in ohmsE. ω - Angular velocity in radians per second (NOTE: ω also equals 2π f.)F. Z - Impedance in ohmsG. 2π - Radians in one cycle (NOTE: 2π equals approximately 6.28.)H. VARS (Volt Amperes Reactive) - Reactive apparent power I. PF - Power factor, the ratio of real power to apparent power


What is Reactance?

Reactance is like resistance for AC circuits Reactance limits, or reduces, current for AC

However, reactance does not use or consume energy in the way that resistance does Energy is stored in the form of an electric or magnetic

field This energy can be released and returned to the circuit

6Copyright © Texas Education Agency, 2014. All rights reserved.

Types of Reactance

There are two types of reactance Capacitive reactance Inductive reactance

Capacitive reactance stores energy in the form of an electric field

Inductive reactance stores energy in the form of a magnetic field


Inductive Reactance Formula

For sinusoidal AC waveforms:

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XL = ω L = 2πfLω: Angular velocity in radians per second (ω = 2πf)L: Inductance in henriesF: Frequency in hertz

Inductive reactance is directly proportional to the rate of change of current or voltage (the frequency) and the amount of inductance


Reactance in an Inductor

In an inductor, an increasing source voltage is temporarily used by (dropped across) the coil

However, this voltage does not create current Voltage is high, current is low for a time

The energy is converted into a magnetic field and temporarily stored

When the source voltage decreases, this stored energy is converted back into current Current is high, voltage is low for a time


Phase Shift

Voltage and current in a reactive device are not related the way they are in a resistive device

These effects are based on time and frequency The time effects are exponential, not linear

The energy is stored first and returned later This creates something called a phase shift

between voltage and current In an inductive device, voltage leads current


Phase Shift Shown Graphically

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Inductor voltage versus current for AC in a pure inductive circuit

voltage

current


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Current and Voltage Relationship in a R-L Circuit

A. Current lags voltage by 90º in a pure inductive circuit

B. Current and voltage are in phase in a pure resistive circuit

C. In an R-L circuit, current lags voltage between 0º and 90º depending upon

1. Relative amounts of R and L present 2. Frequency of applied voltage or current (angular

velocity)


Impedance

A circuit with a reactive device will also usually have a resistor as well There is always some amount of resistance in a

reactive device Resistance is the same for DC and AC Reactance is NOT the same for DC and AC

The equivalent resistance of a circuit with both reactance and resistance is called impedance

This combination of resistance and reactance does not directly add to create impedance


Series R – L Circuit

With DC voltage

The instant switch S1 is closed; the source voltage is dropped across the inductor

Current is initially zero but will begin to rise as the magnetic field reaches maximum strength

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RLS1VS


Series R – L Circuit

With DC voltage

After the magnetic field reaches maximum strength, no voltage is dropped across the inductor because there is no change in the field

Current reaches a maximum value

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RLS1VS

I = Copyright © Texas Education Agency, 2014. All rights reserved.

Series R – L Circuit (DC)

The current increase follows this curve

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RLS1VS

I


Circuit Response Time (DC)

This curve shows that current is changing over time

The time is defined in terms of a time constant It takes a time equal to five time constants for current to reach the maximum value

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I


R – L Time Constant (DC)

The value of the time constant is determined by circuit resistance and inductance values

The formula for the time constant is:

And the formula for the time response of the current is

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τ = (The Greek symbol tau (τ) is the symbol for the time constant.)

It = Copyright © Texas Education Agency, 2014. All rights reserved.

R – L Time Constant (DC)

The value of the time constant is determined by circuit resistance and inductance values

The formula for the time constant is:

And the formula for the time response of the current is

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τ = (The Greek symbol tau (τ) is the symbol for the time constant.)

It = Copyright © Texas Education Agency, 2014. All rights reserved.

This term is an exponent.


During one time constant the current reaches 63.2% of maximum value

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IIt = It = It = It =



During one time constant the current reaches 63.2% of maximum value

During the next time constant current reaches 63.2% of the rest of the way to maximum current, or 86.5% of maximum

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I



During the next time constant the current reaches 63.2% of the rest of the way

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IIt = It = It = It =


Series R – L Circuit (AC)

With AC voltage

AC voltage is constantly changing When the voltage is rising, some of the electrical

energy goes into increasing the magnetic field

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RLS1VS



With AC voltage

When the voltage is falling, energy from the magnetic field is returned to the circuit in the form of current

Current reaches a maximum value when the voltage across the inductor is zero

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RLS1VS


Phase Relationship

Recall this phase relationship between voltage and current for Alternating Current (AC)

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voltage

current



With AC, both voltage and current are constantly changing

Inductor magnetic field strength is also constantly changing

This means the inductor always has an AC resistance called Inductive Reactance

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RLS1VS


AC Inductive Reactance

Recall the formula for Inductive Reactance

XL adds to the opposition of AC current flow depending on the frequency of the AC As frequency changes, XL changes, current changes,

and voltage drops change The phase difference between voltage and current

also changes27

XL = ω L = 2πfL


AC Circuit Analysis

What is the current?

XL = 2πfL = 6.28(60)(.05) = 18.85 Ω It seems straightforward, but it is not Because current and voltage are out of phase,

they do not reach peak values at the same time28

R = 20 ΩL = 50 mHS1VSVS = 10 V,60 Hz


AC Circuit Analysis


XL and R have the same units (Ohms), but they cannot be directly added

They combine to form impedance using the impedance formula

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R = 20 ΩL = 50 mHS1VSVS = 10 V,60 Hz

Z = Copyright © Texas Education Agency, 2014. All rights reserved.

AC Circuit Analysis


= 27.5 Ω I = = = 0.364 A

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R = 20 ΩL = 50 mHS1VSVS = 10 V,60 Hz

Z = =


The Impedance Triangle

VL is 90° out of phase with current Current is in phase with voltage in a resistor This means that XL is 90° out of phase with R This 90° phase shift gives us something called the

impedance triangle

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XL(and VL)

R (and VR)Z

20 Ω18.5 Ω



Because this circuit has both resistance and reactance (impedance, Z); the phase angle between voltage and current is not 90° It is between 0° and 90°

We can use trigonometry to calculate the phase difference

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XL(and VL)

R (and VR)Z

20 Ω18.5 Ω

θ

R is the adjacent side, XL is the opposite side, and Z is the hypotenuse



We have three trigonometric formulas

Because we often only have XL and R, use Tan

solve for θ, θ =

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XL(and VL)

R (and VR)Z

20 Ω18.5 Ω

θ

θ = θ = 42.77°

Sine θ = Cos θ = Tan θ = Tan θ = Tan-1( )

Tan-1( ) = Tan-1( )


Power and Impedance

Only true resistance consumes power Inductors store energy in a magnetic field

This means they absorb energy to build the magnetic field

But return the energy later as the magnetic field collapses

This means power in an inductive circuit is not consumed the same way as power in a resistive circuit


Three Types of Power

1. True power is the power consumed by resistance

2. Reactive power is the power stored in a magnetic field by an inductor

3. Apparent power is the combination of true power and reactive power

You cannot directly add true power and reactive power because of the phase difference between voltage and current


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Formulas for Determining True Power

PT = I2R PT = VRIR PT = VIapp cosine θ or VIapp • PF

(where PF is the power factor)

NOTE: True power is the actual power consumed by the resistance and is measured in watts.


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Formulas for Determining Reactive Power

PX = I2X PX = VXIX PX = VI sin θ (where θ = or ) NOTE: Reactive power appears to be used by reactive components, but inductors use no power or energy, they take from the circuit to create a magnetic field but return it to the circuit when current direction reverses.


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Formulas for Determining Apparent Power

PA = VI PA = I2Z PA = NOTE: Apparent power is the power that appears to be used and is measured in volt-amperes.


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Formulas for Determining Power Factor

PF = PT / PA (true power divided by apparent power)

PF = VR / VS PF = R / Z PF = cos θ (where θ is the angle between current and voltage )


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Formula for Determining Quality Factor (Q) or Figure of Merit of an Inductor

Q = XL / RS (where XL is inductive reactance in ohms of an inductor

and RS is series resistance in ohms)

(NOTE: the quality factor (Q) or figure of merit is themeasure of a coil’s energy-storing ability.)


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Presentation Summary Terms and Definitions Symbols and Definitions Factors needed to compute inductive reactance, XL

Formula for computing inductive reactance (sinusoidal waveforms) Current and voltage relationships in RL circuits Computing applied voltage and impedance in series RL circuits Formulas for determining

true power apparent power reactive power power factor quality factor (Q) or figure of merit of an inductor

Inductive time constants Universal time constant chart


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Electronics Inductive Reactance Copyright © Texas Education Agency, 2014. All rights reserved