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

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Terms and definitions
Symbols and definitions
Factors needed to compute inductive reactance, X
L
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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Copyright © Texas Education Agency, 2014. All rights reserved.

A.
B.
C.
D.
E.
F.
Resistance- opposition to current flow, which results in energy dissipation.
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.)
Impedance- opposition to current including both resistance and reactance. (NOTE: Resistance, reactance, and impedance are all measured in ohms.)
Inductive reactance- the opposition to a change in current caused by inductance.
Power- the rate of energy consumption in a circuit (true power).
Reactive power- the product of reactive voltage and current in an AC circuit.
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I.
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.
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.
K.
Angular velocity- the rate of change of cyclical motion.
(NOTE: angular velocity is expressed in radians per second.)
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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F.
G.
A.
B.
C.
D.
E.
H.
I.
X - Reactance in ohms
X
L
- Inductive reactance in ohms f - Frequency in hertz
R - Resistance in ohms
ω - Angular velocity in radians per second (NOTE: ω also equals 2π f .)
Z - Impedance in ohms
2π - Radians in one cycle (NOTE: 2π equals approximately
6.28.)
VARS (Volt Amperes Reactive) - Reactive apparent power
PF - Power factor, the ratio of real power to apparent power
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Copyright © Texas Education Agency, 2014. All rights reserved.

 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
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
Energy is stored in the form of an electric or magnetic field
This energy can be released and returned to the circuit
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 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
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 For sinusoidal AC waveforms:
X
L
= ω L = 2π f L
ω: Angular velocity in radians per second ( ω = 2πf )
L: Inductance in henries
F: Frequency in hertz
 Inductive reactance is directly proportional to the rate of change of current or voltage (the frequency) and the amount of inductance
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 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
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 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
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 Inductor voltage versus current for AC in a pure inductive circuit voltage current
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A.
B.
C.
Current lags voltage by 90º in a pure inductive circuit
Current and voltage are in phase in a pure resistive circuit
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)
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 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
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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
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 With DC voltage
R
S
1
V
S
L
 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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 With DC voltage
R
S
1
V
S
L


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 I =
V s
R
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 The current increase follows this curve
R
S
1
V
S
L
I
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 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
I five time constants for current to reach the maximum value
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 The value of the time constant is determined by circuit resistance and inductance values
 The formula for the time constant is:
L
τ =
R
(The Greek symbol tau ( τ ) is the symbol for the time constant.)
 And the formula for the time response of the current is
I t
=
V
S
R
(1 − 𝑒 −
τ )
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 The value of the time constant is determined by circuit resistance and inductance values
 The formula for the time constant is:
L
τ =
R
(The Greek symbol tau ( τ ) is the symbol for the time constant.)
 And the formula for the time response of the current is
I t
=
V
S
R
(1 − 𝑒 −
τ )
This term is an exponent.
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 During one time constant the current reaches
63.2% of maximum value
I t
=
V
R
S (1 − 𝑒 −
τ )
I
I t t
=
V
S
R
=
V
S
R
I t
=
V
R
S
(1 − 𝑒
− 1
)
(1 − .368)
(.632)
I
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 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
I
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 During the next time constant the current reaches 63.2% of the rest of the way
I t
=
V
R
S
I t
=
V
R
S
I t
I t
=
V
S
R
=
V
R
S
(1 − 𝑒
(1 − 𝑒
(1 − .135)
(.865)
−
τ
− 2 )
)
I
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 With AC voltage
R
S
1
V
S
L
 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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 With AC voltage
R
S
1
V
S
L
 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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 Recall this phase relationship between voltage and current for Alternating Current (AC) voltage current
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 With AC, both voltage and current are constantly changing
R
S
1
V
S
L
 Inductor magnetic field strength is also constantly changing
 This means the inductor always has an AC resistance called Inductive Reactance
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 Recall the formula for Inductive Reactance
X
L
= ω L = 2π f L
 X
L adds to the opposition of AC current flow depending on the frequency of the AC
 As frequency changes, X
L and voltage drops change changes, current changes,
 The phase difference between voltage and current also changes
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 What is the current?
R = 20 Ω
S
1
V
S
= 10 V,
60 Hz
V
S
L = 50 mH
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X
L
= 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 time
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 What is the current?
R = 20 Ω
S
1
V
S
= 10 V,
60 Hz
V
S
L = 50 mH
 X
L and R have the same units (Ohms), but they cannot be directly added
 They combine to form impedance using the impedance formula
Z = 𝑅 2 + 𝑋
𝐿
2
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 What is the current?
R = 20 Ω
S
1
V
S
= 10 V,
60 Hz
V
S
L = 50 mH

Z = = 20 2 + 18.85
2 = 27.5 Ω
 𝑅 2
𝑉
𝑍
𝑆 =
10 𝑉
2
27.5 Ω
= 0.364 A
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

V
L is 90° out of phase with current
Current is in phase with voltage in a resistor
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
This means that X
L is 90° out of phase with R
This 90 ° phase shift gives us something called the impedance triangle
X
L
(and V
L
)
18.5 Ω
Z
20 Ω R (and V
R
)
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 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 °
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We can use trigonometry to calculate the phase difference
X
L
(and V
L
)
R is the adjacent side, X
L is the opposite side, and Z is the hypotenuse
18.5 Ω
Z
θ
20 Ω R (and V
R
)
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 We have three trigonometric formulas
Sine θ = opp hyp adj
Cos θ = hyp
Tan θ = opp adj
 Because we often only have X
L
 Tan θ = opp adj and R, use Tan solve for θ, θ = Tan -1 ( opp adj
)
 θ = Tan -1 ( opp adj
) = Tan -1 (
18.5
20
)
X
L
(and V
L
)
 θ = 42.77
°
18.5 Ω
Z
θ
20 Ω R (and V
R
)
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 Only true resistance consumes power
 Inductors store energy in a magnetic field
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
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
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1.
2.
True power is the power consumed by resistance
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
Copyright © Texas Education Agency, 2014. All rights reserved.
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
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P
T
= I 2 R
P
T
= V
R
I
R
P
T
= VI app cosine θ or VI
(where PF is the power factor) app
• PF
NOTE: True power is the actual power consumed by the resistance and is measured in watts.
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

P
X
= I 2 X
P
X
= V
X
I
X
 P
X
= VI sin θ
(where θ =
V
A or
X
)
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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
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
P
A
= VI
P
A
= I 2 Z
P
A
=
V 2
Z
NOTE: Apparent power is the power that appears to be used and is measured in volt-amperes.
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
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PF = P
T
/ P
A
(true power divided by apparent power)
PF = V
R
/ V
S
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 = X
L
/ R
S
(where X
L
R
S is inductive reactance in ohms of an inductor and is series resistance in ohms)
(NOTE: the quality factor ( Q ) or figure of merit is the measure of a coil’s energy-storing ability.)
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Copyright © Texas Education Agency, 2014. All rights reserved.










Terms and Definitions
Symbols and Definitions
Factors needed to compute inductive reactance, X
L
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
Copyright © Texas Education Agency, 2014. All rights reserved.
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