Transcript Inductive reactance

Electronics
Inductive Reactance
Copyright © Texas Education Agency, 2014. All rights reserved.
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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2
Terms and Definitions
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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Terms and Definitions (cont’d.)
G.
H.
I.
J.
K.
Apparent power- the product of volts and amperes (or the
equivalent) in an AC circuit.
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.)
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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Symbols and Units
A.
B.
C.
D.
E.
F.
G.
H.
I.
X - Reactance in ohms
XL - 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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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
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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
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Inductive Reactance Formula

For sinusoidal AC waveforms:
XL = 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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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


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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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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

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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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Phase Shift Shown Graphically

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.
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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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

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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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Series R – L Circuit

With DC voltage
R
S1
VS


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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Series R – L Circuit

With DC voltage
R
S1
VS


L
After the magnetic field reaches maximum
strength, no voltage is dropped across the
inductor because there is no change in the field
Vs
Current reaches a maximum value I =
R
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15
Series R – L Circuit (DC)

The current increase follows this curve
R
S1
VS
L
I
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Circuit Response Time (DC)



This curve shows that current is changing over
time
The time is defined in
terms of a time constant
I
It takes a time equal to
five time constants for
current to reach the
maximum value
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17
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:
τ=

L
(The Greek symbol tau (τ) is the symbol for the time constant.)
R
And the formula for the time response of the
current is
It =
VS
R
(1 −
−t τ
𝑒
)
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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:
τ=

L
(The Greek symbol tau (τ) is the symbol for the time constant.)
R
And the formula for the time response of the
current is
It =
VS
R
(1 −
−t τ
𝑒
)
This term is an exponent.
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Circuit Response Time (DC)

During one time constant the current reaches
63.2% of maximum value
It =
It =
It =
It =
VS
R
VS
R
VS
R
VS
R
(1 −
−t τ
𝑒
(1 − 𝑒 − 1 )
)
I
(1 − .368)
(.632)
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Circuit Response Time (DC)


During one time constant the current reaches
63.2% of maximum value
During the next time
constant current reaches
I
63.2% of the rest of the
way to maximum current,
or 86.5% of maximum
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Circuit Response Time (DC)

During the next time constant the current
reaches 63.2% of the rest of the way
It =
It =
It =
It =
VS
R
VS
R
VS
R
VS
R
(1 −
−t τ
𝑒
(1 − 𝑒 − 2 )
)
I
(1 − .135)
(.865)
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22
Series R – L Circuit (AC)

With AC voltage
R
S1
VS


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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Series R – L Circuit (AC)

With AC voltage
R
S1
VS


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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Phase Relationship

Recall this phase relationship between voltage
and current for Alternating Current (AC)
voltage
current
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Series R – L Circuit (AC)

With AC, both voltage and current are constantly
changing
R
S1
VS


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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AC Inductive Reactance

Recall the formula for Inductive Reactance
XL = ω L = 2π f L

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 changes
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AC Circuit Analysis

What is the current?
R = 20 Ω
S1
VS = 10 V,
60 Hz



VS
L = 50 mH
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 time
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AC Circuit Analysis

What is the current?
R = 20 Ω
S1
VS = 10 V,
60 Hz


VS
L = 50 mH
XL and R have the same units (Ohms), but they
cannot be directly added
They combine to form impedance using the
impedance formula
Z = 𝑅2 + 𝑋𝐿 2
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AC Circuit Analysis

What is the current?
R = 20 Ω
S1
VS = 10 V,
60 Hz

Z=
L = 50 mH
VS
= 202 + 18.852 = 27.5 Ω
𝑉𝑆
10 𝑉
2
 I𝑅
= +=𝑋𝐿 2
𝑍
27.5 Ω
= 0.364 A
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The Impedance Triangle
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
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
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
XL
(and VL)
18.5 Ω
Z
20 Ω
R (and VR)
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The Impedance Triangle

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
X
(and V )
R is the adjacent side, XL
is the opposite side, and Z 18.5 Ω Z
is the hypotenuse
L

L
θ
20 Ω
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R (and VR)
32
The Impedance Triangle

We have three trigonometric formulas
adj
Cos θ =
hyp
opp
Sine θ =
hyp

Tan θ =
opp
adj
Because we often only have XL and R, use Tan
opp
 Tan θ =
adj
solve for θ,
opp
Tan-1(

θ=

θ = 42.77°
adj
)
θ = Tan-1(
18.5
-1
= Tan (
20
)
opp
adj
)
XL
(and VL)
18.5 Ω
Z
θ
20 Ω
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R (and VR)
33
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
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Three Types of Power
1.
2.
3.
True power is the power consumed by
resistance
Reactive power is the power stored in a
magnetic field by an inductor
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 = VR
VA or
R
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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Formulas for Determining
Apparent Power



PA = VI
PA = I2Z
2
V
P =
A
Z
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 = X L / 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 the
measure of a coil’s energy-storing ability.)
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40
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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41