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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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3
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
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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:
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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
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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 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 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
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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)
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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
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
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
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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
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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 It takes a time equal to five time constants for current to reach the maximum value
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I
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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:
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.
Circuit Response Time (DC)
During one time constant the current reaches 63.2% of maximum value
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IIt = It = It = It =
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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 63.2% of the rest of the way to maximum current, or 86.5% of maximum
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I
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Circuit Response Time (DC)
During the next time constant the current reaches 63.2% of the rest of the way
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IIt = It = It = It =
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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
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Series R – L Circuit (AC)
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
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Phase Relationship
Recall this phase relationship between voltage and current for Alternating Current (AC)
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voltage
current
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Series R – L Circuit (AC)
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
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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
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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
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AC Circuit Analysis
What is the current?
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
What is the current?
= 27.5 Ω I = = = 0.364 A
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R = 20 ΩL = 50 mHS1VSVS = 10 V,60 Hz
Z = =
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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
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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
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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
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The Impedance Triangle
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( )
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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. 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
Copyright © Texas Education Agency, 2014. All rights reserved.