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2 Presentation Overview 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 Copyright © Texas Education Agency, 2014. All rights reserved.

3 Terms and Definitions A. 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. Copyright © Texas Education Agency, 2014. All rights reserved.

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

5 Symbols and Units A. X - Reactance in ohms B. X L - Inductive reactance in ohms C. f - Frequency in hertz D. R - Resistance in ohms E. ω - Angular velocity in radians per second (NOTE: ω also equals 2π f.) F. Z - Impedance in ohms G. 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 Copyright © Texas Education Agency, 2014. All rights reserved.

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 6 Copyright © 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 7 Copyright © Texas Education Agency, 2014. All rights reserved.

Inductive Reactance Formula For sinusoidal AC waveforms: 8 X L = ω L = 2πfL 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 Copyright © Texas Education Agency, 2014. All rights reserved.

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

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

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

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

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 14 R L S1S1 VSVS Copyright © Texas Education Agency, 2014. All rights reserved.

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 15 R L S1S1 VSVS Copyright © Texas Education Agency, 2014. All rights reserved.

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 17 I 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 18 (The Greek symbol tau ( τ ) is the symbol for the time constant.) 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 19 (The Greek symbol tau ( τ ) is the symbol for the time constant.) 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 During the next time constant current reaches 63.2% of the rest of the way to maximum current, or 86.5% of maximum 21 I Copyright © Texas Education Agency, 2014. All rights reserved.

Circuit Response Time (DC) During the next time constant the current reaches 63.2% of the rest of the way 22 I Copyright © Texas Education Agency, 2014. All rights reserved.

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 23 R L S1S1 VSVS Copyright © Texas Education Agency, 2014. All rights reserved.

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 24 R L S1S1 VSVS Copyright © Texas Education Agency, 2014. All rights reserved.

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 26 R L S1S1 VSVS Copyright © Texas Education Agency, 2014. All rights reserved.

AC Inductive Reactance Recall the formula for Inductive Reactance X L adds to the opposition of AC current flow depending on the frequency of the AC As frequency changes, X L changes, current changes, and voltage drops change The phase difference between voltage and current also changes 27 X L = ω L = 2πfL Copyright © Texas Education Agency, 2014. All rights reserved.

AC Circuit Analysis What is the current? 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 28 R = 20 Ω L = 50 mH S1S1 VSVS V S = 10 V, 60 Hz Copyright © Texas Education Agency, 2014. All rights reserved.

AC Circuit Analysis What is the current? X L and R have the same units (Ohms), but they cannot be directly added They combine to form impedance using the impedance formula 29 R = 20 Ω L = 50 mH S1S1 VSVS V S = 10 V, 60 Hz Copyright © Texas Education Agency, 2014. All rights reserved.

AC Circuit Analysis 30 R = 20 Ω L = 50 mH S1S1 VSVS V S = 10 V, 60 Hz Copyright © Texas Education Agency, 2014. All rights reserved.

The Impedance Triangle V L is 90° out of phase with current Current is in phase with voltage in a resistor This means that X L is 90° out of phase with R This 90° phase shift gives us something called the impedance triangle 31 X L (and V L ) R (and V R ) Z 20 Ω 18.5 Ω Copyright © Texas Education Agency, 2014. All rights reserved.

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 32 X L (and V L ) R (and V R ) Z 20 Ω 18.5 Ω θ R is the adjacent side, X L is the opposite side, and Z is the hypotenuse Copyright © Texas Education Agency, 2014. All rights reserved.

The Impedance Triangle We have three trigonometric formulas Because we often only have X L and R, use Tan solve for θ, θ = 33 X L (and V L ) R (and V R ) Z 20 Ω 18.5 Ω θ θ = θ = 42.77° Copyright © Texas Education Agency, 2014. All rights reserved.

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

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

36 Formulas for Determining True Power P T = I 2 R P T = V R I R P T = VI app cosine θ or VI app PF (where PF is the power factor) NOTE: True power is the actual power consumed by the resistance and is measured in watts. Copyright © Texas Education Agency, 2014. All rights reserved.