REACTANCE. By: Enzo Paterno Date: 03/2013

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1 REACTANCE REACTANCE By: Enzo Paterno Date: 03/2013 5/2007 Enzo Paterno 1

2 RESISTANCE - R i R (t R A resistor for all practical purposes is unaffected by the frequency of the applied sinusoidal voltage or current (i.e. assumes f < 100 khz As such, R can be treated as a constant and using Ohm s law: Z: Impedance R: Resistance Let v V I R R R : Z i R Z ( t ( t i + v R (t R R ( t I I P I P P [ Ω] sinωt R I Rsinωt sinωt [ A] Rsinωt R 0 [ V ] [ Ω] For a purely resistive element, v R (t and I R (t are in phase, with their peak values related by Ohm s law. 5/2007 Enzo Paterno 2 P

3 RESISTANCE - R 5/2007 Enzo Paterno 3

4 RESISTANCE - R 5/2007 Enzo Paterno 4

5 RESISTOR I Vs V i(t For a resistor V R (t & I R (t are in phase I P V R I P P IN PHASE V P v(t 0 5/2007 Enzo Paterno 5

6 INDUCTIVE REACTANCE - X L Let : v L V I L L ( t Z i L ( t I ω L L [ H ] i L (t + v L (t P X sinωt [ A] dil ( t L ω L I P cosωt ω L I P sin( ωt + 90 [ V ] dt ω L I P sin( ωt + 90 Z ω L 90 jω L [ Ω] I sinωt P L [ Ω] X L : Inductive Reactance For an inductor, the voltage is: v L ( t dil ( t L dt For an inductor v L (tleads i L (t by 90, or i L (t lags v L (t by 90 with their peak values related by Ohm s law. 5/2007 Enzo Paterno 6

7 INDUCTIVE REACTANCE - X L The Phase Relationship Between Inductor Voltage and Current: Voltage leads current by 90 Current lags voltage by 90 X L V I P P For an inductor V L leads I L by 90, or I L lags V L by 90 5/2007 Enzo Paterno 7

8 INDUCTIVE REACTANCE - X L i L (t L [ H ] + v L (t An inductor has a reactance which is a resistance that varies as a function of frequency. As such you might think of an inductor as a variable resistor whose resistance is controlled by the signal frequency applied to the inductor. XL( f 2πfL ωl [ Ω] f 0 f f 0 dc : f 0 X L 0 ( inductor acts as a short for f X L ( inductor acts as an open 5/2007 Enzo Paterno 8

9 INDUCTIVE REACTANCE - X L X L [ Ω ] Linear function XL { 1 2 L } 2πfL kf π R Increasing L 0 f [ Hz ] 5/2007 Enzo Paterno 9

10 INDUCTIVE REACTANCE - X L Example: Calculate the total current for the circuit below: I V X S L 12V 1kΩ 5/2007 Enzo Paterno 10

11 INDUCTIVE REACTANCE - X L Example: Calculate X L for the circuit below f 50 khz L 1 mh X L 2πfL 2π (50x10 3 (1x Ω 5/2007 Enzo Paterno 11

12 INDUCTIVE REACTANCE - X L Example: Calculate the total current for the circuit below 10 V rms f 5 khz L 33 mh 3 3 X L 2πfL 2π (5x10 (33x I VS 10V X 1.04 kω L ma kω 5/2007 Enzo Paterno 12

13 Let : dvc ( t ic ( t C ω CVP cosωt ω CVP sin( ωt + 90 dt VC VP sinωt 1 1 Z 90 j I ω CV sin( ωt + 90 ω C ω C C Z v CAPACITIVE REACTANCE - X C C ( t V 1 ω C i C (t P sinωt P X C + [ Ω] [ V ] C [ F ] + v C (t [ V ] [ Ω] For a capacitor i C (t leads v C (t by 90, or v C (t lags i C (t by 90 with their peak values related by Ohm s law. 5/2007 Enzo Paterno 13 For a capacitor, the current is: i ( t X C : Capacitive Reactance C C dv C dt ( t

14 CAPACITIVE REACTANCE - X C The Phase Relationship Between Capacitor Current and Voltage: Current leads voltage by 90 Voltage lags current by 90 V X C I P P For a capacitor I C leads V C by 90, or V C lags I C by 90 5/2007 Enzo Paterno 14

15 CAPACITIVE REACTANCE - X C i(t C [ F ] + f 0 f f 0 + v(t A capacitor has a reactance which is a resistance that varies as a function of frequency. As such you might think of a capacitor as a variable resistor whose resistance is controlled by the signal frequency applied to the capacitor. dc : for f f Xc 0 X c X c 1 2π f C 1 ωc ( capacitor 0 ( capacitor [ Ω] acts act as an open as a short 5/2007 Enzo Paterno 15

16 CAPACITIVE REACTANCE - X C Xc [ Ω ] Non Linear function Xc 1 k π 2π f C f { 1 2 C R } Increasing C 0 f [ Hz ] 5/2007 Enzo Paterno 16

17 CAPACITIVE REACTANCE - X C Example: Calculate the total current below 1 X C 2πfC 2π 1 ( 60 Hz( 22 μf 121Ω I C V X S C 10 V 121Ω 8.26 ma 5/2007 Enzo Paterno 17

18 CAPACITIVE REACTANCE - X C Series and Parallel Values of X C 5/2007 Enzo Paterno 18

19 REACTANCE X C - Summary Case X v I Purely Resistive R In phase In phase Purely Inductive ωl Leads I by 90 Lags v by 90 Purely Capacitive 1 Lags I by 90 Leads v by 90 ωc 5/2007 Enzo Paterno 19

20 EXAMPLE The voltage across a resistor is indicated. Find the sinusoidal expression for the current when the resistor is 10Ω. ( v( t 100sin t 60 Solution : Voltage and current, for a resistor, are in phase v 100sin(377t i( t sin(377t + R 10Ω 10 i( t 10sin(377t /2007 Enzo Paterno 20

21 EXAMPLE The current through a 5Ω resistor is indicated. Find the sinusoidal expression for the voltage across the resistor if: ( i( t 40sin t 30 Solution : For a resistor the voltage and current are in phase v( t ir 40v (5Ωsin(377t v( t 200sin(377t sin(377t /2007 Enzo Paterno 21

22 EXAMPLE The current through a 0.1H coil is indicated. Find the sinusoidal expression for the voltage across the coil if: ( 377 i( t 7sin t 70 Solution : X V L For an inductor, v leads i by 90 v( t v( t (7A (37.7Ω ωl 377 rad/s (0.1H sin(377t sin(377t V L IL 263.9v X 37.7Ω 90 L 5/2007 Enzo Paterno 22

23 EXAMPLE The voltage across a 1 µf capacitor is indicated. Find the sinusoidal expression for the current through the capacitor if: v( t 30sin 400t Solution : X I C For an capacitor, i leads v by 90 i( t V X 1 ωc C 12x rad/s (1x 10 30v 2500Ω 3 12mA sin(400t 1 F 2500Ω 5/2007 Enzo Paterno I C V X C C

24 EXAMPLE Given the voltage across a device and the current through this device, determine whether the device is a resistor, capacitor or inductor. v( t 100sin( ωt + 40 i( t 20sin( ωt + 40 Solution : Since v(t and i(t are in phase, V 100v R 5Ω I 20 A the device is a resistor 5/2007 Enzo Paterno 24

25 EXAMPLE Given the voltage across a device and the current through this device, determine whether the device is a resistor, capacitor or inductor. v( t 1000sin(377t + 10 i( t 5sin(377t 80 5/2007 Enzo Paterno 25

26 EXAMPLE Given the voltage across a device and the current through this device, determine whether the device is a resistor, capacitor or inductor. v( t 1000sin(377t + 10 i( t 5sin(377t 80 Solution : Since v(t leads i(t by 90, the device is an inductor V 1000v X L 200Ω I 5 A 200Ω X L ωl L H 377 rad/s 5/2007 Enzo Paterno 26

27 EXAMPLE Given the voltage across a device and the current through this device, determine whether the device is a resistor, capacitor or inductor. v( t 500sin(157t + 30 i( t 1sin(157t /2007 Enzo Paterno 27

28 EXAMPLE Given the voltage across a device and the current through this device, determine whether the device is a resistor, capacitor or inductor. v( t 500sin(157t + 30 i( t 1sin(157t Solution : Since i(t leads v(t by 90, the device is a capacitor V 500v X C 500Ω I 1A 1 1 X C C 12.74µ F ωc 157 rad/s (500Ω 5/2007 Enzo Paterno 28

29 EXAMPLE At what frequency will the reactance of a 200 mh inductor match the resistance level of a 5 kω resistor? 5/2007 Enzo Paterno 29

30 EXAMPLE At what frequency will the reactance of a 200 mh inductor match the resistance level of a 5 kω resistor? Solution : X L f X 2πfL L 2πL 5000 Ω 5000 Ω khz 5/2007 Enzo Paterno 30

31 EXAMPLE At what frequency will an inductor of 5 mh have the same reactance as that of a 0.1 µf capacitor? 5/2007 Enzo Paterno 31

32 EXAMPLE At what frequency will an inductor of 5 mh have the same reactance as that of a 0.1 µf capacitor? Solution : X L 1 2πfL 2πfC f 2 f X C 1 2 4π LC 1 2π LC 7.12 khz 5/2007 Enzo Paterno 32

33 AVERAGE POWER AVERAGE POWER DUE TO AC SINUSOIDAL INPUTS 5/2007 Enzo Paterno 33

34 AVERAGE POWER i(t i( t I sin v( t V sin ( ωt + θ ( ωt + θ i v { P - + v(t L O A D p iv IV sinα sin β we get : sin We use the trig.identity : 1 2 ( ωt + θ sin( ωt + θ i [ cos( α β cos( α + β ] v p IV 2 cos IV 2 ( ( θ θ cos 2ωt + θ + θ v i v i 5/2007 Enzo Paterno 34

35 AVERAGE POWER p IV cos 2 IV v i t v θ 2 ( θ θ cos( 2ω + θ + i constant time-varying f(t P AVG 0 p AVG 5/2007 Enzo Paterno 35

36 AVERAGE POWER IV p AVG cos θ v θ i 2 IV 2 cosα Phase difference between I(t and v(t For Purely Resistive networks: i(t and v(t are in phase Get a maximum P AVG : P AVG IV / 2 For Purely Inductive or Capacitive networks: i(t and v(t are out of phase by 90 Get a minimum P AVG : P AVG 0 The average power provides a net transfer of energy It represents the power delivered to and dissipated by the load. 5/2007 Enzo Paterno 36

37 EXAMPLE P avg (V m I m / 2 cos θ v θ i Let α θ v θ i We get VmI m Vm I m P avg cosα cosα RMS value of power equals peak since we are Dealing with average power (dc Which expresses P avg in terms of the effective RMS values P V avg rms I rms cosα 5/2007 Enzo Paterno 37

38 EXAMPLE Find the average power dissipated in a network having the following input voltage and current: v( t 100sin(377t + 40 i( t 20sin(377t + 70 α IV p AVG cos θ v θi 2 IV 2 (100(20 p AVG cos30 2 cosα 866 5/2007 Enzo Paterno 38 W

39 EXAMPLE A series RLC circuit has the following impedance: Z R + JX L JX C Determine the frequency at which the average power is maximum. Solution: It occurs when the network is purely resistive with Z R Z R + for Z JX R L 1 2πfL 2πfC 1 f 2π LC JX X C L R X 5/2007 Enzo Paterno 39 C + J ( X 0 L X X L C X C

40 POWER FACTOR POWER FACTOR 5/2007 Enzo Paterno 40

41 POWER FACTOR In the equation P avg (V m I m / 2 cos α, (i.e. α θ v θ i, the factor that has significant control over the delivered power level is the cos α. No matter how large the voltage or current. When cos α 0, then P avg P min 0 When cos α 1, then P avg P max (V m I m / 2 Since cos α controls the power, the expression is named power factor and is defined by: 2Pavg Power factor Fp cosα V I m m V P rms avg I rms 5/2007 Enzo Paterno 41

42 POWER FACTOR For a purely resistive load, the phase angle between v and i is α 0 giving a power factor of 1 (F p cos α cos 0 1. The delivered power is then the maximum value of (V m I m / 2 watts. For a purely reactive load, (inductive or capacitive, the phase angle between v and i is α 90 giving a power factor of 0 ( F p cos α cos The delivered power is then the minimum value of zero watts. When the load is a combination of resistive and reactive elements, the power factor will vary between 0 and 1. The more resistive the total impedance, the closer the power factor is to 1; the more reactive the total impedance, the closer the power factor is to 0. 5/2007 Enzo Paterno 42

43 POWER FACTOR The terms leading and lagging are often written in conjunction with the power factor. They are defined by the current through the load. If the current leads the voltage across a load, the load has a leading power factor. If the current lags the voltage across the load, the load has a lagging power factor. In general: Capacitive networks have leading power factors Inductive networks have lagging power factors. 5/2007 Enzo Paterno 43

44 POWER FACTOR Determine the power factor of the load given below, and indicate whether it is leading or lagging: i leads the voltage 5/2007 Enzo Paterno 44

45 POWER FACTOR Determine the power factor of the load given below, and indicate whether it is leading or lagging: i lags the voltage 5/2007 Enzo Paterno 45

46 POWER FACTOR Determine the power factor of the load given below, and indicate whether it is leading or lagging: Load is resistive F p neither leads or lag 5/2007 Enzo Paterno 46

47 POWER FACTOR wikipedia.org PHASORS 5/2007 Enzo Paterno 47

48 PHASORS As part of performing the analysis of an AC circuit, it will be required to perform mathematical operations with sinusoidal voltages and/or currents represented in the time domain. One lengthy but valid method of performing this operation is to place both sinusoidal waveforms on the same set of axes and add algebraically the magnitudes of each at every point along the abscissa. Long & Tedious process 5/2007 Enzo Paterno 48

49 PHASORS To alleviate this long and tedious process, one technique to perform such mathematical operations is to use PHASORS. A phasor is the polar form representation of the time domain voltage or current: Time domain Phasor domain Magnitude: The peak values I or V are converted to RMS Angle: Phase θ Time domain i(t I m v(t V m sin Peak value sin Example : ( ωt + θ m ( ωt + θ v θ 5/2007 Enzo Paterno 49 i RMS value Note : I m θ 2 V Phasor domain

50 PHASORS Phasors can also be converted back to their time domain format: Phasor domain i v(t I θ RMS value v V θ Example : i(t 2 I 2 m V sin Peak value m ( ωt + θ sin Note : 2 ( ωt + θ Time domain 5/2007 Enzo Paterno 50

51 PHASORS Phasor domain Time domain 5/2007 Enzo Paterno 51

52 PHASORS Convert the following from the time to the phasor domain: Convert the following from the phasor to the time domain when f 60 Hz: 5/2007 Enzo Paterno 52

53 PHASORS Find the input voltage of the circuit below if: v v a b ( t ( t 50sin(377t 30sin(377t Va - + Vin Vb - Solution : KVL Vin Va + Vb Va ( Vb ( j17.68 j /2007 Enzo Paterno 53

54 PHASORS KVL Vin Va + Vb Vin Vin ( j j j36. l Thus; Vin 2(54.76 sin(377t /2007 Enzo Paterno 54

55 PHASORS v v a b ( t ( t 50sin(377t 30sin(377t Vin 2 (54.76sin(377t /2007 Enzo Paterno 55

56 COMPUTER ANALYSIS PSPICE 5/2007 Enzo Paterno 56

57 COMPUTER ANALYSIS Plots: P C V C I C 5/2007 Enzo Paterno 57

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