Topics in Electrical and Computer Engineering Phasors Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Copyright © 2008-10 by Douglas Wilhelm Harder. All rights reserved. Phasors Outline In this topic, we will look at – – – – – – – – The necessary background Sums of sinusoidal functions The trigonometry involved Phasor representation of sinusoids Phasor addition Why phasors work Phasor multiplication and inverses Phasors for circuits Phasors Background Consider any periodic sinusoid with T = 1/f = 2p/w f = 1/T = w/2p w = 2pf = 2p/T – Period – Frequency – Angular frequency It is possible to write such a sinusoid as v cos(wt + f) where – v is the amplitude – f is the phase shift f v Phasors Background As v cos(wt + f + 2p) = v cos(wt + f), restrict –p < f ≤ p Engineers throw an interesting twist into this formulation – The frequency term wt has units of radians – The phase shift f has units of degrees: –180° < f ≤ 180° For example, V cos(377t + 45°) V cos(377t – 90°) Phasors Background A positive phase shift causes the function to lead of f For example, –sin(t) = cos(t + 90°) leads cos(t) by 90° Compare cos(377t) and cos(377t + 45°) Blue peaks first - Blue is leading Phasors Background A negative phase shift causes the function to lag by f For example, sin(t) = cos(t – 90°) lags cos(t) by 90° Compare cos(377t) and cos(377t – 90°) Red peaks first - Blue lags behind Phasors Background If the phase shift is 180°, the functions are out of phase E.g., –cos(t) = cos(t – 180°) and cos(t) are out of phase Compare cos(377t) and cos(377t – 180°) Phasors Background Suppose we add a number of sinusoidal voltages with: – Equal frequencies – Different amplitudes (voltages) and phase shifts E.g., 3 cos377t 75 10 cos377t 50 4 cos377t 15 5 cos377t 42 12 cos377t 90 It may not be obvious, but the result will be another sinusoid of the form A cos377t f Phasors Background These are the five sinusoids and their sum 14.81cos377t 3.36 Phasors Derivations (The Hard Way) We will now show that the sum of two sinusoids with – The same frequency, and – Possibly different amplitudes and phase shifts is a sinusoid with the same frequency Our derivation will use trigonometric formula familiar to all high-school students – Later, we will see how exponentials with complex powers simplify this observation! Phasors Example Derivation (The Hard Way) Consider the sum of two sinusoids: 3 coswt 9 2 cos wt 45 Use the rule cosa b cosa cosb sin a sin b Thus we expand the terms 9 2 coswt p2 9 2 coswt cos45 9 2 sin wt sin 45 9 coswt 9 sin wt Phasors Example Derivation (The Hard Way) Therefore 3 coswt 9 2 coswt 45 12coswt 9 sinwt We wish to write this as v coswt f v coswt cosf v sinwt sin f We therefore deduce that v cosf 12 v sin f 9 Phasors Example Derivation (The Hard Way) Given v cosf 12 v sin f 9 Square both sides and add: v cosf v sin f 2 2 122 9 2 v 2 cos2 f sin 2 f 225 2 2 Because cos f sin f 1 it follows that v 225 15 Phasors Example Derivation (The Hard Way) Again, given v cosf 12 v sin f 9 Take the ratio: v sin f 9 tanf 0.75 v cosf 12 Therefore f ≈ 36.87° Phasors Example Derivation (The Hard Way) It follows that 3 coswt 9 2 coswt 45 15coswt 36.87 This is independent of the frequency: w2 w 5 Phasors Full Derivation (The Hard Way) Consider the sum of two sinusoids: v1 coswt f1 v2 coswt f2 Using the rule cosa b cosa cosb sin a sin b we can expand v1 cos wt 1 v2 cos wt 2 v1 cos wt cos 1 v1 sin wt sin 1 v2 cos wt cos 2 v2 sin wt sin 2 cos wt v1 cos 1 v2 cos 2 sin wt v1 sin 1 v2 sin 2 Phasors Full Derivation (The Hard Way) Now, given v1 coswt f1 v2 coswt f2 coswt v1 cosf1 v2 cosf2 sin wt v1 sin f1 v2 sin f2 Suppose we can write this in the form v coswt f First, v coswt f v coswt cosf v sinwt sin f and thus v cosf v1 cosf1 v2 cosf2 v sinf v1 sinf1 v2 sin f2 Phasors Full Derivation (The Hard Way) First, given v cosf v1 cosf1 v2 cosf2 v sin f v1 sin f1 v2 sin f2 square both sides and add v 2 v12 2v1v2 cosf1 cosf2 sin f1 sin f2 v2 2 v v12 2v1v2 cos f1 cos f 2 sin f1 sin f 2 v2 2 Phasors Full Derivation (The Hard Way) Similarly, given v cosf v1 cosf1 v2 cosf2 v sin f v1 sin f1 v2 sin f2 take the ratio sin f v1 sin f1 v2 sin f2 tanf cosf v1 cosf1 v2 cosf2 v1 sin f1 v2 sin f 2 f tan v1 cosf1 v2 cosf 2 1 Phasors Full Derivation (The Hard Way) Adding n sinusoids must be done one pair at a time: v1 cos wt f1 v2 cos wt f2 v3 cos wt f3 v1, 2 coswt f1, 2 v3 coswt f3 v1, 2,3 coswt f1, 2,3 Phasors Using Phasors Rather than dealing with trigonometric identities, there is a more useful representation: phasors Assuming w is fixed, associate v coswt f vf where vf is the complex number with magnitude v and argument f The value vf is a phasor and is read as vee phase fee Phasors Using Phasors The addition of sinusoids is equivalent to phasor addition Function of Time n v k coswt fk k 1 trigonometry v coswt f Phasors n v f k k k 1 complex number addition vf We transform the problem of trigonometric addition into a simpler problem of complex addition Phasors Using Phasors Recalling Euler’s identity vf ve jf vcosf j sin f Therefore n n v f v k k 1 k k 1 k cosfk j n v k 1 k sin fk Phasors Using Phasors Given our motivating example 3 coswt 9 2 coswt 45 – Using phasors: 30 9 245 3 9 9 j 12 9 j – The result is 15cos wt 36.87 1536.87 Phasors Using Phasors For example, given our first sum: 3 cos377t 75 10 cos377t 50 4 cos377t 15 5 cos377t 42 12 cos377t 90 Using phasors 375 1050 4 15 542 12 90 3e 75 j 10e 50 j 4e 15 j 5e 42 j 3e 14.78 0.87 j 14.813.36 we get the sum 14.81cos377t 3.36 90 j Phasors Using Phasors These graphs show: – The individual phasors – The sum of the phasors Phasors Using Phasors A plot of the sum 3 cos377t 75 10 cos377t 50 4 cos377t 15 5 cos377t 42 12 cos377t 90 is identical to a plot of 14.81cos377t 3.36 Phasors Why Phasors Work To understand recall that ve j wt f v coswt f jv sinwt f j wt f Therefore v coswt f ve Plot the complex exponential 14.81e j 377t 3.36 The real part: 14.81 cos(377t + 3.36°) The imaginary part: 14.81sin(377t + 3.36°) Phasors Why Phasors Work Note that v1e j wt f1 v2e j wt f2 v1e jwt e jf1 v2 e jwt e jf2 v1e jf1 v2e jf2 e jwt v1f1 v2f2 e jwt and because w z w z, it follows v1 coswt f1 v2 coswt f2 v1e j wt f1 v2 e j wt f2 v1e jwt e jf1 v2 e j wt e jf2 v1e jwt e jf1 v2 e jwt e jf2 v1f1 v2 f2 e jwt Phasors Derivation (The Easy Way) Previously, we used trigonometry to show that the sum of two sinusoids with the same frequency must continue to have the same frequency Using complex exponentials, this becomes obvious v1e j wt f1 v e 2 j wt f2 v f v f e jwt 1 1 2 2 Phasors Phasor Multiplication Multiplication may be performed directly on phasors: – The product of two phasors r1f1 and r2 f2 is the phasor r r f f 1 2 1 2 – For example, 545 2 60 10 15 – This generalizes to products of n phasors: r f r f r r f 1 1 n n 1 n 1 fn Phasors Phasor Multiplication Calculating the inverse is also straight-forward: – The inverse of the phasor rf is the phasor – For example, 1 f r 545 1 0.2 45 – Using multiplication, we note the product is 1: 1 1 rf f r f f 10 r r Phasors Linear Circuit Elements When dealing with alternating currents (AC): – The generalization of the resistance R is the complex impedance Z = R + jX – The generalization of the conductance G is the complex admittance Y = G + jB Ohm’s law easily generalizes: V = IZ Phasors Linear Circuit Elements With AC, the linear circuit elements behave like phasors Inductor with inductance L Z jw L w L90 Resistor with resistance R Z R0 1 1 90 Capacitor with capacitance C Z jwC wC – Note that inductance and capacitance are frequency dependant We can use this to determine the voltage across an linear circuit with AC: V = IZ Phasors Linear Circuit Elements Consider three circuit elements in series: with R= 2W L = 50 mH C = 750 mF The total impedance is: 1 Z 20 w 0.05090 90 0.750w • E.g., if w 1, it follows that Z 2.376 32.69 Phasors Linear Circuit Elements If the current across this circuit is I = 10 cos(t) A, we may compute the voltage: V IZ 100 2.376 32.69 23.76 32.69 or V = 23.76 cos(t – 32.69°) V Phasors Linear Circuit Elements If the current across this circuit is I = 10 cos(377t) A, we need to recalculate the impedance: Z 20 377 0.05090 0.7501377 90 18.9583.94 Now V IZ 100 18.9583.94 189.583.94 or V = 189.5 cos(377t + 83.94°) V Phasors Linear Circuit Elements Consider three circuit elements in parallel with R= 2W L = 50 mH C = 750 mF The total admittance is: 1 1 1 1 Y 90 Z 20 w 0.05090 w 0.750 E.g., if w 1, it follows that Y 0.50 20 90 0.7590 18.95 83.94 Z 0.0527683.94 Phasors Linear Circuit Elements If the current across this circuit is I = 10 cos(t) A we may compute the voltage: V IZ 100 0.0527683.94 0.527683.94 or V = 0.5276 cos(t + 83.94°) V Phasors Linear Circuit Elements If the current across this circuit is I = 10 cos(377t) A we need to recalculate the impedance: 1 1 1 Y Z 20 377 0.05090 282.789.90 or Z 0.003537 89.90 1 1 3770.750 90 Phasors Linear Circuit Elements Thus, with the current I = 10 cos(377t) A and impedance Z 0.003537 89.90 we may now calculate V IZ 100 0.003537 89.90 0.03537 89.90 or V = 0.03537 cos(377t – 89.90°) V This should be intuitive, as the capacitor acts as essentially a short-circuit for high frequency AC Phasors Linear Circuit Elements A consequence of what we have seen here is that any linear circuit with alternating current may be replaced with – A single resistor, and – A single complex impedance • A capacitor or an inductor R22 w 2 X 22 R22 w 2 X 22 R1 , w X1 wX2 R2 Phasors Final Observations These techniques do not work if the frequencies differ The choice of 377 in the examples is deliberate: – North American power is supplied at 60 Hz: 2p 60 ≈ 376.99111843 – Relative error: 0.0023% – European power is supplied at 50 Hz: 2p 50 = 100p ≈ 314.159265358 Phasors Final Observations Mathematicians may note the peculiar mix of radians and degrees in the formula 10cos377t 50 Justification: – The phase shifts are 180° or less (p radians or less) – Which shifts are easier to visualize? 120° 60° 45° 30° 2.094 rad or 2p/3 rad 1.047 rad or p/3 rad 0.785 rad or p/4 rad 0.524 rad or p/6 rad – Is it obvious that 2.094 rad and 0.524 rad are orthogonal? – Three-phase power requires the third roots of unity: • These cannot be written easily with radians (even using p) Phasors Summary In this topic, we will look at – – – – Sum of sinusoidal functions The trigonometry is exceptionally tedious The phasor representation v coswt f vf The parallel between phasor addition and trigonometric summations – Phasor multiplication and inverses – Use of phasors with linear circuit elements Phasors Acknowledgments • I would like to thank Prof. David Nairn for assistance in answering questions and Hua Qiang Cheng for pointing out errors in a previous version of these slides Usage Notes • These slides are made publicly available on the web for anyone to use • If you choose to use them, or a part thereof, for a course at another institution, I ask only three things: – that you inform me that you are using the slides, – that you acknowledge my work, and – that you alert me of any mistakes which I made or changes which you make, and allow me the option of incorporating such changes (with an acknowledgment) in my set of slides Sincerely, Douglas Wilhelm Harder, MMath dwharder@alumni.uwaterloo.ca