Phasors and Complex Numbers

1. Phasors and Complex Numbers Alan Murray

2. Alan Murray – University of Edinburgh Complex NumbersRevision j = v(-1), so j2 = -1 2,3, -4.75 are real numbers 2j, 3j, -4.75j are imaginary numbers (3.5 - 7.2j) is a complex number With real and imaginary parts +3.5 and -7.2j respectively This lecture section is a small mathematical detour, to remind you about complex numbers, how they behave, and to prepare us to use them to do electronic things. First of all – here is what we mean by a complex number.This lecture section is a small mathematical detour, to remind you about complex numbers, how they behave, and to prepare us to use them to do electronic things. First of all – here is what we mean by a complex number.

3. Alan Murray – University of Edinburgh Complex Arithmetic C1 = a + jb C2 = c + jd Adding two complex numbers … C1 + C2 = (a + c) + j(b + d) add Re() and Im() parts separately Multiplying two complex numbers …  C1 x C2 = (a + jb) x (c + jd) = (ac – bd) + j(bc + ad) But … C1 x C1 = (a + jb) x (a + jb) = (a2 – b2) + j x 2ab So |C1| is NOT v(C12) as it is for real numbers Some reminders as to how the real and imaginary parts of complex numbers work. Adding is easy – just like adding the components of a vector, we add the real parts to make the real part of the sum and do the same with the imaginary parts. Multiplying complex numbers isn’t so easy, though. If we simply multiply directly, we get another complex number, which will not be helpful if we want to, for example, take the square root of the square of a complex number in order to find its magnitude (i.e. how big it is!). Let’s note this little problem and set it aside for a few slides – once we’ve built up a picture of complex numbers as pictures it will be easy to come up with the correct way to multiply complex numbers together.Some reminders as to how the real and imaginary parts of complex numbers work. Adding is easy – just like adding the components of a vector, we add the real parts to make the real part of the sum and do the same with the imaginary parts. Multiplying complex numbers isn’t so easy, though. If we simply multiply directly, we get another complex number, which will not be helpful if we want to, for example, take the square root of the square of a complex number in order to find its magnitude (i.e. how big it is!). Let’s note this little problem and set it aside for a few slides – once we’ve built up a picture of complex numbers as pictures it will be easy to come up with the correct way to multiply complex numbers together.

4. Alan Murray – University of Edinburgh Complex Numbers as Pictures

5. Alan Murray – University of Edinburgh Complex Numbers as Pictures

6. Alan Murray – University of Edinburgh Complex Numbers as Pictures As we noted a couple of slides back, taking the magnitude of a complex numbers slightly more complex (sic) than taking the magnitude of a real number – we must multiply a number, not by itself, but by it’s complex conjugate (i.e. by itself with the sign of all imaginary bits reversed, so that, for example, x+jy has x-jy as its conjugate). Then we can take the v of the result (which will be real) and bingo – the magnitude of the original compelx number you first thought of.As we noted a couple of slides back, taking the magnitude of a complex numbers slightly more complex (sic) than taking the magnitude of a real number – we must multiply a number, not by itself, but by it’s complex conjugate (i.e. by itself with the sign of all imaginary bits reversed, so that, for example, x+jy has x-jy as its conjugate). Then we can take the v of the result (which will be real) and bingo – the magnitude of the original compelx number you first thought of.

7. Alan Murray – University of Edinburgh Complex Numbers as Pictures Spot the difference …Spot the difference …

8. Alan Murray – University of Edinburgh AC Circuit analysis Trigonometry is conceptually simple Phasor diagrams are nice to analyse sin(A+B)+cos(C+D) etc is a pain! Phasor diagrams for complex circuits look like this Back to lots of nasty trigonometry We can use complex numbers to make the mechanics of AC circuit analysis better. Trig. Is superficially, relatively attractive, as we’ve all llived with sin() and cos() etc. for years. Phasors seems a nice, comfortable half-way-house … and provide (a) a nice way to analyse simple circuits and (b) a nice bridge of understanding towards complex number methods. However, phasor diagrams for complicated circuits are a mess and re-introduce a lot of nasty trigonometry! So – we are going to bite the bullet NOW and introduce the use of complex number notation for electronic signals. You will not like it, initially, so we’ll stick to very simple examples, to keep the maths unthreatening! It is one of the most powerful and elegant links between maths, physics and engineering, however, so don’t write it off as “just a load of useless theory”. It is not. It is fundamental to much of what will follwos in the next 3-4 years and, once you’ve got your head around it, is very satisfying. End of sales pitch/sermon!We can use complex numbers to make the mechanics of AC circuit analysis better. Trig. Is superficially, relatively attractive, as we’ve all llived with sin() and cos() etc. for years. Phasors seems a nice, comfortable half-way-house … and provide (a) a nice way to analyse simple circuits and (b) a nice bridge of understanding towards complex number methods. However, phasor diagrams for complicated circuits are a mess and re-introduce a lot of nasty trigonometry! So – we are going to bite the bullet NOW and introduce the use of complex number notation for electronic signals. You will not like it, initially, so we’ll stick to very simple examples, to keep the maths unthreatening! It is one of the most powerful and elegant links between maths, physics and engineering, however, so don’t write it off as “just a load of useless theory”. It is not. It is fundamental to much of what will follwos in the next 3-4 years and, once you’ve got your head around it, is very satisfying. End of sales pitch/sermon!

9. Alan Murray – University of Edinburgh Rationale Argand Diagrams = Phasor Diagrams? Can we use complex number mathematics (L) to make geometrical things simpler to do (J)? No pain, no gain!

10. Alan Murray – University of Edinburgh Remember how phasors work? Think back to the whirly arrows. We were able to use actual angles on a diagram to represent phase differences between voltage and currents. We were also able to “spin” the arrows to allow us to extract the time-dependednt behaviour of the circuit. The method for getting real voltages and currents out was easy – simply project the arrows, at an instant in time, on to the x-axis (or y-axis, it doesn’t matter in phasor diagrams) and out pops the sinusoidal variation of everything – all nice related in phase as it should be. The complex-number equivalent should be easy, as the diagrams are so similar …Think back to the whirly arrows. We were able to use actual angles on a diagram to represent phase differences between voltage and currents. We were also able to “spin” the arrows to allow us to extract the time-dependednt behaviour of the circuit. The method for getting real voltages and currents out was easy – simply project the arrows, at an instant in time, on to the x-axis (or y-axis, it doesn’t matter in phasor diagrams) and out pops the sinusoidal variation of everything – all nice related in phase as it should be. The complex-number equivalent should be easy, as the diagrams are so similar …

11. Alan Murray – University of Edinburgh Translation into complex numbers and Argand diagrams? And here are the rules. Angles on the diagram and phase angles between signals are equivalent as in a phasor diagram. Spinning is equivalent to letting the angle Ø run 0?360°?720°?1080° etc. Projecting on to the x-axis is even easier – just take the real part of everything. The only tricky bit is accepting the mathematical tool that complex number methods represent.And here are the rules. Angles on the diagram and phase angles between signals are equivalent as in a phasor diagram. Spinning is equivalent to letting the angle Ø run 0?360°?720°?1080° etc. Projecting on to the x-axis is even easier – just take the real part of everything. The only tricky bit is accepting the mathematical tool that complex number methods represent.

12. Alan Murray – University of Edinburgh So we need to represent C = a+jb in terms of Ø and |C| = C0 So let’s take an example, of C = a+jb and represent it in terms of magnitude and angle. If we do this, we find that |C| = v(a2+b2) and that a = |C|cos(Ø), b = |C|sin(Ø), so C can be written as |C|[cos(Ø)+jsin(Ø)]. So far, so good … only a little trigonometry and nothing truly unpleasant in the maths.So let’s take an example, of C = a+jb and represent it in terms of magnitude and angle. If we do this, we find that |C| = v(a2+b2) and that a = |C|cos(Ø), b = |C|sin(Ø), so C can be written as |C|[cos(Ø)+jsin(Ø)]. So far, so good … only a little trigonometry and nothing truly unpleasant in the maths.

13. Alan Murray – University of Edinburgh There is a neater, if superficially nastier, notation C = a+jb = C0[cos(Ø)+jsin(Ø)] It turns out, can be written:- cos(Ø)+jsin(Ø) = ejØ So C = C0ejØ (or C0ÐØ) Means “C0 at an angle of Ø°” ejØ does all the right things, when we take the real part to get a real number (voltage or current, in our case) out of it. Our numbers will be real voltages and currents Just as conventional 2D vectors can be written in “polar form”, so can a complex number – and here it is. We can write cos(Ø)+jsin(Ø) = ejØ and everything becomes much tidier and compact … for example, C = C0[cos(Ø)+jsin(Ø)] becomes C = C0ejØ … takes abit of getting used to, but it looks nicer, doesn’t it?Just as conventional 2D vectors can be written in “polar form”, so can a complex number – and here it is. We can write cos(Ø)+jsin(Ø) = ejØ and everything becomes much tidier and compact … for example, C = C0[cos(Ø)+jsin(Ø)] becomes C = C0ejØ … takes abit of getting used to, but it looks nicer, doesn’t it?

14. Alan Murray – University of Edinburgh cos(Ø)+jsin(Ø) = ejØ … why? This derivation/proof is not examinable – but there are many of us who don’t like bits of maths that are as important as this left completely unexplained. As an electrical engineer, you’ll never have to reproduce this!This derivation/proof is not examinable – but there are many of us who don’t like bits of maths that are as important as this left completely unexplained. As an electrical engineer, you’ll never have to reproduce this!

15. Alan Murray – University of Edinburgh cos(Ø)+jsin(Ø) = ejØ … proved. It’s not hard to see that, by sticking “jØ” in the place of “x” everywhere and multipliying out j2 = 1 everywhere, the ejØ series is simply the sum of the one that is equal to cos(Ø) plus that equal to sin(Ø), multiplied by “j”. This shows that ejØ = cos(Ø) + jsin(Ø) …. Known as De moivre’s Theorem, and very useful! Remember, this proof is for interest only – not examinable.It’s not hard to see that, by sticking “jØ” in the place of “x” everywhere and multipliying out j2 = 1 everywhere, the ejØ series is simply the sum of the one that is equal to cos(Ø) plus that equal to sin(Ø), multiplied by “j”. This shows that ejØ = cos(Ø) + jsin(Ø) …. Known as De moivre’s Theorem, and very useful! Remember, this proof is for interest only – not examinable.

16. Alan Murray – University of Edinburgh Some examples for you C1 = 3 + 4j C2 = 5 – 2j C1+C2 = C1*C2= C1*C1 = |C1| =

17. Alan Murray – University of Edinburgh Some examples for you

18. Alan Murray – University of Edinburgh Some examples for you

19. Alan Murray – University of Edinburgh And ×j does something interesting Take a complex number ejØ ejØ = cos(Ø)+jsin(Ø) Multiply by j j×ejØ = jcos(Ø)+j2sin(Ø) = -sin(Ø)+jcos(Ø) Multiply by j again j×j×ejØ = j2cos(Ø)-jsin(Ø) = -cos(Ø)-jsin(Ø) Multiply by j again j×j×j×ejØ = j3cos(Ø)-j2sin(Ø) = sin(Ø) -jcos(Ø) And finally j×j×j×j×ejØ = ejØ =cos(Ø)+jsin(Ø) Now comes the payoff for all that complex nastiness. If we buy into the use of V0ejØ (for example) to represent a voltage of amplitude V0 at a phase angle of Ø at a particular moment in time, then simply multipliying by j has an interesting effect. Each time we do a “multiply-by-j”, we get a phase shift in the voltage (think of it as a phasor diagram again …) of 90°. This is going to prove to be one of the most useful mathematical “tools” in our toolbox, once you have got your head around it. It will allow us to represent the “CIV” in CIVIL by a factor of 1/j and the “VIL” in CIVIL by a factor of xj. Now comes the payoff for all that complex nastiness. If we buy into the use of V0ejØ (for example) to represent a voltage of amplitude V0 at a phase angle of Ø at a particular moment in time, then simply multipliying by j has an interesting effect. Each time we do a “multiply-by-j”, we get a phase shift in the voltage (think of it as a phasor diagram again …) of 90°. This is going to prove to be one of the most useful mathematical “tools” in our toolbox, once you have got your head around it. It will allow us to represent the “CIV” in CIVIL by a factor of 1/j and the “VIL” in CIVIL by a factor of xj.

20. Alan Murray – University of Edinburgh And ×j does something interesting ejØ = cos(Ø)+jsin(Ø) Multiply by j j×ejØ = jcos(Ø)+j2sin(Ø) = -sin(Ø)+jcos(Ø) Multiply by j j×j×ejØ = j2cos(Ø)-jsin(Ø) = -cos(Ø)-jsin(Ø) Multiply by j j×j×j×ejØ = j3cos(Ø)-j2sin(Ø) = sin(Ø)-jcos(Ø) Multiply by j j×j×j×j×ejØ = ejØ =cos(Ø)+jsin(Ø) So let’s look at the effect of this on a real signal, by doing it all over again while studing the real part of ejØ, which will, of course, be a sine-like wave, from what we have just looked at … ejØ = cos(Ø) + jsin(Ø). Lo and behold. Each time we multiply by j, the wave moves 90° to the left – a phase advance. If we do it 4 times, we get back to where we started, because 4x 90° = 360°, which is the same as 0°. So let’s look at the effect of this on a real signal, by doing it all over again while studing the real part of ejØ, which will, of course, be a sine-like wave, from what we have just looked at … ejØ = cos(Ø) + jsin(Ø). Lo and behold. Each time we multiply by j, the wave moves 90° to the left – a phase advance. If we do it 4 times, we get back to where we started, because 4x 90° = 360°, which is the same as 0°.

21. Alan Murray – University of Edinburgh Summary …complex numbers …and some little cunning stunts e jØ = cos(Ø) + jsin(Ø) xj rotates a complex number by 90° i.e. xj advances Re(e jØ) by 90° e jp/2 = cos(p/2) + jx sin(p/2) = 0 + jx 1 So e jp/2 = +j That’s enough stuff about complex numbers, simply as mathematical objects. Before we actually use them in earnest, here is a summary of what we need to know and a couple of cute little “stunts” to learn about that show us that:- e jp = +j And 1/j = -jThat’s enough stuff about complex numbers, simply as mathematical objects. Before we actually use them in earnest, here is a summary of what we need to know and a couple of cute little “stunts” to learn about that show us that:- e jp = +j And 1/j = -j

22. Alan Murray – University of Edinburgh What does this have to do withV = V0cos(?t) etc? Set ? = ?t And here’s how we are going to use it. We will write real voltages and currents in the form ej?t and put phase angles in the form ej?t+90° , ej?t-38° etc and it will all, then come out in the mathematical wash. Just as in a phasor diagram, where we took the projection on to the x-axis to recover a “snapshot” of the voltage or current in question, we will take the real part from the complex number for to achieve the same end.And here’s how we are going to use it. We will write real voltages and currents in the form ej?t and put phase angles in the form ej?t+90° , ej?t-38° etc and it will all, then come out in the mathematical wash. Just as in a phasor diagram, where we took the projection on to the x-axis to recover a “snapshot” of the voltage or current in question, we will take the real part from the complex number for to achieve the same end.