Review of Phasor Notation

Home , Phase (waves), Phasor, Root mean square

Chapter 1

ECE 130a

Introduction to Electromagnetics

ejjq =+cosq sinq Complex Vectors and Time Harmonic Representation (Chapter 1.5, 1.6 Text)

Complex Numbers Imaginary i) Rectangular/Cartesian Representation cajb=+ b ii) Polar Representation c φ jf 22 Real cce= , where cab=+ a = −1 b f tan dia cajbcec=+ =jf =cosff + jc sin

real part imaginary part magnitude phase iii) Addition/Subtraction Cartesian representation most convenient cajb=+ dejh=+ Then: cd+=a ae ++f jbha +f −= −+ − cda aef jbha f iv) Multiplication/Division 2 Note that afj =−1 22 2 cc* =+= a b c , =− where cajb* , the complex conjugate. v) Cartesian Representation cd=+afafafaf a jb e + jh = ae − bh + j ah + be + + − ++ − c = ajb= ajbejh= afafae bh j be ah + + − 22+ d ejh ejhejh eh vi) Polar Representation (more convenient for multiplication and division)

c j ff− jf jf j diff+ c di12 cce= 1 dde= 2 cd= c d e 12 = e d d vii) Complex Representation of Time-harmonic Scalars Consider:

vtaf=+ Acosbgwfo t

amplitude phase angular/radian 1 frequency magnitude aaa$$$= * 2 ωπ= $ and 2 f amplitude Re a1

radian frequency frequency Sinusoidally Time Varying Fields (1.6 Text) F p I 1) fA=+=+cosafwf t110 cos 120 p t H 4 K p FNote: radians =∞45 I H 4 K is a scalar field that depends on time only, but not on position (location). This is not a wave. For example, this is the voltage in any electric household plug. Note that the phase f simply alters the starting point.

p t Fin units of I H wK voltage

The circular frequency ωπ= 2 f , where f is the frequency in Hz. Here, f = 60 hertz (Hz) and ωπ==120 377 rad / sec 2) Waves -- travelling waves (main subject of this course) z vV=-cosw F t I o H vK

=-Vtkzo cosaw f or Vtzo cosawb- f F z I =-Vftcos2p →+λ o H l K Repeats zz wp22f p k aor bf ∫= = propagation constant vv l v = λ f λ = wavelength 21π T ≡≡≡period v = phase velocity ω f

π 2 = λ β

F p I z G in units of J H b K with t = 0

This is a scalar traveling wave. 3) Vector Travelling Waves -- electromagnetic waves Examples: r F yI (i)EeE=-$ cosww t =-eE$ cosa t kyf xoH vK xo (x-oriented field, + y propagation) r F 1 $$3 I (ii)Ee=+G xyo eEtkzJ cosafw - H 2 2 K $ y eE $$1 3 $ 3 eeEx=+ e y 1 2 2 2 x 1 r 2 $ ∞ (iii)EeE=-+xocoscw tkz45 h (x-oriented, + z propagation) r p p (iv)HeH=-$ sinF xtzeHI cosafwb-+$ cosF xtzI sin afwb- xoH a K z1 H a K (waveguide example) Example: (x-directed planewave, - y propagation) r Eytaf, =++ e$ 100coscwb t y 45 ∞ h ω x = β L y ∞ O v =+et$ 100cos w F I + 45 x NM H vK QP This is a vector travelling wave field.

t = 0

p t = 4w

F p I y G in units of J H b K Propagates in the − y direction

At ty=+045:coscb ∞ h p At ty=+=-:coschbb90 ∞ sin y 4w Note that this vector field: $ (i) always points in the ex direction (reverses, of course). (ii) propagates in the y direction. (iii)spatially depends only on the y coordinate (which is the direction of propagation). z z uniform y in x-y y plane x x The magnitude E, that is, the density of these lines (or the length of the arrows), varies sinusoidally, both in time and in the variable y direction of propagation.

Ex 100 π 3π 3 ω ω At y =− λ t 8 2π ω r 3 23p p At y =− λ : Ee=-+$ cos100 Fw t l I 8 x H l 84K p =-etet$ 100cosFw I = $ 100sinw xxH 2 K

r p At y = 0: Ee=+$ cos100 Fw t I x H 4 K

100 π 3π ω ω t 2π ω Method of Phasors (“Set your phasors on ‘Stun,’ Mr. Spock!”) Background material needed: Review of complex algebra Eulers Law: exjxjx =+cos sin Reexexjx==cos , Im jx sin Suppose we have a field that varies sinusoidally in time and/or space. For a scalar field, voltage, for example, we have: = ω vVo cos t (a sinusoidal voltage)

vV=-o cosaw tkzf (a voltage wave on a transmission line)

These are real measurable quantities. We represent them by writing as follows:

jtw vVe= Remro Now we do either of (a) or (b):

= jtω (a) Drop “Re ”. The remaining v Veo is called the complex voltage (including time dependence) or voltage phasor (including time dependence).

jtw jtω (b) vVe= Remro Drop “Re ” and “e ” = The resultant v Vo is called the (complex) phasor. Note that, for this example, the complex phasor happens to be real. (PHASORS are written in bold type- face.)

If v =+Vto cosawff jtafwf+ jtwf j ==ReVeo Re Veo e jjtfw = Re Veo e

jf Veo is a complex quantity in polar representation. Im ejjf =+cosff sin jf 22 Vo e =+=cosff sin 1 φ Re Graphical Picture of Complex Fields Consider a vector field with harmonic time dependence: r p L j O $ ∞ $ jtw 6 EeE=+∫xocoscw t30 h Re M eEee xo P N Q p j $ ∞ 6 This is another notation for eExo– 30 e . r $ ∞ jtw Complex form: EeE=–xo30 e

phasor amplitude (also a vector)

If we picture the entire time dependent complex field in the complex plane, we see that it rotates at angular velocity ω . π Im ω t = = 3ω t 0 Eo ° 30 Re π = t ω

Note that the direction of the complex field on this complex plane has no relation- $ ship to its direction in real physical space, which here is constant ex . The entire complex field, including e jtω time dependence, is often called “Rotating Phasor.” The real field is the projection on the Re axis. Amplitude, Time Average, Mean Square and Root Mean Square (a) Amplitude is the maximum value of the time oscillation (real number). r $ (i) Ee=-+x100cosafwbf t z real r $ jf E = eex100 phasor

Amplitude is 100 rr 1 2 Amplitude ≡ magnitude of phasor ≡ E = EE* −+ββ (ii) v =−100eejz 100 jz 2 waves in opposite direction, phasor form L eejx- - jx O MUse sin x = P N 2 j Q 200 j =-ee+-jzbb - jz =-200 jzsin b ⇐ This is the phasor 2 j (simplified). Amplitude of the total wave is v =-200jzsinbbbb =afaf - 200 jz sin 200 jz sin = 200 sin z function of z! Now, real form vzta ,sinsinf = 200 bw z t because: vzta ,sinf =-Re 200 jb zejtw =−Re 200jzsinβω cos t + 200 sinβ zt sinω = βω 200sinzt sin arrows show time oscillations z

(b) Time average of a harmonic quantity is always zero. jf ← vta f =+ Vo cosawf t f v = Veo phasor 1 T = 1 vtaf=+∫Vto cos awf f 0 T T z0 f

T 221 2 vtaf=+Vtdto cos awf f T z0 T 112 L 1 O =++Vtdto cosaf22wf T z0 NM2 2 QP V 2 = o 2 Using phasors for v 2 cannot give correct answers since phasors are not valid for a nonlinear situation. Try it and see: 222= ω 2222jtwf+ j vVo cos t , but v = Veo Does not give cos 2 when real part is taken for the time average. But the following “trick” works: For the time average: (Note: It gives only the time average.) 1 1 V 2 v2() t==Revv * Re Vejff Ve- jo = 2 22o o (This works for vector phasors, too!) r $$ If: E =+eExx eE yy r rr 2 1 Etza ,*f = Re diEE◊ Vector “dot” product r 2 EEee= - jkz $ A A o x phasor The Mean Square is related to Average Power (averaged over whole cycles). The same “trick” works for any other second order quantity. 1 Note: The""Re is necessary here, vtitafaf = Re vi* 2 since v andi may have a phase difference. (d) Root Mean Square (RMS)

2 vt2a f or Eta f

1 For harmonic fields, RMS value = amplitude. 2 Example: r If: Ee=-$ 100cosafwb t z r r cylindrical coordinates arz,,f f $ He=--∞f 03.cosafwb t z 60 rr Calculate EH¥ , using both real fields and phasors. Solution: (a) Using real fields rr $ EH¥= ez 30cosafawb t - z cos wb t - z -∞ 60 f 1 Use product of cosAB◊ cos=-++mr cosafaf ABAB cos rr 2 EH¥= e$ cos ∞+ cos1560af 2wb t - 2 z -∞ 60 rrz $ EH¥= ez 75. + 0

(b) Using phasors r $ E = er100 Phasors correspondingrr to p rr rr r - j 1 $ 3 the real Etza ,,f Htza ,f EH¥=RediEH ¥* H = eef 03. 2 p + j 1 $ 3 EH¥=* eez 15◊ 2 1 Re EH¥=* ee$ 15cos 60 ∞=$ 7. 5 2 zz

Phasor examples with waves (1) Space dependence is due to propagation only (a) Uniform plane EM wave -- vector (b) Transmission line wave -- scalar voltage or current (1a) Propagation is only in one direction r $ R F z I U $ Real field: Eztaf, =- eExocosSwf t + V =-+eE xocos a wbf t z f T H vK W ω ω Propagation is in +z$ direction β = v = v β r $ jtafwbf-+ z Complex field: Ezta , f = eEexo $ jf jtafwb- z = eEexo e

Phasor field: (two options possible)

jtafwb- z (i) Drop entire e dependence r Phasor:E = eEe$ jf (algebraic)∫–eE$ f (schematic) r xo xo $$∞∞j 30 For example, E =–∫eeexx100 30 100 r p $ jtdiwb-+ z 6 Etzaf, = Re ex 100 e $ =-+∞etzx 100cosafwb 30 Another example: jj=+cosff sin , when f =∞90 r a f $$ E ==–∞ejyy50 e 50 90 r p $$jtafwb- z jtchwb-+ z 2 Etzaf, ==Re ey 50 je ey Re 50 e $ $ =-50etzetzyysinafwb - = 50cos a wb - + 90 ∞ f r Another example: E =-c eje$$h100 r xy $$jtafwb- z Etzaf, =-Re 100ch exy jee $ $ =-+-100 etzetzxycosawbf sin a wbf rr 2 1 1 2 $$$$ e ==-+Re 2 EE◊◊* 2af100 chch exyxy je e je 1 22 =+=2afafaf100 1 1 100

(ii) Drop only e jtω dependence r Eztaf, =-+ eE$ cos awbf t z f r xo jtafwbf-+ z Eztaf, = e$ Re Ee r xo $ -+jzjbf E = eEexo We can see that − βz and f both are phases. The phase changes with propagation. The phase rotates in the complex plane. For waves travelling in one direction only, we don’t usually use this. Instead, jtafwb- z we drop off of e -- usually (but not always). Example: (1b) Transmission line wave Waves travelling in two opposite directions

vtza ,f =-+++ Voo cosawb t zf G V cosa wbf t z f -+jzbw jt jjzjt f bw vtz(), =+Re Veo eG Veo e e We only have one option: Drop e jtω dependence, since βz dependence is not the same on the two parts.

-+jzbfb j jz Phasor: v =+Veo G e e ~ =+− jzββΓ~ 2 jz Γ jf Veo 1 e =Ge = complex reflection coefficient Im ~ part 1 + Γe 2 jzb ~ Γ jf Γ = G e called “Crank φ 1 Re part Diagram”

diagram of [ ] quantity in complex plane (2) Space dependence may exist perpendicular to the direction of propagation; for example, laser beams, guided electromagnetic waves, optical waves in fibers. Not the subject of ECE 130A, but used in ECE 130B. Example of (2) with (i) propagation in one direction: E.M. wave in rectangular waveguide

p jtafwb- z Eafxyzt,,, = eE$ sinF xeI xa< yo H a K jtafwb- z Drop e : p E = eE$ sinF xI xa< yo H a K Another example: pp H =-eH$ sinF xjeHI - $ cosF xI xa< xoH a K z1 H a K Phasor: Significance of j : x$ and z$ components are 90° out of phase Real field is: r p p HeH=-$ sinF xtzeHI cosawb-+f $ cosF xtzI sinawb- f xoH a K z1 H a K Similar fields exist in any guided wave, such as waves in optical fibers! (3) Modulated (time dependent amplitude) waves All the examples brought in (1) and (2) were waves of a single sinusoidal frequency and amplitude constant in time. In communications, we impress information onto a single frequency wave by modulating its amplitude in time. The original sinusoid is the carrier and the modulation of the amplitude contains the information. Example: Pulse code modulation unmodulated carrier r z t EeE=-$ cosw F t I xo H vK

z = 0 a single pulse impressed onto a carrier r --bgt z 2 / 2s 2 z t EeEe=-$ v o cosFw t I xo H o vK

r 22 $ - t / 2 s o At z = 0: EeEe= xo cosw o t

r 22 $ - tjt/ 2 swoo Phasor form: EeEe= xo e

amplitude envelope carrier This envelope function is a Gaussian.

t 2 2 − t 22/σ ==⋅ o EEEEe* o The modulated wave is no longer a single frequency. By Fourier analysis, it is easy ω to show that a band of frequencies about o are present. Frequency Domain Representation unmodulated wave a single pulse modulated wave E()w E()w

w w w o w o 1 +• +• Etzaf, ==0 E afww ejtw d EEtedtafw = af- jtw z-• 2p z-• 2 s 2 --o bgww 2 o For the single pulse modulated field: EEeafwps= 2 oo 2 2 2 22--swwoobg EEeafwps= 2 oo Such a pulse is in fact a very good approximation to an actual one travelling in an optical fiber part of an optical communication system. Schematic of such a system:

Many km of optical fiber square law detector Electronic Diode Photo- Amplifier Modulator Laser Detector

Encoder Processor

Responds to envelope of wave Information only since light frequency is too fast \ is demodulator