1a.1 Lecture 1a Comple numbers, phasors and vectors Introduction This course will require ou to appl several concepts ou learned in our undergraduate math courses. In some cases, such as comple numbers and phasors, ou have probabl used these concepts regularl. In other cases, such as vector calculus and series solutions of differential equations, this ma be the first time ou have had to appl them. ccordingl our first few lectures will be devoted to a mathematical review. s this is a review, we will not derive most of the results and theorems, but merel state them. If ou wish to review the derivations, please refer to our undergraduate math/phsics tets. Comple numbers The imaginar unit j has the propert 1 j = 1 (1) general comple number can be epressed as =+ j () where and are real numbers. We call them the real and imaginar parts of and we write =Re{} =Im{} We can also epress a comple number in a polar format b writing =ρ cosϕ =ρ sin ϕ where ρ and ϕ, the modulus and argument (or the magnitude and phase), are real numbers related to and b ρ= + ϕ=tan 1 ( / ) Using (4) we can epress a comple number in the form (3) (4) (5) =ρ(cos ϕ+ j sin ϕ) =ρ e jϕ (6) where in the last step we've used Euler's formula Some important values are e j ϕ =cos ϕ+ j sin ϕ (7) 1 Electrical engineers use j for the imaginar unit instead of i as is common in math and phsics tets. The inverse tangent must be a four-quadrant version such as the atan(,) function in Matlab. e ± j 0 = 1 e ± jπ/ =± j e ± j π = 1 From Euler's formula it follows that cosϕ= e j ϕ +e jϕ sin ϕ= e j ϕ e j ϕ j We sometimes use the following notation to signif the magnitude and phase of a comple number We can then write ρ= ϕ= (8) (9) (10) = e jð (11) s a shorthand we can leave out the e j. For eample, we might write =1.5Ð 3 (some calculators use this tpe of notation). This means =1.5e j 3 =1.5[cos(3 )+ j sin (3 )]. The conjugate of a comple number is denoted b. It is obtained b changing the sign of the imaginar part Since = j (1) ρe j ϕ = j (13) we can also obtain the conjugate b changing the sign of the argument. In general the change j j in an epression produces the conjugate. Let = + j and w=u+ j v be two comple numbers. Their sum and difference are ±w=(±u)+ j(±v) (14) The real and imaginar parts of the sum/difference of two comple numbers are the sum/difference of the real and imaginar parts of the comple numbers. The product of two comple numbers is In polar notation we have w=(+ j)(u+ jv) =(u v)+ j( v+ u) (15) w= e j w e j w = w e j( + w) (16) The magnitude of the product is the product of the magnitudes and the phase of the product is the sum of the phases. Note that For division we have = = + (17) EE518: dvanced Electromagnetic Theor Scott Hudson 015-0-0
1a. in rectangular form or w = w w u+ v = u +v + j u v u +v w (18) w = w e j (Ð Ðw) (19) in polar form. The magnitude of the quotient is the quotient of the magnitudes and the phase of the quotient is the difference of the phases. The eponential of a comple number is e =e e j =e (cos + jsin ) (0) When taking the product of eponentials we add the (comple) eponents Fourier Transforms and Phasors e e w =e +w (1) The Fourier transform of a function of time f (t) is F (ω)= f (t)e j ωt dt () We often call F (ω) the spectrum of f (t) ; ω is the radian frequenc. The inverse Fourier transform is f (t)= F (ω)e j ωt d ω π (3) This shows that an function f ( t) can be represented as a superposition of functions of the form e j ωt weighted b the spectrum F (ω). Linear, time-invariant (LTI) sstems have the important propert that if a sinusoid of a given frequenc is applied as the input then the output is a sinusoid at the same frequenc. ll that can change is the magnitude and phase of the sinusoid, not the frequenc. This propert is ver useful for analing phsical sstems. Since we will find ourselves using sinusoidal signals almost eclusivel, a bookkeeping technique call phasor notation will prove ver convenient. general sinusoidal signal can be written v (t)=a cos(ω t+ϕ) (4) where a is the magnitude, ϕ the phase and ω the radian frequenc. We can write acos(ω t+ϕ)=re{a e j(ωt+ϕ) } =Re{[ ae jϕ ] e jω t } We define the phasor of v (t) to be the comple number Then (5) V =a e j ϕ (6) v (t)=re {V e jω t } (7) In phasor analsis we use the comple constant V as a representation of the time-domain signal v (t). We develop equations and solutions in the phasor domain. n time we want the time-domain signal we multipl b e j ωt and take the real part. For a non-sinusoidal signal v (t), we can consider a superposition of phasors at different frequencies: v(t)= Re {V (ω)e j ωt } d ω π =Re{ V (ω)e j ωt d ω π } (8) where V (ω), the phasor amplitude as a function of frequenc, is the spectrum of v (t). Vectors We refer to a single number, either real or comple, as a scalar. Scalars can represent phsical properties that are specified b an amplitude (and possibl phase), such as pressure or voltage. For properties that have both amplitude/phase and direction, such as force, we emplo vectors. We will use bold-face letters to represent vectors, for eample. When writing b hand one tpicall represents a vector using a bar or arrow above a letter, as in or. nother notation is to underline the letter, as in, which represents boldface. In rectangular coordinates we represent a vector b its three components in the, and directions =(,, ) (9) The magnitude of a vector is a scalar and in rectangular coordinates is given b the Pthagorean formula = + + (30) We will often represent the magnitude of a vector b the corresponding italic letter, as in. Location in space is usuall represented b the position vector r=(,, ) (31) In this case r represents distance from the origin to the point r. dding or subtracting vectors is accomplished b adding or subtracting their components ±B=( ±B, ±B, ±B ) (3) This is illustrated below. It can be useful to think of a vector difference as B=+( B), that is, we flip the direction of the second vector and add. EE518: dvanced Electromagnetic Theor Scott Hudson 015-0-0
1a.3 If B=0 for two non-ero vectors, we sa the are orthogonal. Since 0, B 0, we must have cosθ B =0, or θ B =π/=90 o. B Cross product The cross product of two vectors produces another vector. +B B = a ( B B )+ a ( B B )+ â ( B B ) (39) Notice the permutations of from top to bottom. You can use these to memorie the above formula. The cross product B is orthogonal to both and B, and the magnitude of the cross product is This is illustrated below. B = B sin θ B (40) -B Figure 1: Vector addition and subtraction. Unit vectors If we divide a vector b its magnitude we obtain a vector that has unit magnitude. We will use the following notation for unit vectors: â = (33) The three unit vectors corresponding to the rectangular coordinate aes are ^a =(1,0,0) ; ^a =(0,1,0) ; ^a =(0,0,1) (34) These provide an alternate wa to represent a vector = a a a (35) Later we will consider vectors in clindrical, spherical, and other coordinate sstems. In these cases the unit coordinate vectors might be functions of position. Multipling a vector b a scalar results in the components of the vector being multiplied b the scalar Dot product k =(k,k, k ) (36) There are two useful was to combine vectors through multiplication. The dot product produces a scalar In terms of magnitudes we have B = B + B + B (37) B = B cosθ B (38) where θ B is the angle between the vectors. The direction of the cross product is given b the right hand rule. Point the fingers of our right hand in the direction of. Sweep our fingers to the director of B. Your right thumb then points in the direction of B. The cross product is ero if θ B =0,π, that is, if the two vectors are parallel. Normal and tangential components We are often interested in the orientation of a vector relative to some surface. ssume the surface normal is â n. This is a unit vector which is perpendicular to the surface. We can break a vector into normal and tangential components where B θ B Figure : Cross product of two vectors B = n + t (41) EE518: dvanced Electromagnetic Theor Scott Hudson 015-0-0
1a.4 This is illustrated below. n =â n ( â n ) t = n =â n â n (4) Note that the substitution an a n leaves n, t unchanged. Therefore, it doesn't matter in which direction we define the surface normal. Scalar fields scalar field is a mapping that assigns a scalar to ever spatial position in some spatial domain. We denote this with an epression of the form f ( r). In rectangular coordinates this would be epressed as f (,, ). scalar field can be comple in which case we represent it as the sum of its real and imaginar parts f (r)= f r (r)+ j f i (r) (43) scalar field, either real or comple, can also be a function of time. We then write f (r;t). This means that at an given point r 0 the field is a function of time given b g(t )= f (r 0 ;t). If at ever point in space the field is a real sinusoidal function of time with radian frequenc ω then we can appl the phasor concept. The result is a scalar phasor field f (r) where Vector fields â n n t surface Figure 3: Components of a vector normal and tangent to a surface. f (r;t)=re{ f (r)e jω t } (44) vector field is a mapping that assigns a vector to ever spatial position in some spatial domain, and we write (r). Breaking this into rectangular components we have (r)=â (,, )+â (,, )+â (,, ) (45) vector field defines three scalar fields, one for each vector component. The scalar fields, and therefore the vector field, ma be comple. We ma emplo a coordinate sstem other than rectangular coordinates. We will consider general orthogonal coordinate sstems in a subsequent lecture. We can combine the vector field concept with the phasor concept to arrive at a vector phasor field. In this case n eample is where we assume 0 is real. (r ;t)=re {(r)e j ωt } (46) (r ;t)=re {â 0 e j ϕ e jβ e jω t } =â 0 cos(ωt β+ϕ) (47) We will almost alwas represent the electric and magnetic fields as vector phasor fields. Indeed, most of this course will involve manipulations of, and relations between, vector phasor fields. Time-average dot product Consider two time-varing, real vectors representing some phsical fields and ( cos(ω t+ϕ ), cos(ωt+ϕ ), cos(ω t+ϕ )) (48) (B cos(ω t+θ ), B cos(ω t+θ ), B cos(ω t+θ )) (49) The dot product is B cos(ω t+ϕ )cos(ω t+θ )+ B cos(ω t+ϕ )cos(ωt+θ )+ B cos(ωt+ϕ )cos(ωt+θ ) (50) In most EM problems it is the time-average of such quantities that are usuall of interest. Using a basic trig identit B cos(ω t+ϕ )cos(ω t+θ ) = 1 B [cos(ϕ θ )+cos( ωt+ϕ +θ )] (51) Time-averaging will eliminate the last term. So the timeaverage dot product is B cos(ϕ θ )+ B The corresponding phasors are cos(ϕ θ )+ B cos(ϕ θ ) (5) =( e jϕ, e j ϕ, e j ϕ ) (53) B=(B e jθ, B e jθ, B e j θ ) (54) B inspection we can verif that the time-average dot product we derived above is equal to 1 Re ( B )= 1 Re ( B ) (55) Therefore, when epressions like 1/ Re ( B ), where and B are vector phasors, comes up in our analsis, we can identif this as the time-average of the dot product of the corresponding real, phsical fields that the phasors represent. EE518: dvanced Electromagnetic Theor Scott Hudson 015-0-0
1a.5 Line, surface and volume integrals Two important integration operations that can be performed on a vector field are line integrals and surface integrals. Suppose we have a vector field and we define a path L between two points P 1, P, as shown below. dl P â n ds S P 1 Figure 4: Line integral Figure 5: Surface integral. The line integral of over path L is a scalar denoted b L dl. The integrand is dl= d+ d+ d (56) The displacement dl is tangent to the curve. If the path is a closed path ( P 1 =P ) then we use the notation L dl. Now suppose we define some surface S, as illustrated below. The surface integral of over S is a scalar denoted b S ds. The integrand is ds= ^a n ds= n ds (57) The vector ds is normal to the surface and has area ds. The surface integral sums the normal component of over the surface. If the surface is a closed surface, we use the notation S ds. Finall, a volume integral of a scalar field is denoted b V f dv. References 1. Churchill, Brown and Verhe, Comple Variables and pplications, 3 rd Ed., McGraw Hill, 1976, ISBN 0-07- 010855-.. Carrier, Krook and Pearson, Functions of a comple variable, Hod Books, 1983, ISBN 07-010089-6. 3. Kresig, dvanced Engineering Mathematics, 4 th Ed.,Wile, 1979, ISBN 0-471-00140-7. Useful vector identities The following identities can be readil verified b carring out the indicated operations in rectangular coordinates. B = B (58) (B C)=( C) B ( B)C (59) ( B C)=B (C )=C ( B) (60) ^a n ( B)= ^a n ( t B t ) (61) This last identit sas that the normal component of a cross product is equal to the cross product of the tangential components. EE518: dvanced Electromagnetic Theor Scott Hudson 015-0-0