Electromagnetics Primer

ELECTROMAGNETICS PRIMER

Cylindrical Coordinates
, , 

Solenoidal field:

Can also be written as  ⋅ d  0 ⇔ $ ⋅  0 ⇔ ∀ '
, ,  

$%(

.  .  ̂ - - - 1 $ ≝ ̂    ̂  - - - + 1 1 * * ̂      -  sin
 - )-

Gradient, Divergence, and Curl , * *

 d ̂  d   d ̂ dℓ d  dd ̂  dd   dd ̂ d  ddd

Spherical Coordinates
, , 

 d ̂  d    sin
 d  dℓ d    sin
 dd̂   sin
 dd  dd d    sin
 ddd

Line Integral



Line integral along  ≝ 
, ,  ⋅ dℓ

Conservative fields:



Can also be  0 ⇔ $ %  0 ⇔ written as  ⋅ dℓ ∀  $&
, ,  

Surface Integral

Surface integral through  ≝ 
, ,  ⋅ d

'

Cartesian Cylindrical Spherical

Gradient: Operates on a scalar field and outputs a vector field. It is always normal to a constant value surface (e.g., 0
, ,   3) and always points in the direction of maximum change in the scalar function. grad
0 ≝ $0 -0 -0 -0 , .  .  ̂ Cartesian - - * - *-0 1 -0 -0 * ̂    ̂ Cylindrical -  - -  -0 1 -0 + ̂   * -  - Spherical * 1 -0 *   )  sin
 -

Divergence: Operates on a vector field and outputs a scalar field. A positive divergence implies a source of flow, and a negative divergence implies a sink. ∮ ⋅ d

Div7 8 ≝ lim ? ' B  $ ⋅  ;<→> Δ - C - D - E ,   Cartesian - - - * 1- G - E *1 * -
 F    -  - Cylindrical * 1 -  
 F    - + 1 *  Spherical *  sin
 -
sin
 H  * - G 1 *  )  sin
 -  7$ ⋅ 8d   ⋅ d

Divergence Theorem: I

'

where  is the surface that encloses the volume .

Curl: Operates on a vector field and outputs a vector field.

∮  ⋅ dℓ curl
  ≝ lim ?  BN  $ %  ;L→> ΔM . Q - C

. , * - * D * * ̂  * 1 QQ   - - + F G * * ̂  * 1 QQ *  * sin
 - - F H )

OPC

̂ Q - E

Cartesian

̂ QQ - E

'

 V< ⇔ U ⋅ d  Wenc 1. $ ⋅ U

Postulates: Cylindrical

 sin
  QQ -  sin
 G

Spherical



where  is the line enclosing the surface . Laplacian

Operates on either a scalar or vector and outputs the same. For scalars: - - -  $ ≝ $ ⋅ $      - - -    - 0 - 0 - 0 ,   Cartesian -  -  -  * *1 -0 1 - 0 - 0 R S    Cylindrical * - -  -   - * 1 -  -0 $ 0  R S   - - + 1 -0 * Spherical *    sin
 - Rsin
 -S * 1 - 0 *    )  sin
 -  For vectors: $  $7$ ⋅ 8  $ % 7$ % 8  $ C .  $ D .  $ E ̂ for Cartesian $ %
$&  0 (the curl of a gradient is 0) $ ⋅ 7$ % 8  0 (the divergence of a curl is 0)

Important Vector Identities

A vector field is uniquely defined (within an additive constant) by specifying its curl and divergence. A corollary to this is that any vector field can be decomposed into the gradient of a scalar field and the curl of a vector field:  −$&  $ %

( Electrostatics



7$ % 8 ⋅ d   ⋅ dℓ

Stokes’s Theorem:

Helmholtz Theorem

'

 0 2. $ % Y  0 ⇔ Y ⋅ dℓ 

Postulate 1 is Gauss’s Law. It can be used for two types of cases: 1.Calculation of the electric field intensity or electric flux density from known, symmetric charge distributions. 2.Calculation of the equivalent charge in a volume, provided the electric field is known everywhere in space, particularly where Gauss’s law is applied. From postulate 2, we can define the electrostatic potential : P d



Y  −$  − Z ⇒ P\  −  Y ⋅ dℓ d 1-D \

Force on charge ] from electric field: ^  ]Y .

Coulomb’s Law: 1 ]b ] 1 ] ^  ̂ ⇒ Y  ̂  4`a  4`a   1 Vc dℓ 1 VL d ⇒ dY  ̂  ̂ 4`a   4`a   Conductors in electrostatics: 1.Charges in the conductor are allowed to move freely within the conductor. 2.All charge on a conductor resides on its surface. 3.Charges distribute themselves on the conductor to ensure there is zero electric field in the conductor, regardless of its shape. 4.The electric field on the surface of the conductor must be perpendicular to the surface (no tangential component), and hence the electrostatic potential on at the surface is constant. Dielectrics in electrostatics: Electrons are bound to their molecules, but the molecules can be polarized as a dipole due to an externally applied electric field. Electric displacement field/flux density:  a> Y  d

U

If the material is linear and isotropic, then: d  a> ee Y

where ee is the electric susceptibility and is zero in is: free space, and so U  a>
1  ee Y  a> ar Y  aY

U Electric Field Boundary Conditions: f.b % 7Y b − Y  8  0 ⇒ YbT  YT UTb UT ⇒  ab a



7Ub − U 8 ⋅ f.b  VL ⇒ UbN − UN  VL ⇒ ab YbN − a Y2N  VL For a conductor, set the electric fields to 0.

W  To calculate, place imaginary charge W on one surface, calculate electrostatic potential, and plug in. Energy stored by capacitor: 1 kl  j  2 Energy density of electric field: 1 1 U 1 1 ⋅ Y  aY   km  U  U ⋅ $  V<  2a 2 2 2 2 The force on a charge can also be written as: ^  −$km Capacitance:

j

Poisson’s/Laplace’s Equation: V< $   − ⇒ $   0 in charge-free regions a Electric Currents n  o ⋅ d

'

o  pY ⇒   nq, where q 

Ohm’s Law:

Vℓ



ℓ p

s ro ⋅ d s = n d   Y ⋅ o d = rY ⋅ dℓ

Joule’s Law: I



'

Continuity Equation: The divergence of the current density is equal to the negative rate of change of volume charge density anywhere in space. dV< ∇ ⋅ o = − dt Magnetostatics Postulates:

= 0 ⇔ ( ⋅ d = 0 1. ∇ ⋅ ( '

= nenc = o ⇔ u ⋅ dℓ 2. ∇ × u 

From postulate 1, which states that there are no magnetic monopoles (unlike for electrostatics, which can have point charges), we can define the magnetic vector potential : = ∇ ×

( Postulate 2 is Ampere’s Law. It can be used to calculate the magnetic field in symmetric configurations such that the magnetic field can be pulled out of the integral as constant including: 1.Current in an infinite wire. 2.Current in an infinitely long solid or hollow cylindrical conductor. 3.Infinite sheet of current. 4.Multiple conductors in a symmetrical configuration. 5.Nonsymmetric current distributions which are a superposition of symmetric current distributions. v ndw × ̂ 4`   Magnetic fields in materials: = v> 7u + x 8 ( and In an isotropic material, relationship between x is usually: u = eO u

x where eO is the magnetic susceptibility and is positive in paramagnets (attracted to magnetic field) and negative in diamagnets (repulsed from magnetic field). Usually, |eO | < 10{| , so most materials like for this is then: this are hardly magnetic. ( = v>
1 + eO u = v> vF u

( In ferromagnets such as iron, the relationship looks something like the graph to the right, and the relationship between ( and u is nonlinear, although usually ferromagnets are approximated as linear as above since the relationship is such as long as the magnetization does not saturate. Biot-Savart Law:

= d(

The magnetization in a material can be thought of as an equivalent magnetization volume current density o O< when the magnetization in a material is nonuniform and an equivalent magnetization surface current density o OL regardless of the magnetization’s uniformity:

and o OL = x × f. o O< = ∇ × x The magnetic field can be calculated from these . currents rather than x

Magnetic Field Boundary Conditions b − (  8 ⋅ f.b = 0 ⇒ (b} = (} 7( ⇒ v u} = vb ub} b − u  8 = o L ⇒ ub~ − u~ = oL f.b × 7u (b~ (~ ⇒ − = oL vb v

⋅ ‚

Φ  ' ( = n n To calculate, place imaginary n through inductor, calculate flux through the inductor, and plug in. Energy stored in inductor: 1 k = n  2 Energy density of magnetic field: 1 1 1 ( ⋅ u = (u = vu  = kƒ = ( 2 2 2 2v Inductance:

=

Force on a charged particle in magnetic field:

^ = ] × ( A charged particle with a component of its velocity perpendicular to a uniform magnetic field will gyrate with a radius and a frequency of, respectively: …† ]( „ = and ‡„ = ]( … The guiding center will drift with a velocity:

Y × ( „ = ∥ + ( Time-Varying Electromagnetic Fields = V< ⇔ U ⋅ d = W‰Š„ 1. ∇ ⋅ U

Postulates:

'



-( d = − r( ⋅ d s 2. ∇ × Y = − ⇔ Y ⋅ dℓ -t dt '  = 0 ⇔ ( ⋅ d = 0 3. ∇ ⋅ ( '





-U -U =  ro + = o + ⋅ dℓ 4. ∇ × u ⇔ u s ⋅ d

-t -t  ' = aY

5. U = vu

6. ( 8 7. ^ = ]7Y +  × ( Postulates 1 through 4 are Maxwell’s Equations. Postulates 5 and 6 are the constitutive relations that

relate the fields to materials. Postulate 7 is the Lorentz force equation. Faradays Law:

emf = −

dΦ d ⋅ d s = − r( dt dt '

= 7 × (

8 ⋅ dℓ = Y ⋅ dℓ 



Two ways to solve problems: 1.Find the change in magnetic flux in a closed circuit either because the magnetic field or area is varying. 2.If the emf is caused by motion, plug in to the equation on the bottom. The force opposing the motion is:

^ = ndw × ( 

Lenz’s Law: Direction of current generated by Faraday’s law is such that the magnetic field it creates for increasing ( or opposes the change (opposite ( for decreasing ( or ). and in the same direction as (

The boundary conditions do not change from the static case when time dependence is taken into account. Electromagnetic Uniform Plane Waves

Time harmonic wave equation: ∇ Y = Œ‡v7pY + Œ‡aY 8 = Œ‡v7pu + Œ‡au 8 ∇ u

If p = 0, the solution is given by (for +̂ propagation): Y
 = Y>  {ŽE . Y
 = >  {ŽE . u  where ‘ = ‡√va is the wavenumber and v = 1⁄√va is the intrinsic impedance. The wave travels with a phase velocity and has a wavelength of, respectively: ‡ 1 2` ” = = and • = ‘ √va ‘

If p ≠ 0, the electric permittivity is represented as a complex number: p a„ = a − Œ = a — − Œa —— = a
1 − Œ tan
™„  ‡ where p tan
™„  = ‡a is the loss tangent. The wavenumber is complex and is given by: p ‘„ = ‡šva ›1 − Œ œ ‡a In this case, a complex propagation constant  is

defined as:  = Œ‘„ = žŒ‡v
p + Œ‡a = Ÿ + Œ  where Ÿ and   are called the attenuation constant and phase constant, respectively, such that the solution is now: Y
 = Y>  {¡E . Y
 = >  {¡E . u „ where „ is now a complex intrinsic impedance and is given by: v „ = ¢ p a−Œ ‡ The phase velocity and wavelength are now given by: ‡ 2` ” = and • =     The velocity at which the wave’s energy propagates is given by the group velocity, defined as: 1 1 £ = = ‚‘⁄‚‡ ‚  ⁄‚‡ Materials are often classed as one of the following cases below to make simplifying assumptions:
Lossless case (p = 0 ⇒ tan
™„  = 0 ⇒ a only real): v Ÿ = 0,   = ‘ = ‡žva, „ =  = š a
Low-loss case (p ≪ ‡a ⇒ tan
™„  ≪ 1):  | Use Taylor series of √1 +  = 1 + − + ⋯ 2 8 p v 1 p  ⇒ Ÿ = š ,   = ‡žva R1 + › œ S 2 a 8 ‡a v Œp „ = š R1 + S a 2‡a Note: set p = 0 in   and „ for very low-loss
Intermediate case (p ≈ ‡a ⇒ tan
™„  ≈ 1): va p  Ÿ = ‡š ¨¢1 + › œ − 1 2 ‡a   = ‡š „ = ¢

va p ¨¢1 + › œ + 1 2 ‡a v

p a−Œ‡




Good conductor (p ≫ ‡a ⇒ tan
™„  ≫ 1): ‡vp 1 + Œ ‡v š Ÿ= =š = ž`0vp, „ = 2 p √2 1 Penetration/skin depth: ‚ = ™ = Ÿ

1 ‡v =š p™ 2p Converting between Np/m and dB/m for Ÿ: 20 ŸdB = − Ÿ = −8.686ŸNp ln
10 Np Polarization: the figure traced by the tip of the electric field vector as a function of time at a fixed point in space. Three types: 1.Linear: Electric field has only one component or all of its components are in phase. 2.Circular: Electric field has at least two components, and the two components have equal magnitudes out of phase by 90°. 3.Elliptical: Any wave that is not linearly or circularly polarized, either because the components are not 90° out of phase or do not have the same magnitude. Circular and Elliptical can be either right-handed or left-handed. To determine, the polarization is handed if the thumb of the  hand traces out the electric field as t increases. See the diagram above for a wave propagating in the +̂ direction. Note: Linear and circular polarization are special cases of elliptical polarization, and waves of a given polarization can often be written as the superposition of two waves with a different polarization. Surface Resistance: qL =

Poynting’s Theorem

8 ⋅ d = 7Y × u ¬­­­®­­­¯ ' Power leaving '

vu  aY  − r + s d -t I 2 2 ¬­­­­­­­®­­­­­­­¯ Time rate of decrease in stored energy within '

−  Y ⋅ o d ¬­­®­­¯ I

Power from sources (° ±° ²³´ ) or dissipated as loss (° ±µm )

The Poynting vector is defined as:

 = Y × u and represents the power density of an electromagnetic wave. It points in the direction of power flow (also wave propagation). In the time harmonic case, it is a time-averaged quantity:

1 ~
tœ dt  ›Y
t × u ¶ > 1 ∗
‡œ = ℜ ›Y
‡ × u 2 1 Y 1 1 = a¹Y  = = v¹u  = u  2 2 2 2

 P<£ = P<

Reflection and Transmission of Plane Waves

Normal Incidence: Yr c − bc Yt 2c Γ= = and » = = Yi c + bc Yi c + bc Note: » = 1 + Γ. The waves form a combination of a standing wave and a propagating wave. For a PEC,  = 0, so Γ = −1 and » = 0, and the wave is just a standing wave. Oblique Incidence: Reflection and transmission coefficients change to more complicated expressions. Snell’s Law of Refraction and Reflection: sin
¼  pb vr ar f = =¢ = and i = r sin
½  p vbr abr fb

Two special cases: 1.Brewster’s Angle: Incident angle for which there is no reflection (i.e., Γ = 0 so » = 1). Depends on whether incident wave has “parallel polarization” (Y

is in the plane of incidence) or “perpendicular polarization” (Y is perpendicular to the plane of incidence). a a v ab − vb a ¢ sin
\∥  = ¢ = ab + a vb ab − a if vb = v

v v ab v − a vb ¢ sin
\†  = ¢ = vb + v ab vb − v if ab = a Because most dielectrics do not have different permeability, the Brewster angle is most often associated with parallel polarization. 2.Total Internal Reflection: Occurs when Snell’s law results in a sine greater than 1, i.e., when: fb sin
t  = sin
i  > 1 f The critical angle above which an incident wave will be totally internally reflected is: sin
ic  =

pb vr ar f =¢ = p vbr abr fb

Transmission Lines Defined by four parameters: 1. : inductance per unit length 2. j: capacitance per unit length 3. q: resistance per unit length 4. À: conductance per unit length

These parameters are related by: pd À vd ad = j and = ad j where vd , ad , and pd are the permeability, permittivity, and conductivity, of the dielectric.

Governing Equations (Telegraphers’ Equations): -
, t -n
, t = −qn
, t −  - -t -n
, t -
, t = −À
, t − j - -t Time Harmonic Analysis of Transmission Lines

Wave equation: d  − 
q + Œ‡
À + Œ‡j = 0 dt  d n − n
q + Œ‡
À + Œ‡j = 0 dt  Solution (with  measured from load): 
 =  Á  Á¡E +  {  {¡E  Á Á¡E  { {¡E n
 = n Á  Á¡E + n {  {¡E =  −  Â> Â> where:  = Ÿ + Œ  = ž
q + Œ‡
À + Œ‡j Characteristic impedance: Â> ≝

Á { q + Œ‡ = − =¢ Á { n n À + Œ‡j

The phase velocity and wavelength are the same as for uniform plane waves: ‡ 2` p = and • =    

Distortionless Lines: For a transmission line to be distortionless (i.e., the attenuation constant, phase velocity, and characteristic impedance of the line are constant), the line parameters must satisfy: q À =  j For this case, ‡ 1 p = =   √j j  = q ¢ + Œ ‡√j ¬®¯ ⇒  Å Ã Â> = ¢ j Ä

Note that a lossless line is also distortionless, so the preceding also applies for them except Ÿ = 0.

Load reflection coefficient: ÂL − Â> ΓL = = |ΓL | Ž∠
ÈL ÂL + Â> The voltage and current on the line can be written in terms of ΓL as: 
 =  Á
 Á¡E + ΓL  {¡E   Á Á¡E
 n
 = − ΓL  {¡E  Â> Line impedance: 
 ÂL + Â> tanh
 Â
 = = Â> n
 Â> + Â tanh


Standing Wave Ratio: max nmax 1 + |ΓL | SWR ≝ = = min nmin 1 − |ΓL | The wave on the transmission line will be the sum of a forward propagating wave and a standing wave as with plane waves. The voltage magnitude will have maxima and minima at: •
∠
ΓL  + 2`f, f = 0,1,2, … max = 4` •
∠
ΓL  +
2f − 1`, f = 0,1,2, … min = 4` (so note that the maxima and minima are separated by a quarter wavelength.) Special cases for Γ :
ÂL = Â> ⇒ ΓL = 0 ⇒ ∠
ΓL  = 0, SWR = 0. The wave is purely forward propagating.
ÂL = 0 ⇒ ΓL = −1 ⇒ ∠
ΓL  = `, SWR = ∞. The wave is purely standing with minima at the load and every half wavelength from it. The impedance is purely imaginary.
ÂL = ∞ ⇒ ΓL = +1 ⇒ ∠
ΓL  = 0, SWR = ∞. The wave is purely standing with maxima at the load and every half wavelength from it. The impedance is purely imaginary.
ÂL ∈ Ñ (i.e., ÂL is purely imaginary): SWR = ∞. The wave is purely standing.
ÂL = qL ∈ ℝ (i.e., ÂL is purely real): ∠
ΓL  = 0 (if qL > Â> , and the voltage has a maximum at the load) or ∠
ΓL  = ` (if qL < Â> , and the voltage has a minimum at the load). The minimum and maximum impedance magnitudes are given by: Â> Âmax = Â> SWR and Âmin = SWR Power on the line: | Á | ÄE
 d
 = − |ΓL |  {ÄE  cos7∠
Â> 8 2|Â> | For lossless lines tanh becomes Œtan and Ÿ = 0.

As t → ∞, a transmission line can be replaced with a single wire. When a change happens (e.g., load/generator is connected/disconnected/changed), a voltage and current wave will propagate and possibly reflect with coefficients: qL − Â> qG − Â> ΓL = ΓG = qL + Â> qG + Â> As long as the line is distortionless (see above), the wave will move with a constant velocity regardless of its shape. The voltage and current at any point on the line is given by the sum of all the waves that have passed by that point at that time. Transient Analysis of Transmission Lines