UCD EEC130A Formulasheet
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Transcript of UCD EEC130A Formulasheet
EEC 130A : Formula Sheet
Updated: Mar. 18th 2012
1 Waves and Phasors
The general expression for a one-dimensionalwave
A0e−αx cos
(2π
Tt± 2π
λx+ θ0
)or
A0e−αx cos (ωt± βx+ θ0)
In phasor form
A0e−αxej(±βx+θ0)
2 Transmission Lines
Telegrapher’s Equations (phasor form)
−dV (z)
dz= (R′ + jωL′) I(z)
−dI(z)
dz= (G′ + jωC ′) V (z)
Wave equation (expressed in voltage)
d2V (z)
dz2− γ2V (z) = 0
Propagation constant
γ = α + jβ =√
(R′ + jωL′) (G′ + jwC ′)
Characteristic Impedance
Z0 =
√R′ + jωL′
G′ + jwC ′
For lossless transmission lines
β = ω√L′C ′ =
2π
λ
Phase velocity
up =ω
β= fλ
For lossless transmission lines
up =1√L′C ′
For most TEM transmission lines
up =1√µε
=1
√µ0µrε0εr
For quasi-TEM transmission lines on non-magnetic substrates, the above is approxi-mated by
up =1
√µ0µrε0εeff
=c
√εeff
Voltage expression on a lossless line for awave propagating in the z direction (Solu-tion to the wave equation)
V (z) = V +0 e−jβz + V −0 e
jβz
Current expression on a line for a wave prop-agating in the z direction
I(z) = I+0 e−jβz + I−0 e
jβz
1
Reflection coefficient from load ZL
ΓL =ZL − Z0
ZL + Z0
Voltage standing wave ratio on a line withreflection coefficient ΓL
SWR =|V |max|V |min
=1 + |Γ|1− |Γ|
Position of voltage maximum
dmax =θrλ
4π+nλ
2,
where n = 0, 1, 2, . . . if θr ≥ 0, and n =1, 2, . . . if θr < 0.
Position of voltage minimum
dmin = dmax ±λ
4.
depending on whether dmax is greater or lessthan λ/4.
Input impedance of a transmission line seenat a distance l from a load ZL
Z(l) = Z0ZL + jZ0 tan(βl)
Z0 + jZL tan(βl)
Input reflection coefficient of a lossless trans-mission line seen at a distance l from a loadZL
Γin = Γe−j2βl
3 Electrostatics
Force on a point charge q inside a static elec-tric field
F = qE
Gauss’s law∮S
D · dS = Q or ∇ ·D = ρ
Electrostatic fields are conservative
∇× E = 0 or
∮C
E · dl = 0
Electric field produced by a point charge qin free space
E =q (R−Ri)
4πε0 |R−Ri|3
Electric field produced by a volume chargedistribution
E =1
4πε
∫V ′
R′ρv dV
′
R′2
Electric field produced by a surface chargedistribution
E =1
4πε
∫S′R′ρs ds
′
R′2
Electric field produced by a line charge dis-tribution
E =1
4πε
∫l′R′ρl dl
′
R′2
Electric field produced by an infinite sheetof charge
E = zρs2ε
Electric field produced by an infinite line ofcharge
E =D
ε= r
Dr
ε= r
ρl2πεr
Electric field - scalar potential relationship
E = −∇V or V2 − V1 = −∫ P2
P1
E · dl
2
Electric potential due to a point charge(with infinity chosen as the reference)
V =q
4πε0 |R−Ri|
Poisson’s equation
∇2V = −ρε
Constitutive relationship in dielectric mate-rials
D = ε0E + P
where P is the polarization.
P = ε0χeE
Electrostatic energy density
we =1
2εE2
Boundary conditions
E1t = E2t or n× (E1 − E2) = 0
D1n −D2n = ρs or n · (D1 −D2) = ρs
Ohm’s law
J = σE
Conductivity
σ = ρvµ
where µ stands for charge mobility.
Joule’s law
P =
∫E · J dv
4 Magnetostatics
Force on a moving charge q inside a magneticfield
F = qu×B
Force on an infinitesimally small current el-ement Idl inside a magnetic field
dFm = Idl×B
Torque on a N -turn loop carrying current Iinside a uniform magnetic field
T = m×B
where m = nNIA.
Gauss’s law for magnetism
∇ ·B = 0 or
∮S
B · dS = 0
Ampere’s law
∇×H = J or
∮C
H · dl = I
Magnetic flux density — magnetic vectorpotential relationship
B = ∇×A
Magnetic potential produced by a currentdistribution
A =µ
4π
∫V ′
J
R′dV ′
Vector Poisson’s Equation
∇2A = −µJ
Magnetic field intensity produced by an in-finitesimally small current element (Biot-Savart law)
3
dH =I
4π
dl× R
R2
Magnetic field produced by an infinitely longwire of current in the z-direction
H = φI
2πr
Magnetic field produced by a circular loopof current in the φ-direction
H = zIa2
2(a2 + z2)3/2
Constitutive relationship in magnetic mate-rials
B = µ0H + µ0M
Magnetization
M = χmH
Boundary conditions
B1n = B2n or n · (B1 −B2) = 0
H1t −H2t = Js or n× (H1 −H2) = Js
Magnetostatic energy density
wm =1
2µH2
5 Maxwell’s Equations
Integral form ∮S
D · ds = Q
∮C
E · dl = −∫S
∂B
∂t· ds
∮S
B · ds = 0
∮C
H · dl =
∫S
(J +
∂D
∂t
)· ds
6 Useful Integrals
∫dx√x2 + c2
= ln(x+√x2 + c2)∫
dx
x2 + c2=
1
ctan−1
x
c∫dx(
x2 + c2)3/2 =
1
c2x√
x2 + c2∫x dx√x2 + c2
=√x2 + c2∫
x dx
x2 + c2=
1
2ln (x2 + c2)∫
x dx
(x2 + c2)3/2= − 1√
x2 + c2∫dx
(a+ bx)2= − 1
b(a+ bx)
7 Constants
Free space permittivity
ε0 = 8.85× 10−12 F/m
Free space permeability
µ0 = 4π × 10−7 H/m
4