Exam reference sheet

  • Electromagnetics
  • FE Exam
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posted on 01 Sep 2016

Physical Constants {#physical-constants .unnumbered}


 Speed of light            $c$             $3 \times 10^{8} \text{ m/s}$
 Boltzmann const           $K$          $1.38 \times 10^{-23} \text{ J/K} $
Elementary charge          $e$               $1.6 \times 10^{-19} \ C$
  Permittivity       $\varepsilon_0$   $8.85 \times 10^{-12} \ \text{ F/m} $
  Permeability           $\mu_0$       $4 \pi \times 10^{-7} \ \text{ H/m}$    Electron rest-mass         $m_e$          $9.11 \times 10^{-31} \text{ kg}$
Proton rest mass          $m_p$         $1.67 \times 10^{-27} \text{ kg} $
 Plank constant            $h$           $6.63 \times 10^{-34} J \cdot s$    intr-imped-free-spc      $\eta_0$                  $376.7 \Omega $   --------------------- ----------------- ---------------------------------------

Unit Prefixes {#unit-prefixes .unnumbered}


Multiple    Prefix   Symbol    $10^{18}$     exa       E    $10^{15}$     peta      P    $10^{12}$     tera      T
$10^{9}$     giga      G
$10^{6}$     mega      M
$10^{3}$     kilo      k
$10^{2}$    hecto      h
  $10$       deka      da    $10^{-1}$     deci      d    $10^{-2}$    centi      c    $10^{-3}$    milli      m    $10^{-6}$    micro    $\mu$    $10^{-9}$     nano      n    $10^{-12}$    pico      p    $10^{-15}$   femto      f    $10^{-18}$    atto      a   ------------ -------- --------

Complex Numbers {#complex-numbers .unnumbered}

Euler’s Identity {#eulers-identity .unnumbered}

\[\begin{aligned} e^{j \theta} &= \cos{\theta} + j \sin{\theta} \\ \sin{\theta} &= \frac{e^{j \theta} - e^{-j \theta}}{2j} \\ \cos{\theta} &= \frac{e^{j \theta} - e^{-j \theta}}{2} \\ \end{aligned}\]

Polar to Rectangular Relations

\[z= x + jy = \abs{z}e^{j \theta}\] \[z= x + jy = \abs{z}e^{-j \theta}\] \[\begin{aligned} x &= \Re(z) = \abs{z} \cos{\theta} \\ y &= \Im(z) = \abs{z} \sin{\theta} \\\end{aligned}\] \[\abs{z} = \sqrt{z z^{*}} = \sqrt{x^2 + y^2}\] \[z^n = \abs{z}^n e^{jn \theta}\] \[\theta = \tan^{-1}{\left(\frac{y}{x}\right)}\] \[z^{\frac{1}{2}} = \pm \abs{z}^{\frac{1}{2}} e^{\frac{j \theta}{2}}\]

Complex Algebra

\[z_1 = x_1 + jy_1\] \[z_2 = x_2 + jy_2\] \[z_1 = z_2 \text{ iff } x_1 = x_2 \text{ and } y_1 = y_2\] \[z_1 + z_2 = (x_1 + x_2) + j(y_1 + y_2)\] \[z_1z_2 = \abs{z_1} \abs{z_2} e^{j \left( \theta_1 + \theta_2 \right)}\] \[\frac{z_1}{z_2} = \frac{\abs{z_1}}{\abs{z_2}} e^{j \left( \theta_1 - \theta_2 \right)}\]

Properties of $j$ and $e$

\[-1 = e^{j \pi} = e^{-j \pi} = 1 \phase{\pm 180^{\circ}}\] \[j = e^{\frac{j \pi}{2}} = 1 \phase{90^{\circ}}\] \[-j = e^{-\frac{j \pi}{2}} = 1 \phase{-90^{\circ}}\] \[\sqrt{j} = \pm e^{\frac{j \pi}{4}} = \pm \frac{\left( 1+j \right)}{\sqrt{2}}\] \[\sqrt{-j} = \pm e^{-\frac{j \pi}{4}} = \pm \frac{\left( 1-j \right)}{\sqrt{2}}\] \[e^{j4500}\]

The angle $\theta$ is the argument of $z$. It is defined for all $z \neq 0$: \(\arg(z) = \theta = \begin{cases} \arctan{\left( \frac{y}{x} \right)} & \text{if } x \ge 0 \\ \arctan{\left( \frac{y}{x} \right)} + \pi & \text{if } x < 0 \text{ and } y \ge 0 \pm 2n\pi \\ \arctan{\left( \frac{y}{x} \right)} - \pi & \text{if } x < 0 \text{ and } y < 0 \\ \end{cases}\)

Trigonometric Identities {#trigonometric-identities .unnumbered}

\(\sin{x \pm y} = \sin{x} \cos{y} \pm \cos{x} \sin{y}\) \(\cos{x \pm y} = \cos{x} \cos{y} \mp \sin{x} \sin{y}\) \(2 \sin{x} \sin{y} = \cos{(x-y)} - \cos{(x + y)}\) \(2 \sin{x} \cos{y} = \sin{(x+y)} + \sin{(x - y)}\) \(2 \cos{x} \cos{y} = \cos{(x+y)} + \cos{(x - y)}\) \(\sin{2x} = 2 \sin{x} cos{x}\) \(\cos{2x} = 1 - 2 \sin^2{x}\) \(\sin{x} + \sin{y} = 2 \sin{\left( \frac{x+y}{2} \right) \cos{\left( \frac{x-y}{2} \right)}}\) \(\sin{(-x)} = -\sin{x}\) \(e^{jx} =\cos{x} + j \sin{x}\) \(\sin{x} = \frac{e^{jx} - e^{-jx}}{2j}\) \(\cos{x} = \frac{e^{jx} + e^{-jx}}{2}\) \(A = \frac{1}{2}ab\sin C\) \(a^2 = b^2 + c^2 - 2bc\cos A\) \(\frac{a} = \frac{b} = \frac{c}\)

\(\sin{x \pm y} = \sin{x} \cos{y} \pm \cos{x} \sin{y}\) \(\cos{x \pm y} = \cos{x} \cos{y} \mp \sin{x} \sin{y}\) \(2 \sin{x} \sin{y} = \cos{(x-y)} - \cos{(x + y)}\) \(2 \sin{x} \cos{y} = \sin{(x+y)} + \sin{(x - y)}\) \(2 \cos{x} \cos{y} = \cos{(x+y)} + \cos{(x - y)}\) \(\sin{2x} = 2 \sin{x} cos{x}\) \(\cos{2x} = 1 - 2 \sin^2{x}\)

\[\sin{x} + \sin{y} = 2 \sin{\left( \frac{x+y}{2} \right) \cos{\left( \frac{x-y}{2} \right)}}\]

\(\sin{(-x)} = -\sin{x}\) \(e^{jx} =\cos{x} + j \sin{x}\) \(\sin{x} = \frac{e^{jx} - e^{-jx}}{2j}\) \(\cos{x} = \frac{e^{jx} + e^{-jx}}{2}\)

Time to Frequency Domain

$z(t)$ $\tilde{Z}$ ———————————————————— ——————- ———————————————- $A \cos{\omega t}$ $\leftrightarrow$ $A$ $A \cos{(\omega t + \phi_0)}$ $\leftrightarrow$ $Ae^{j \phi_0}$ $A \cos{(\omega t + \beta x + \phi_0)}$ $\leftrightarrow$ $Ae^{\beta x + \phi_0}$ $A e^{-\alpha x} \cos{(\omega t + \beta x + \phi_0})$ $\leftrightarrow$ $A e^{-\alpha x}e^{\beta x + \phi_0}$ $A \sin{\omega t}$ $\leftrightarrow$ $A e^{-j \pi / 2}$ $A \sin{\omega t + \phi_0}$ $\leftrightarrow$ $A e^{j(\phi_0 - \pi/2)}$ $ \frac{d}{dt}(z(t)) $ $\leftrightarrow$ $j \omega \tilde{Z}$ $ \frac{d}{dt} \left[ A \cos{(\omega t + \phi_0)} \right]$ $\leftrightarrow$ $j \omega A e^{j \phi_0}$ $\int{z(t) dt}$ $\leftrightarrow$ $\frac{1}{j \omega} \tilde{Z}$ $\int{A \sin{(\omega t + \phi_0)} dt}$ $\leftrightarrow$ $\frac{1}{j \omega} A e^{j(\phi_0 - \pi/2)}$

Maxwell’s Equations


     Gauss’s Law                           $\nabla \cdot \vec{D} = \rho_{\nu}$
    Faraday’s Law            $\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$    Gauss’s Law (magnetism)                         $ \nabla \cdot \vec{B} = 0$
     Ampre’s Law         $\nabla \times \vec{H} = \vec{J} + \frac{\partial \vec{D}}{\partial t}$   ------------------------- -------------------------------------------------------------------------

Vector Identities

\(\vec{A} \cdot \vec{B} = AB\cos\theta_{AB}\) \(\vec{A} \times \vec{B} = \hat{n} AB\sin\theta_{AB} \: \hat{n}\text{ normal to plane containing } \vec{A} \text{ and } \vec{B}\) \(\vec{A} \cdot \left( \vec{B} \times \vec{C} \right) = \vec{B} \cdot \left( \vec{C} \times \vec{A} \right) = \vec{C} \cdot \left( \vec{A} \times \vec{B} \right)\) \(\vec{A} \times \left( \vec{B} \times \vec{C} \right) = \vec{B} \left( \vec{A} \cdot \vec{C} \right) - \vec{C} \left( \vec{A} \times \vec{B} \right)\) \(\nabla \left( U + V \right) = \nabla U + \nabla V\) \(\nabla(UV) = U\nabla V + V \nabla U\) \(\nabla \cdot \left( \vec{A} + \vec{B} \right) = \nabla \cdot \vec{A} + \nabla \cdot \vec{B}\) \(\nabla \cdot \left(U \vec{A} \right) = U \nabla \cdot \vec{A} + \vec{A} \cdot \nabla U\) \(\nabla \times \left(U \vec{A} \right) = U \nabla \times \vec{A} + \nabla U \times \vec{A}\) \(\nabla \times \left(\vec{A} + \vec{B} \right) = \nabla \times \vec{A} + \nabla \times \vec{B}\) \(\nabla \cdot \left(\vec{A} \times \vec{B} \right) = \vec{B} \cdot \left(\nabla \times \vec{A} \right) - \vec{A} \cdot \left(\nabla \times \vec{B} \right)\) \(\nabla \cdot \left( \nabla \times \vec{A} \right) = 0\) \(\nabla \times \nabla V = 0\) \(\nabla \cdot \nabla V = \nabla^2V\) \(\nabla \times \nabla \times \vec{A} = \nabla \left(\nabla \cdot \vec{A}\right) - \nabla^2 \vec{A}\) \(\int\limits_V \left( \nabla \cdot \vec{A} \right) dV = \oint\limits_S \vec{A} \cdot d \vec{s} \: (S \text{ encloses } V)\) \(\int\limits_V \left( \nabla \times \vec{A} \right) \cdot d\vec{s} = \oint\limits_C \vec{A} \cdot d \vec{l} \: (S \text{ bounded by } C)\)

Table of Derivatives {#table-of-derivatives .unnumbered}

Function $\leftrightarrow$ Derivative
$x^n$ $\leftrightarrow$ $nx^{n-1}$
$e^x$ $\leftrightarrow$ $e^x$
$ln (x)$ $\leftrightarrow$ $\frac{1}{x}$
$sin (x)$ $\leftrightarrow$ $cos (x) $
$cos (x)$ $\leftrightarrow$ $-sin (x) $
$-sin (x)$ $\leftrightarrow$ $-cos (x) $
$-cos (x)$ $\leftrightarrow$ $sin (x) $
\(\frac{d}\left[ {f\left( u \right)} \right] = \frac{d}\left[ {f\left( u \right)} \right]\frac\) Chain Rule: $\frac{d}{dx} \left( {A(B(x)} \right) =\frac{dA(B)}{dB} \frac{dB(x)}{dx} $
Product Rule: $\frac{d}{dx} \left( A(x) B(x) \right) = A(x) \frac{dB}{dx} + \frac{dA}{dx} B(x)$
Quotient Rule: $\frac{d}{dx} \left( \frac{A(x)}{B(x)} \right) = \frac{B(x) \frac{dA}{dx} - A(x)\frac{dB}{dx}}{B^2(x)}$\

Table of Integrals {#table-of-integrals .unnumbered}

Logarithms {#logarithms .unnumbered}

\(y = \log _b \left( x \right){\textrm{ iff }}x = b^y\) \(\log _b \left( x \right) = \log _b \left( c \right)\log _c \left( x \right) = \frac\) \(\log _b \left( {\frac{x}{y}} \right) = \log _b \left( x \right) - \log _b \left( y \right)\) \(\log _b \left( {x^n } \right) = n\log _b \left( x \right)\) \(\log _b \left( 1 \right) = 0\) \(\log _b \left( b \right) = 1\) \(\log _b \left( {xy} \right) = \log _b \left( x \right) + \log _b \left( y \right)\)

Other EM Laws {#other-em-laws .unnumbered}

Biot-Savart: