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The Unofficial ELEC3105 Final Exam Formula Sheet Edit

Magnetic Circuits Edit

Ampere's Law $\oint H \cdot dl = NI$

$\phi = \phi_{max} \sin(\omega t)$

$v = N \frac{d \phi}{d t}$

$H = \frac{1}{\mu}B$

$\Re = \frac{l}{\mu_0 S}$

$\phi = \frac{N I}{\Re}$

Magnetising current $I_m = (\phi_c \Re_c + \phi_g \Re_c) \frac{1}{N} = (\frac{\phi_c l_c}{\mu_c S_c} + \frac{\phi_g l_g}{\mu_g S_g}) \frac{1}{N}$

Magnetising inductance $L_m = \frac{N \phi_c}{I_m} = \frac{\lambda}{i_m}$

$\mu_0 = 4 \pi \times 10^{-7}$

AC Power Computation Edit

Single Phase Edit

Real power $P = V I \cos \phi$

Reactive power $Q = V I \sin \phi$ (in VAR)

Complex power $= S = V I^* = P + jQ$

Apparent power $= |S| = \sqrt{P^2 + Q^2} = V I$

Three Phase Edit

$P = 3 V_p I_p \cos \phi = \sqrt{3} V_l I_l \cos \phi$

where $V_l$ and $I_l$ are line-to-line voltage and current for delta-connected loads, and $V_p$ and $I_p$ are phase voltage and current for star-connected loads.

Transformers Edit

$E = 4.44 N f \phi_{max} = 4.44 N f B_{max} S_c$

Referring to primary side:

$a = \frac{N_1}{N_2}$

$V'_2 = a V_2$

$I'_2 = \frac{1}{a} I_2$

$R'_2 = a^2 R_2$

$L'_2 = a^2 L_2$

Efficiency $\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{loss}}$

% Voltage regulation = $\frac{V_{1} - V_{2rated}}{V_{2rated}} \times 100$

Electromechanical Energy Conversion Edit

In a rotary system:

$Torque = \frac{1}{2} i^{2} \frac{dL}{d\theta}$

where:

$i$ = current

$L$ = inductance

$\theta$ = angle of displacement

DC Machines Edit

$E = k_{E}' \phi \omega = K_{E} \omega$ where $K_{E}$ = (area of coil) x B x (number of coils)

$T = k_{T}' \phi i_a = K_T i_a$ where $K_{T}$ = (area of coil) x B x (number of coils)

The mechanical power used by the load is

$P = T \omega$

and the power delivered to the load is

$P = E i_a$

The total loop equation for the armature is

$V = R_a i_a + k_{E}' \phi \omega$

and the equation for the field coil is simply $V = R_f i_f$.

Induction Machines Edit

$n_s = \frac{f_s}{p}$

Thevenin equivalent circuit:

$V_{Th} = \frac{X_m V_1}{\sqrt{R_1^2 + ( X_1 + X_m )^2}} \approx \frac{X_m V_1}{X_1 + X_m}$

$Z_{Th} = \frac{j X_m (R_1 + jX_1)}{R_1 + j(X_1 + X_m)}$ but that is pretty bad, try $R_{Th} \approx R_1$ and $X_{Th} \approx X_1$

Slip for maximum produced torque:

$s_{Tmax} = \frac{R_2'}{\sqrt{R_{Th}^2 + (X_{Th} + X_2')^2}}$

Power Edit

(Input power) - (Stator copper loss) = (Air gap power)

(Air gap power) - (Rotor copper loss) = (Developed mechanical power)

(Dev. mechanical power) - (Windage and friction losses) = (Output Power)

(Windage and friction losses) = $\frac{3 {I'}_{nl}^{2} {R'}_{2} ( 1 - s_{nl} ) }{s_{nl}}$

Synchronous Machines Edit

As with induction machines, $n_s = \frac{f_s}{p}$

The synchronous impedance is calculated from the open circuit test voltage and short-circuit test current: $Z_s = \frac{E_{OCC}}{I_{SCC}} = R_a + j X_s$.

Often, $R_a \ll X_s$ so $Z_s \approx X_s$.

The important power equation is:

$P = \frac{3 V_a E_a}{X_s} \sin \delta$

where $\delta$ is the load angle.

Power Electronics Edit

For buck converter:

$V_o = D V_d$

where D is the duty cycle.

For boost converter:

$V_o = \frac{V_d}{1-D}$