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Why three-phase electric power is awesome

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}

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