Midterm 2 Cheat sheet: front back
Three Phase Power
Y
(Common in residential home; has neutral)
$$
\begin{aligned}
\text{I}_{\phi} &= \text{I}_{L} \quad \text{(Phase Current = Line Current)}\\[1em]
\text{V}_{\phi} &= \frac{1}{\sqrt{3}} \text{V}_{L} \quad \text{(Phase voltage less than line voltage)} \\[1em]
208 &\text{V}_{LL} = 120 \text{V}_{\phi}
\end{aligned}
$$
Delta
$$
\begin{aligned}
\text{I}_{\phi} &= \frac{1}{\sqrt{3}} \text{I}_{L} \quad \text{(Phase current less than line current)}\\[1em]
\text{V}_{\phi} &= \text{V}_{L} \quad \text{(Phase voltage = Line Voltage)}
\end{aligned}
$$
\begin{gathered}
\text{Total} \underbrace{\text{Apparent Power}}_{S=\sqrt{\text{P}^2 + \text{Q}^2}} \quad \boxed{\quad \text{S}_{tot} = 3 \cdot \text{V}_{\phi} \cdot \text{I}_{\phi} \quad [\text{VA}] \quad}
\end{gathered}
Total power delivered is always 3 times of each individual phase.
Complex Power Review
\begin{gather}
1000\text{kVA} \ @ \ 0.8 \text{pf (lagging)} \\[1em]
\text{S}_{tot} = 1000\text{k} \ \angle \ \cos^{-1} (0.8) \\[1em]
\text{S}_{tot} = \underbrace{1000\text{k} \cos(37^\circ)}_{\text{Real Power}} + \underbrace{j1000\text{k} \sin(37^\circ)}_{\text{Reactive Power}}
\end{gather}
Capacitive Ground Current
\begin{aligned}
\text{I}_{\text{gnd}} &= \frac{\text{V}_{\text{rms}}}{|\text{Z}_{c}|} \\[1em]
&= \frac{120}{ \frac{1}{2 \pi 60 \ (13.28 \text{nF})}} \\[1em]
&= 0.6 \text{mA}
\end{aligned}
\begin{aligned}
\text{Z}_{\text{tot}} &= \text{R} \, – \, j \text{X}_c \\[1em]
|\text{Z}_{\text{tot}}| &= \sqrt{\text{R}^2 + \text{X}^2_c}
\end{aligned}
Dot Convention
1. Only winding, injection current into dot creating flux, matters during dot placement
2. +/- of voltages align with dot
3. Current into dot drive flux around the core in the same direction
Transformer Model
$$
\begin{gathered}
{\scriptsize \text{Transformer voltage}} \\[0.1em]
\boxed{\enspace \frac{V_1}{N_1} = \frac{V_2}{N_2} \enspace}
\end{gathered}
\qquad
\begin{gathered}
{\scriptsize \text{Transformer current}} \\[0.1em]
\boxed{\enspace N_1 \ i_1 = N_2 \ i_2 \enspace}
\end{gathered}
$$
1. Current into dotted primary = current out of dotted secondary
2. More coil = more voltage = less current
Magnetizing Inductance
$$
\begin{gathered}
{\scriptsize \text{Magnetizing Inductance}} \\[0.1em]
\boxed{\enspace\text{L}_{\mu} = \frac{\text{N}^2}{\mathbb{R}} \enspace}
\end{gathered}
$$
$$
\begin{gathered}
{\scriptsize \text{Reluctance}} \\[0.1em]
\boxed{\enspace \mathbb{R} = \frac{l_c}{\mu_c \ A_c} \enspace}
\end{gathered}
$$
$$
\begin{gathered}
{\scriptsize \text{Reflected L}_{\mu}} \\[0.1em]
\boxed{\enspace \text{L}_{\mu_2} = \left( \frac{N_2}{N_1} \right) ^2 \text{L}_{\mu_1} \enspace}
\end{gathered}
$$
$$
\begin{aligned}
\text{L} &\propto \mu_c \ {\scriptsize \text{(magnetic permeability)}} \\[1em]
\text{L} &\propto N^2 \ {\scriptsize \text{(number of turns on that side)}}
\end{aligned}
$$
Volt-Seconds Saturation of the Transformer Core
$$
\begin{gathered}
\int v \ dt = N A_c B_c \\[1em]
\text{Must keep} \ \int v \ dt \leq N A_c \underbrace{B_{\text{sat}}}_{\approx \text{2T}} \\[1em]
v \cdot t = N A_c B_c
\end{gathered}
$$
$$
\begin{gathered}
{\scriptsize \text{Inductor Voltage}} \\[0.1em]
\boxed{\enspace \text{V}_{L} = L \frac{\Delta i}{\Delta t}}
\end{gathered}
$$
Useful formula to find unknown L on the same side as performing a DC test and measuring current over time on the same side.
$$
\text{Must keep B < B}_{sat} \\[1em]
\text{At Max B}_{sat}, \mu_c = \mu_0 \\[1em]
\text{(very small magnetic permeability)}
$$
$$
\text{Higher freq AC has less} \int v \ dt, \\[1em]
\text{so we can use smaller core in} \\[1em]
\text{transformer. (400hz in planes)}
$$
Impedance Changing Property
$$
{\scriptsize \text{Impedance Changing Property}} \\[0.1em]
\boxed{\enspace \text{Z}_{in} = \left( \frac{N_1}{N_2} \right) ^2 \text{Z}_{load} \enspace} \\[3em]
P_{max} = \frac{V_{\text{s,rms}}^2}{R_{\text{antenna}}} = \frac{4}{50} = 80 \text{mW} \\[2em]
$$
Say we want 200mW, we need to change Zin.
$$
Z_{\text{in}} = \frac{V_{\text{s,rms}}^2}{P_{\text{tx}}} = \frac{4}{0.2} = 20 \Omega \\[2em]
$$
We need 20 Ω impedance
$$
\begin{gathered}
Z_{\text{in}} = \left( \frac{N_1}{N_2} \right) ^2 \text{R}_{antenna} \\[1em]
\frac{N_1}{N_2} = \sqrt{ \frac{R_p} {R_{\text{antenna}}} } = \sqrt{\frac{20} {50}} = 0.63
\end{gathered}
$$