The Class Notes

Notes for reviewer Be sure all 'l' have been replaced with $\ell$

4 jan 2010

$$\begin{gathered} Limit \\ A \\ \to \mathrm{\infty }: \end{gathered}$$

$$\begin{gathered} X_0 \\ =\frac{1}{\beta }{X}_s \\ \Rightarrow V_0=\frac{R_1+R_2}{R_1}{V}_{in} \end{gathered}$$

$$\begin{gathered} X_i \\ =0 \end{gathered}$$

Picture drawn by Kirk Betz based on drawing by Dr. Frohnes, lecture Jan. 4, 2010

$$\begin{gathered} V_{in} \\ =V_0(\frac{R_1}{R_1+R_2}) \end{gathered}$$.

$$\begin{gathered} V_0 \\ =\frac{1}{(\frac{R_1}{R_1+R_2})}{V}_{in} \end{gathered}$$

Magnetic Circuits

jan 6 2010

$$\begin{gathered} \overrightarrow{F} \\ =q\overrightarrow{v}\times \overrightarrow{B} \end{gathered}$$

$$\begin{gathered} d\overrightarrow{F} \\ =Id\overrightarrow{\ell }\times \overrightarrow{B} \end{gathered}$$

${ℱ}=H{\ell }_1+H{\ell }_2$

$$\begin{gathered} V \\ =R_{1}I+R_{2}I \end{gathered}$$

Picture drawn by Kirk Betz based on drawing by Dr. Frohnes, lecture Jan. 6, 2010

Magnetic Equations

$\int \overrightarrow{H}d\overrightarrow{\ell }={ℱ}$

$\oint \overrightarrow{H}d\overrightarrow{\ell }=Ni=\sum _{n}H\ell +Ni=0$

$\oint \overrightarrow{B}d\overrightarrow{s}=0$

$$\begin{gathered} \int \overrightarrow{B}d\overrightarrow{s}=\varphi \approx B{A}_{rea} \\ Magnetic \\ Flux \end{gathered}$$

$$\begin{gathered} {ℛ}\equiv Reluctance \\ \frac{{ℱ}}{\varphi }=\frac{Ni}{\varphi } \end{gathered}$$

$$\begin{gathered} \overrightarrow{B}=\mu \overrightarrow{H} \\ Assumes \\ Linearity \end{gathered}$$

${ℛ}\frac{\ell }{\mu A}$

Picture drawn by Kirk Betz based on drawing by Dr. Frohnes, lecture Jan. 6, 2010

Pictures drawn by Kirk Betz based on drawing by Dr. Frohnes, lecture Jan. 6, 2010

Magnetic Circuits Examples

What about chancing currents, etc.?

Picture drawn by Kirk Betz based on drawing by Dr. Frohnes, lecture Jan. 8, 2010

$\oint \overrightarrow{H}d\overrightarrow{\ell }=Ni$

Case i)

$$\begin{gathered} \mu =10^4{\mu }_0 \\ in \\ the \\ core \end{gathered}$$ Something about this part doesn't seem right.

$$\begin{gathered} Find \\ \overrightarrow{B} \\ in \\ the \\ gap. \end{gathered}$$

Graph and picture 6

$$\begin{gathered} H\ell \\ =NI \end{gathered}$$, $$\begin{gathered} I \\ \propto H \end{gathered}$$

$$\begin{gathered} NI \\ ={ℱ}\backsim V \end{gathered}$$

${ℛ}=\frac{\ell }{\mu A}\backsim R=\frac{\ell }{\sigma A}$

$$\begin{gathered} \varphi \\ =BA\backsim I=JA \end{gathered}$$

$R_c=\frac{{\ell }_1}{\mu A}$

$R_g=\frac{g}{{\mu }_0(\sqrt{A}+g{)}^2}$

$$\begin{gathered} \varphi \\ =B(\sqrt{A}+g{)}^2=\frac{NI}{R_g+R_c} \end{gathered}$$

$B_g\frac{NI}{(R_g+R_c)(\sqrt{A}+g{)}^2}$

Magnetic Circuits Continued

jan 11, 2010

some random graph here, can't really read it.

Case ii) Include non-linearity & find B in the Gap

$\oint \overrightarrow{H}d\overrightarrow{\ell }=NI=H{\ell }_1+H_g=H({\ell }_1+g)$

$$\begin{gathered} \varphi \\ =\int \overrightarrow{B}d\overrightarrow{s}=BA \end{gathered}$$

picture 7 goes here

$$\begin{gathered} \varphi \\ =\frac{NI-H{\ell }_1}{R_g}=\frac{-1}{R_g}(H{\ell }_1)+\frac{NI}{R_g} \end{gathered}$$ not sure about the -1 here

What energy is list in the hysteresis loop?

$$\begin{gathered} P \\ =vi \end{gathered}$$

$$\begin{gathered} W \\ =\int Pdt \end{gathered}$$

$$\begin{gathered} \oint \overrightarrow{E}d\overrightarrow{\ell }=\frac{-d}{dt}\int \overrightarrow{B}d\overrightarrow{s}Farada{y}^{\prime }s \\ Law \end{gathered}$$

hmm check these

$\overrightarrow{E}=\frac{J}{\sigma }$

$$\begin{gathered} \lambda \\ =Li \end{gathered}$$

e is voltage

$e=\frac{d\lambda }{dt}=L\frac{di}{dt}$

$$\begin{gathered} \lambda \\ =N\varphi \end{gathered}$$

$$\begin{gathered} N\equiv number \\ of \\ turns \end{gathered}$$

$\varphi \equiv Flux$