Class Wiki - User contributions [en]

Kirk Betz

2010-02-17T07:30:55Z

Kirkbetz:

List of Articles in Draft

:[[The Class Notes]]

List of Articles in Finished

:[[Example Problem - Toroid]]

Contact info
Kirk.Betz@wallawalla.edu

:[[Conference attendance]]

:[[Points for reviews]]

[[Study]]

Links

[http://hyperphysics.phy-astr.gsu.edu/HBASE/electric/farlaw.html Magnets and Coils]

Kirk Betz

2010-02-17T07:30:03Z

Kirkbetz:

List of Articles in Draft

:[[The Class Notes]]

List of Articles in Finished

:[[Example Problem - Toroid]]

Contact info
Kirk.Betz@wallawalla.edu

:[[Conference attendance]]

:[[Points for reviews]]

[[Study]]

Links

http://hyperphysics.phy-astr.gsu.edu/HBASE/electric/farlaw.html

The Class Notes

2010-01-28T00:32:59Z

Kirkbetz: /* Magnetic Circuits */

The following are the notes as interpreted by [[Kirk Betz]] from ENGR 431 taught by Dr. Rob Frohne.
Electrical Magnetic Conversion is the study of magnetic circuits in all there forms.

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

==EMEC Notes==
''January 4, 2010''

'''Introduction to EMEC'''

Syllabus was handed out and an outline of the class structure what introduce. We where also briefed on what we would be talking about his quarter.

==Magnetic Circuits ==

''January 6 2010''

From circuits we know that V is a function of the E field.

<math> V\ = \int ed\ell </math>

The E field moves along a closed path of length <math> \ell </math>. By integrating E along the path <math> \ell </math> we find the Voltage V as shown in the above equation.

integrated the e field along the path

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

The Class Notes

2010-01-27T23:43:28Z

Kirkbetz: /* Magnetic Circuits */

The following are the notes as interpreted by [[Kirk Betz]] from ENGR 431 taught by Dr. Rob Frohne.
Electrical Magnetic Conversion is the study of magnetic circuits in all there forms.

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

==EMEC Notes==
''January 4, 2010''

'''Introduction to EMEC'''

Syllabus was handed out and an outline of the class structure what introduce. We where also briefed on what we would be talking about his quarter.

==Magnetic Circuits ==

''January 6 2010''

From circuits we know that V is a function of the <math> E\ </math> field.

<math> V\ = \int ed\ell\quad \frac{V}{m} </math>

e field volts per meter

integrated the e field along the path

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

The Class Notes

2010-01-27T23:17:51Z

Kirkbetz: /* Magnetic Circuits */

The following are the notes as interpreted by [[Kirk Betz]] from ENGR 431 taught by Dr. Rob Frohne.
Electrical Magnetic Conversion is the study of magnetic circuits in all there forms.

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

==EMEC Notes==
''January 4, 2010''

'''Introduction to EMEC'''

Syllabus was handed out and an outline of the class structure what introduce. We where also briefed on what we would be talking about his quarter.

==Magnetic Circuits ==

''January 6 2010''

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

The Class Notes

2010-01-27T23:17:14Z

Kirkbetz: /* Magnetic Circuits */

The following are the notes as interpreted by [[Kirk Betz]] from ENGR 431 taught by Dr. Rob Frohne.
Electrical Magnetic Conversion is the study of magnetic circuits in all there forms.

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

==EMEC Notes==
''January 4, 2010''

'''Introduction to EMEC'''

Syllabus was handed out and an outline of the class structure what introduce. We where also briefed on what we would be talking about his quarter.

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

The Class Notes

2010-01-27T23:15:57Z

Kirkbetz: /* EMEC Notes */

The following are the notes as interpreted by [[Kirk Betz]] from ENGR 431 taught by Dr. Rob Frohne.
Electrical Magnetic Conversion is the study of magnetic circuits in all there forms.

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

==EMEC Notes==
''January 4, 2010''

'''Introduction to EMEC'''

Syllabus was handed out and an outline of the class structure what introduce. We where also briefed on what we would be talking about his quarter.

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

The Class Notes

2010-01-27T23:09:30Z

Kirkbetz:

The following are the notes as interpreted by [[Kirk Betz]] from ENGR 431 taught by Dr. Rob Frohne.
Electrical Magnetic Conversion is the study of magnetic circuits in all there forms.

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

==EMEC Notes==
4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

The Class Notes

2010-01-27T23:05:56Z

Kirkbetz:

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

==EMEC Notes==
4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

The Class Notes

2010-01-27T23:05:05Z

Kirkbetz: /* Magnetic Circuits Continued */

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda\ = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ </math>

Ampere's Law

2010-01-27T22:34:49Z

Kirkbetz: /* References */

When current travels through a conductive material it creates a magnetic field. The direction of the magnetic field can be determined by making a fist with your right hand and sticking your thumb out in the direction of the current. Then the direction your fingers curl is the direction of the magnetic field. This also works the other way, you could run current in a loop and curl your fingers in the direction of the current. Your thumb will then be pointing in the direction of the magnetic field.

This is the point at which Ampere's law becomes useful. The intensity of the magnetic field created by a current is found using this law.
Ampere's law states that the sum of currents is equal to the line integral of the magnetic field intensity along a path that encloses those currents. <ref> Electric Drives (our textbook) p. 5-2 </ref>

<math>\oint H\,dl = \sum i </math>

For example:
i1 - i2 + i3 = H

[[Image:ampere_law.jpg]]

==References==
<references/>

==comments==

Isn't Ampere's law derived from Gauss's law?

<math>\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A}</math>

==Reviewers==
Alex Roddy

Tim Rasmussen

Example: Metal Cart

2010-01-27T07:01:29Z

Kirkbetz: /* Review Comments and Status */

==Problem==
A DC generator is built using a metal cart with metallic wheels that travel around a set of perfectly conducting rails in a large circle. The rails are L m apart and there is a uniform magnetic <math>\vec B</math> field normal to the plane. The cart has a penguin,with mass m, and is driven by a rocket engine having a constant thrust <math> F_1 </math>. A wet polar bear, having stumbled out of a shack where he recently had a bad experience with a battery, lays dying across the tracks acting as a load resistance R over the rails. Find The current as a function of time. What is the current after the generator attains the steady-state condition?

[[Image:Emec_cart_polarBear2.png]]

==Solution==

For this Problem the large circle will be represented by a pair of parallel wires and the cart as a single wire. This is illustrated below in the top and end view figures.
[[Image:Emec_cart_topview.png]]

[[Image:Emec_cart_endview.png]]

We have two forces, <math> F_1 </math> being the force from the rocket engine and <math> F_2 </math> being the force caused by the current in the conductor and the Magnetic Field.
The resulting Force <math> F_t </math> is simply the sum of <math> F_1 </math> and <math> F_2 </math>

<math> F_2 </math> can be found using Ampere's Law

<math>\vec F=\int\limits_{c} I ~\vec dl\times \vec B~~~~~\Longrightarrow~~~~~ \vec F_2=\int\limits_{0}^{L} I ~\vec dl\times \vec B ~~~~~\Longrightarrow~~~~~ \vec F_2=- I(t) ~B~L ~~ \hat i</math>

We can also say that <math> I(t)=\frac{-e_m(t)}{R} </math>

And <math>~~ {e_m(t)}= \int\limits_{o}^{L} (\vec v \times \vec B)~\vec dl ~~~~~\Longrightarrow~~~~~ {e_m(t)}=-v~L~B </math>

<math> I(t)=\frac{v~L~B}{R} ~~~~~~~~~~~~~~~~~~~~~~\vec F_2=- I(t) ~B~L ~~ \hat i~~~~~\Longrightarrow~~~~~\vec F_2=- \frac{v~L~B}{R} ~B~L ~~ \hat i ~~~~~\Longrightarrow~~~~~\vec F_2=- \frac{v~L^2~B^2}{R} ~~ \hat i</math>

<math> \dot v=\frac{F_1}{m} ~\hat i~+\frac {F_2}{m}~ \hat i ~~~~~\Longrightarrow~~~~~ \dot v= ~\frac{F_1}{m} ~~\hat i~- \frac{v~L^2~B^2}{R m}~~\hat i ~~~~~\Longrightarrow~~~~~ \dot v~ +~ v \left(\frac{L^2~B^2}{R~m}\right) ~-~\frac{F_1}{m}~=~0 </math>

Now we have a lovely differential equation to work with! To attempt to find the current we will take the Laplace transform.

<math> \mathcal{L}\left\{ \dot v ~+~v\left(\frac{L^2~B^2}{R~m}\right)-\frac{F_1}{m} \right\} ~~~~~\Longrightarrow~~~~~ s~V(s)~+~\frac{L^2B^2}{R~m} V(s) ~-~\frac{F_1}{m~s}</math>

Lets title and substitute in the variable <math>~~~~\psi=\frac{L^2~B^2}{R ~m}~~</math> to simplify things

<math> s~V(s)~+~\psi~V(s)~-~\frac{F_1}{m~s}~=~0</math>

<math> V(s)~(s~+~\psi)~=~\frac{F_1}{m~s} ~~~~~\Longrightarrow~~~~~ V(s)~=~\frac {F_1}{m~s~(s~+~\psi)}</math>

Using partial fraction expansion

<math>\frac {F_1}{m~s~(s~+~\psi)}~~~~~~\Longrightarrow~~~~~~\frac{F_1/m\psi}{s}~-~\frac{F_1/m\psi}{s~+~\psi} </math>

<math>V(s)=\frac{F_1/m\psi}{s}~-~\frac{F_1/m\psi}{s~+~\psi}</math>

<math>V(t)= \mathcal{L}\left\{ \frac{F_1/m\psi}{s}~-~\frac{F_1/m\psi}{s~+~\psi} \right\} ~~~~~\Longrightarrow~~~~~ V(t)=\frac {F_0}{m~\psi}\left(u(t)~-~e^{-\psi~t} ~u(t)\right) ~~~~~\Longrightarrow~~~~~ V(t)= \frac {F_0}{m~\psi}\left(1~-~e^{-\psi~t}\right) </math>

<math> V(t)=\frac {F_0}{m~ \frac{L^2~B^2}{R ~m}}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right) ~~~~~\Longrightarrow~~~~~ V(t)=\frac {F_0~R}{L^2~B^2}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)</math>

We know that
<math>~~ e_m(t)~=~V(t)LB </math>

So we can substitute in V(t) to get

<math> e_m(t)= \frac {F_0~R}{L~B}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)</math>

And we know that <math>~~I(t)~=~\frac{e_m(t)}{R}</math>

<math> I(t)=\frac{\frac {F_0~R}{L~B}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)}{R}~~~~~\Longrightarrow~~~~~ I(t)~=~\frac {F_0}{L~B}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)</math>

To find the steady-state current we simply look at the limit of I(t) as <math>t \rightarrow \infty</math>

<math>\lim_{t\rightarrow \infty} I(t)=\frac{F_0}{L~B} \left( 1- e^{-\infty}\right) ~~~~~\Longrightarrow~~~~~ I(\infty)=\frac{F_0}{L~B} </math>

So the Steady-State Current = <math>\frac{F_0}{L~B}</math>

<math> In~conclusion~it ~can ~be ~seen ~that ~a ~penguin ~driven, ~polar ~bear ~killing ~generator </math>

<math>~would ~be ~a ~viable ~option ~for ~alternative ~energy ~in ~Canada. </math>

==Review Comments and Status==
Kirk Betz- emailed 1.26.10

[[Kirk Betz]] Read and approved 1-26-10

Will Griffith <math> \thickapprox A\ beautiful\ man </math> - emailed 1.26.10

Example: Metal Cart

2010-01-27T06:58:40Z

Kirkbetz: /* Review Comments and Status */

Points for reviews

2010-01-27T06:58:28Z

Kirkbetz:

Paper Points

[[AC Motors]] 53

[[AC vs. DC]] 5

[[magnetic circuits]] 18

[[Example: Metal Cart]] 60

Example: Metal Cart

2010-01-27T06:57:36Z

Kirkbetz: /* Review Comments and Status */

Example: Metal Cart

2010-01-27T06:57:28Z

Kirkbetz: /* Review Comments and Status */

Example: Metal Cart

2010-01-27T06:55:54Z

Kirkbetz: /* Review Comments and Status */

Example: Metal Cart

2010-01-27T06:52:26Z

Kirkbetz: /* Solution */

Example: Metal Cart

2010-01-27T06:50:51Z

Kirkbetz: /* Problem */

==Problem==
A DC generator is built using a metal cart with metallic wheels that travel around a set of perfectly conducting rails in a large circle. The rails are L m apart and there is a uniform magnetic <math>\vec B</math> field normal to the plane. The cart has a penguin,with mass m, and is driven by a rocket engine having a constant thrust <math> F_1 </math>. A wet polar bear, having stumbled out of a shack where he recently had a bad experience with a battery, lays dying across the tracks acting as a load resistance R over the rails. Find The current as a function of time. What is the current after the generator attains the steady-state condition?

[[Image:Emec_cart_polarBear2.png]]

==Solution==

For this Problem we will represent the large circle as a pair of straight parallel wires and the cart as a single wire. This is illustrated below in the top and end view figures
[[Image:Emec_cart_topview.png]]

[[Image:Emec_cart_endview.png]]

We have two forces, <math> F_1 </math> being the force from the rocket engine and <math> F_2 </math> being the force caused by the current in the conductor and the Magnetic Field.
The resulting Force <math> F_t </math> is simply the sum of <math> F_1 </math> and <math> F_2 </math>

<math> F_2 </math> can be found using Ampere's Law

<math>\vec F=\int\limits_{c} I ~\vec dl\times \vec B~~~~~\Longrightarrow~~~~~ \vec F_2=\int\limits_{0}^{L} I ~\vec dl\times \vec B ~~~~~\Longrightarrow~~~~~ \vec F_2=- I(t) ~B~L ~~ \hat i</math>

We can also say that <math> I(t)=\frac{-e_m(t)}{R} </math>

And <math>~~ {e_m(t)}= \int\limits_{o}^{L} (\vec v \times \vec B)~\vec dl ~~~~~\Longrightarrow~~~~~ {e_m(t)}=-v~L~B </math>

<math> I(t)=\frac{v~L~B}{R} ~~~~~~~~~~~~~~~~~~~~~~\vec F_2=- I(t) ~B~L ~~ \hat i~~~~~\Longrightarrow~~~~~\vec F_2=- \frac{v~L~B}{R} ~B~L ~~ \hat i ~~~~~\Longrightarrow~~~~~\vec F_2=- \frac{v~L^2~B^2}{R} ~~ \hat i</math>

<math> \dot v=\frac{F_1}{m} ~\hat i~+\frac {F_2}{m}~ \hat i ~~~~~\Longrightarrow~~~~~ \dot v= ~\frac{F_1}{m} ~~\hat i~- \frac{v~L^2~B^2}{R m}~~\hat i ~~~~~\Longrightarrow~~~~~ \dot v~ +~ v \left(\frac{L^2~B^2}{R~m}\right) ~-~\frac{F_1}{m}~=~0 </math>

Now we have a lovely differential equation to work with! To attempt to find the current we will take the Laplace transform.

<math> \mathcal{L}\left\{ \dot v ~+~v\left(\frac{L^2~B^2}{R~m}\right)-\frac{F_1}{m} \right\} ~~~~~\Longrightarrow~~~~~ s~V(s)~+~\frac{L^2B^2}{R~m} V(s) ~-~\frac{F_1}{m~s}</math>

Lets title and substitute in the variable <math>~~~~\psi=\frac{L^2~B^2}{R ~m}~~</math> to simplify things

<math> s~V(s)~+~\psi~V(s)~-~\frac{F_1}{m~s}~=~0</math>

<math> V(s)~(s~+~\psi)~=~\frac{F_1}{m~s} ~~~~~\Longrightarrow~~~~~ V(s)~=~\frac {F_1}{m~s~(s~+~\psi)}</math>

Using partial fraction expansion

<math>\frac {F_1}{m~s~(s~+~\psi)}~~~~~~\Longrightarrow~~~~~~\frac{F_1/m\psi}{s}~-~\frac{F_1/m\psi}{s~+~\psi} </math>

<math>V(s)=\frac{F_1/m\psi}{s}~-~\frac{F_1/m\psi}{s~+~\psi}</math>

<math>V(t)= \mathcal{L}\left\{ \frac{F_1/m\psi}{s}~-~\frac{F_1/m\psi}{s~+~\psi} \right\} ~~~~~\Longrightarrow~~~~~ V(t)=\frac {F_0}{m~\psi}\left(u(t)~-~e^{-\psi~t} ~u(t)\right) ~~~~~\Longrightarrow~~~~~ V(t)= \frac {F_0}{m~\psi}\left(1~-~e^{-\psi~t}\right) </math>

<math> V(t)=\frac {F_0}{m~ \frac{L^2~B^2}{R ~m}}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right) ~~~~~\Longrightarrow~~~~~ V(t)=\frac {F_0~R}{L^2~B^2}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)</math>

We know that
<math>~~ e_m(t)~=~V(t)LB </math>

So we can substitute in V(t) to get

<math> e_m(t)= \frac {F_0~R}{L~B}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)</math>

And we know that <math>~~I(t)~=~\frac{e_m(t)}{R}</math>

<math> I(t)=\frac{\frac {F_0~R}{L~B}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)}{R}~~~~~\Longrightarrow~~~~~ I(t)~=~\frac {F_0}{L~B}\left(1~-~e^{-\frac{L^2~B^2}{R ~m}~t}\right)</math>

To find the steady-state current we simply look at the limit of I(t) as <math>t \rightarrow \infty</math>

<math>\lim_{t\rightarrow \infty} I(t)=\frac{F_0}{L~B} \left( 1- e^{-\infty}\right) ~~~~~\Longrightarrow~~~~~ I(\infty)=\frac{F_0}{L~B} </math>

So the Steady-State Current = <math>\frac{F_0}{L~B}</math>

<math> In~conclusion~it ~can ~be ~seen ~that ~a ~penguin ~driven, ~polar ~bear ~killing ~generator </math>

<math>~would ~be ~a ~viable ~option ~for ~alternative ~energy ~in ~Canada. </math>

==Review Comments and Status==
Kirk Betz- emailed 1.26.10

Will Griffith- emailed 1.26.10

The Class Notes

2010-01-26T00:09:03Z

Kirkbetz: /* Magnetic Circuits Continued */

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

<math> \phi \equiv Flux\ ???</math>

The Class Notes

2010-01-26T00:07:59Z

Kirkbetz: /* Magnetic Circuits Continued */

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

<math> \lambda\ = N \phi </math>

<math> N \equiv number\ of\ turns</math>

The Class Notes

2010-01-26T00:05:23Z

Kirkbetz: /* Magnetic Circuits Continued */

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

<math> \phi\ = \frac {NI-H\ell_1}{R_g} = \frac{-1}{R_g}(H\ell_1) + \frac{NI}{R_g} </math> not sure about the -1 here

What energy is list in the hysteresis loop?

<math> P\ = vi</math>

<math> W\ = \int Pdt</math>

<math>\oint \vec E d \vec \ell = \frac {-d}{dt} \int \vec B d \vec s \quad Faraday's\ Law</math>

hmm check these

<math> \vec E = \frac {J}{\sigma} </math>

<math> \lambda = L i</math>

e is voltage

<math> e = \frac {d \lambda}{dt} = L \frac{di}{dt}</math>

The Class Notes

2010-01-26T00:02:46Z

Kirkbetz: /* Magnetic Circuits Continued */

The Class Notes

2010-01-25T23:58:33Z

Kirkbetz: /* Magnetic Circuits Continued */

The Class Notes

2010-01-25T23:53:36Z

Kirkbetz: /* Magnetic Circuits Continued */

Notes for reviewer
Be sure all 'l' have been replaced with <math> \ell</math>

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{\ell}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

<math> \phi\ = B(\sqrt{A} + g)^2 = \frac{NI}{R_g + R_c}</math>

<math> B_g \frac{NI}{(R_g + R_c)(\sqrt{A}+g)^2}</math>

==Magnetic Circuits Continued==

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

<math> \oint \vec H d \vec \ell = NI = H \ell_1 + H_g = H(\ell_1 +g) </math>

<math> \phi\ = \int \vec B d \vec s = BA </math>

picture 7 goes here

The Class Notes

2010-01-25T23:46:48Z

Kirkbetz:

The Class Notes

2010-01-25T23:45:09Z

Kirkbetz: /* Magnetic Circuits Examples */

The Class Notes

2010-01-25T23:40:53Z

Kirkbetz: /* Magnetic Circuits Examples */

The Class Notes

2010-01-25T23:37:51Z

Kirkbetz: /* Magnetic Circuits Examples */

The Class Notes

2010-01-25T23:36:51Z

Kirkbetz:

The Class Notes

2010-01-25T23:35:55Z

Kirkbetz:

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec\ell \times \vec B </math>

<math>\mathcal{F} = H\ell_1 + H\ell_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec\ell= \mathcal{F}</math>

<math>\oint \vec Hd \vec\ell= Ni = \sum_{n}H\ell+ Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{\ell}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec\ell= Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> H\ell\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{\ell}{\mu A} \backsim R = \frac{l}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{\ell_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

The Class Notes

2010-01-25T23:28:36Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> Hl\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{l}{\mu A} \backsim R = \frac{l}{\sigma A} </math>

<math> \phi\ = BA \backsim I = JA</math>

<math> R_c= \frac{l_1}{\mu A} </math>

<math> R_g= \frac{g}{\mu_0 (\sqrt{A} + g)^2}</math>

The Class Notes

2010-01-25T23:17:55Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> Hl\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

<math> \mathcal{R} = \frac{l}{\mu A} \backsim R = \frac{l}{\sigma A} </math>

The Class Notes

2010-01-25T23:14:35Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> Hl\ = NI </math>, <math>\ I\ \varpropto H</math>

<math> NI\ = \mathcal{F} \backsim V</math>

The Class Notes

2010-01-25T23:10:44Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> Hl\ = NI </math> <math>,\ I\ \varpropto H</math>

The Class Notes

2010-01-25T23:10:17Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

Graph and picture 6

<math> Hl\ = NI </math>\ <math> I\ \varpropto H</math>

The Class Notes

2010-01-25T23:05:00Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B\ in\ the\ gap. </math>

The Class Notes

2010-01-25T23:04:45Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

<math>Find\ \vec B in\ the\ gap. </math>

The Class Notes

2010-01-25T23:03:43Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

Case i)

<math> \mu = 10^4 \mu_0\ in\ the\ core </math> Something about this part doesn't seem right.

The Class Notes

2010-01-25T23:00:33Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

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

<math> \oint \vec Hd \vec l = Ni </math>

The Class Notes

2010-01-25T23:00:17Z

Kirkbetz: /* Magnetic Circuits Examples */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

[[Image:Magnetloop.JPG‎ ]]

<math> \oint \vec Hd \vec l = Ni </math>

File:Magnetloop.JPG

2010-01-25T23:00:01Z

Kirkbetz: A piece of steel with loops wrapped N times.

A piece of steel with loops wrapped N times.

The Class Notes

2010-01-25T22:57:09Z

Kirkbetz: /* Magnetic Equations */

4 jan 2010

<math>Limit\ A\ \to \infty:</math>

<math>X_0\ = \frac{1}{\beta}X_s\ \Rightarrow V_0 = \frac{R_1 + R_2}{R_1}V_{in}</math>

<math> X_i\ = 0</math>

[[Image:Thebegining.JPG]]

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

<math> V_{in}\ = V_0 (\frac {R_1}{R_1 + R_2})</math>.

<math> V_0\ = \frac {1}{(\frac {R_1}{R_1 + R_2})}V_{in} </math>

==Magnetic Circuits ==

jan 6 2010

<math> \vec F\ = q \vec v \times \vec B </math>

<math>d \vec F\ = I d \vec l \times \vec B </math>

<math>\mathcal{F} = Hl_1 + Hl_2</math>

<math>V\ = R_1I + R_2I</math>

[[Image:Magnetic_cir.JPG]]

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

==Magnetic Equations==

<math>\int \vec Hd \vec l = \mathcal{F}</math>

<math>\oint \vec Hd \vec l = Ni = \sum_{n}Hl + Ni = 0 </math>

<math>\oint \vec Bd \vec s = 0 </math>

<math>\int \vec Bd \vec s = \phi \thickapprox BA_{rea}\ Magnetic\ Flux</math>

<math>\mathcal{R} \equiv Reluctance\ \frac{\mathcal{F}}{\phi} = \frac{Ni}{\phi} </math>

<math> \vec B = \mu \vec H\ Assumes\ Linearity </math>

<math> \mathcal{R} \frac{l}{\mu A}</math>

[[Image:BHField.JPG‎]]

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

[[Image:BFieldsmall.JPG‎]] [[Image:BFsmall.JPG‎ ]]

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

==Magnetic Circuits Examples==

What about chancing currents, etc.?

Picture 5

<math> \oint \vec Hd \vec l = Ni </math>

File:BHField.JPG

2010-01-25T22:56:37Z

Kirkbetz: Histogram for B H field

Histogram for B H field

The Class Notes

2010-01-25T22:55:28Z

Kirkbetz: /* Magnetic Equations */

The Class Notes

2010-01-25T22:55:11Z

Kirkbetz: /* Magnetic Equations */

File:BFsmall.JPG

2010-01-25T22:54:51Z

Kirkbetz: Another view of what is happening inside a material with a present B field.

Another view of what is happening inside a material with a present B field.

The Class Notes

2010-01-25T22:54:04Z

Kirkbetz: /* Magnetic Equations */

File:BFieldsmall.JPG

2010-01-25T22:53:34Z

Kirkbetz: A miniature of what is happening in a material when a B field is present.

A miniature of what is happening in a material when a B field is present.

The Class Notes

2010-01-25T22:41:22Z

Kirkbetz: /* Magnetic Circuits Examples */