# User:Constant314

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Help:Displaying a formula

/What is wrong with the flux cutting model?

Welder Wildlife Foundation

User:Constant314/Magnetic current

Space Cloth

Leapfrog Filter

Magnetic frill generator

Electric vector potential

Magnetic current#Magnetic frill generator

Causality#Fields#Science#Engineering

Frequency dependent negative resistor

Template:Harvard citation text

WP:RX

## To Do

 ********************************************************************************


This topic should always be on the top.

*** some HTML ***

curl E = -B/t

curl H = D/t + Jconduction + Jsource = curl H = D/t + σE + Jsource

where σ = conductivity

User:Constant314/Tow-Thomas active filter

Add alternate explanations of skin effect.

Add magnetic vector potential to the transformer page.

## AWG Formulae

By definition, No. 36 AWG is 0.005 inches in diameter, and No. 0000 is 0.46 inches in diameter. The ratio of these diameters is 1:92, and there are 40 gauge sizes from No. 36 to No. 0000, or 39 steps. Because each successive gauge number increases cross sectional area by a constant multiple, diameters vary geometrically. Any two successive gauges (e.g., A & B ) have diameters in the ratio (dia. B ÷ dia. A) of ${\displaystyle {\sqrt[{39}]{92}}}$ (approximately 1.12293), while for gauges two steps apart (e.g., A, B, & C), the ratio of the C to A is about 1.122932 = 1.26098. The diameter of a No. n AWG wire is determined, for gauges smaller than 00 (36 to 0), according to the following formula:

${\displaystyle d_{n}=0.005~\mathrm {inch} \times 92^{\frac {36-n}{39}}=0.127~\mathrm {mm} \times 92^{\frac {36-n}{39}}}$

(see below for gauges larger than No. 0 (i.e., No. 00, No. 000, No. 0000 ).)

The gauge can be calculated from the diameter using

${\displaystyle n=-39\log _{92}\left({\frac {d_{n}}{0.005~\mathrm {inch} }}\right)+36}$
${\displaystyle n={\frac {-39}{\log _{10}92}}\log _{10}\left({\frac {d_{n}}{0.005~\mathrm {inch} }}\right)+36}$
${\displaystyle n={\frac {-39}{\log _{e}92}}\log _{e}\left({\frac {d_{n}}{0.005~\mathrm {inch} }}\right)+36}$
${\displaystyle n={\frac {-39}{\ln 92}}\ln \left({\frac {d_{n}}{0.005~\mathrm {inch} }}\right)+36}$

## Capacitor Self-Discharge

A discussion on usenet about the self discharge time constant for some tpyes of capacitors, including polypropylene.

## Tansformer talk

In the ideal transformer section it notes that the ideal transformer is lossless, therefore power out = power in. If output volts go down then output current must go up. That’s a conclusion and not an explanation. Unfortunately, the reliable sources tend to write a bunch of equations and conclude that current transforms inversely with the turns ratio. Again it is a conclusion and not an explanation. I have an explanation, but without any reliable sources, it would be WP:OP. I’ll outline it here, hoping that maybe someone else can find a reliable source or determine that it is supportable by the sources already in the article. The gist is this, take a transformer with turns NP and NS and a load on the secondary and try to force a current IP into the primary. If the quantity NP × IP – NS × IS ≠ 0, then you are forcing flux into the nominally infinite (or very large) self-inductance. It responds with an infinite (or very large) voltage seen on both the primary and the secondary. But there is a load on the secondary that would draw an infinite (or very large) current, so you can’t really get an infinite voltage. In fact, the secondary voltage that you can get is just enough to satisfy NP × IP – NS × IS = 0. The transformer, in effect, generates the voltage required to get the secondary current that will satisfy NP × IP – NS × IS = 0. A transformer with an infinite self-inductance is, so to speak, intolerant of NP × IP – NS × IS ≠ 0. You can sort of make a Lenz's law argument that any deviation from NP × IP – NS × IS = 0 would create huge reaction that would oppose the deviation.

Sounds about right. I might have done it in terms of Ampere-Turns, as are commonly used in magnet design problems. Gah4 (talk) 06:19, 22 September 2016 (UTC)

## Skin effect general formula

${\displaystyle J=J_{\mathrm {S} }\,e^{-{(1+j)d/\delta }}}$[1]:362.

where δ is called the skin depth. The skin depth is thus defined as the depth below the surface of the conductor at which the current density has fallen to 1/e (about 0.37) of JS. The imaginary part of the exponent indicates that phase of the current density is shifted 1 radian for each skin depth of penetration. One full wavelength in the conductor requires 2π skin depths, at which point the current density is attenuated to e-2π (-54.6 dB) of its surface value. The wavelength in the conductor is much shorter than the wavelength in vacuum, or equivalently, the speed of light in the conductor is much slower than in vacuum. For example, at 1 MHz in copper, the wavelength is about 0.5 mm and the speed of light is about 500 m/s. As a consequence of Snell's law and this very low speed of light in the conductor, all fields to entering the conductor refract to nearly perpendicular.

When the skin depth is not small with respect to the radius of the wire, current density may be described in terms of Bessel functions. The current density inside round wire away from the influences of other fields, as function of distance from the axis is given by Weeks [2]:38.

${\displaystyle \mathbf {J} _{r}={\frac {k\mathbf {I} }{2\pi R}}{\frac {J_{0}(kr)}{J_{1}(kR)}}}$.

or

${\displaystyle \mathbf {J} _{r}=\mathbf {J} _{R}{\frac {J_{0}(kr)}{J_{0}(kR)}}}$.

where

${\displaystyle \quad \omega }$ = angular frequency of current = 2π × frequency
${\displaystyle \quad r=}$ distance from the axis of the wire
${\displaystyle \quad R=}$ radius of the wire
${\displaystyle \quad \mathbf {J} _{r}=}$ current density phasor at distance, r, from the axis of the wire
${\displaystyle \quad \mathbf {J} _{R}=}$ current density phasor at the surface of the wire
${\displaystyle \quad \mathbf {I} =}$ total current phasor
${\displaystyle \quad J_{0}=}$ Bessel function of the first kind, order 0
${\displaystyle \quad J_{1}=}$ Bessel function of the first kind, order 1
${\displaystyle \quad k={\sqrt {\frac {-j\omega \mu }{\rho }}}={\frac {1-j}{\delta }}}$ the wave number in the conductor
${\displaystyle \quad \delta ={\sqrt {\frac {2\rho }{\omega \mu }}}}$ also called skin depth.
${\displaystyle \quad \rho }$ = resistivity of the conductor
${\displaystyle \quad \mu _{r}}$ = relative magnetic permeability of the conductor
${\displaystyle \quad \mu _{0}}$ = the permeability of free space
${\displaystyle \quad \mu }$ = ${\displaystyle \mu _{r}}$ ${\displaystyle \mu _{0}}$

Since ${\displaystyle k}$ is complex, the Bessel functions are also complex. The amplitude and phase of the current density varies with depth.

The internal impedance of round wire is given by [2]:40:

${\displaystyle \mathbf {Z} _{int}={\frac {k\rho }{2\pi R}}{\frac {J_{0}(kR)}{J_{1}(kR)}}}$.

The internal impedance is complex and may be interpreted as a resistance in series with an inductance. The inductance accounts for energy stored in the magnetic field inside the wire. It has a maximum value of ${\displaystyle {\frac {\mu }{8\pi }}}$ H/m at zero frequency and goes to zero as the frequency increases. The zero frequency internal inductance is independent of the radius of the round wire.

## Magnetic current

Magnetic current is, nominally, a fictitious current composed of fictitious moving magnetic mono-poles. It has the dimensions of volts. Magnetic currents produce an electric field analogously to the production of a magnetic field by electric currents. Magnetic current density is usually represented by the symbol M, which has the units of v/m² (volts per square meter). A given distibution of electric change can be mathematically replaced by an equivalent distribution of magnetic current. This fact can be used to simplify some electromagnetic field problems. [a] [b]

Magnetic current (flowing magnetic mono-poles), M, creates an electric field, E, in accordance with the left-hand rule.

The direction of the electric field produced by magnetic currents is determined by the left-hand rule (opposite to the right-hand rule) as evidenced by the negative sign in the equation curl E = -M. One component of M is the familiar term B/t, which is referred to as the magnetic displacement current or more properly as the magnetic displacement current density. [c] [d] [e]

[7]:286

[7]:286

[f]:291

[g]:291

## Galilean electromagnetism

${\displaystyle E'=E+{\frac {v}{c}}\times B}$

${\displaystyle B'=B-{\frac {v}{c}}\times E}$

${\displaystyle H'=H-v\times D}$

As late as 1963, Purcell offered the following low velocity transformations as suitable for calculating the electric field experienced by a jet plane tranvelling in the Earth's magnetic field.

${\displaystyle E'=E+v\times B}$.
${\displaystyle B'=B-\epsilon _{0}\mu _{0}v\times E}$. [h]:222

In 1973 Bellac and Levy-Leblond state that these equations are incorrect or misleading because they do not correspond to any consistent Galilean limit. Rousseaux gives a simple example showing that a transformation from an initial inertial frame to a second frame with a speed of v0 with respect to the first frame and then to a third frame moving with a speed v1 with respect to the second frame would give a result different from going directly from the first frame to the third frame using a relative speed of (v0 + v1).

Bellac and Levy-Leblond offer two transformations that do have consistent Galilean limits as follows:

The electric limit applies when electric field effects are dominant such as when Faraday's law of induction was insignificant.

${\displaystyle E'=E}$.
${\displaystyle B'=B-\epsilon _{0}\mu _{0}v\times E}$.

The magnetic limit applies when the magnetic field effects are dominant.

${\displaystyle E'=E+v\times B}$.
${\displaystyle B'=B}$.

some claim[i]:12. Something else.

A leapfrog filter is a type of active circuit electronic filter that simulates a passive electronic ladder filter. Other names for this type of filter are active-ladder or multiple feedback filter.[7]:286 The arrangement of feedback loops in the signal flow-graph of the simulated ladder filter inspired the name leapfrog filter.[7]:286 The leapfrog filter maintains the low component sensitivity of the passive ladder filter that it simulates. [j]:291

[k]:291

WP:OR

WP:fringe

WP:NPOV

WP:SYN

WP:RX

WP:SPAM

WP:EXTPROMO

## RFL

http://www.spx.com/en/assets/pdf/T272_OpManual_SPX_v5_DC_en.pdf

3M

http://bellsystempractices.org/Miscellaneous/CableTesting-FaultLocating.pdf

Tempo

http://www.teltec.com.au/images/pdfattachment/pdf_4de824e35e958sidekick%20plus%20user%20guide.pdf

## Wien bridge oscillator control loop stability

Schematic of a Wien bridge oscillator that uses diodes to control amplitude. This circuit typically produces total harmonic distortion in the range of 1-5% depending on how carefully it is trimmed.

In this version of the oscillator, Rb is a small incandescent lamp. Usually R1 = R2 = R and C1 = C2 = C. In normal operation, Rb self heats to the point where its resistance is Rf/2.

http://www.hpl.hp.com/hpjournal/pdfs/IssuePDFs/1960-04.pdf

Small perturbations in the value of Rb cause the oscillating poles to move back and forth across the jω axis. If the poles move into the left half plane, the oscillation dies out exponentially to zero. If the poles move into the right half plane, the oscillation grows exponentially until something limits it. If the perturbation is very small, the magnitude of the equivalent Q is very large so that the amplitude changes slowly. If the perturbations are small and reverse after a short time, the envelope follows a ramp. The envelope is approximately the integral of the perturbation. The perturbation to envelope transfer function rolls off at 6 dB/octave and causes -90° of phase shift.

The light bulb has thermal inertia so that its power to resistance transfer function exhibits a single pole low pass filter. The envelope transfer function and the bulb transfer function are effectively in cascade, so that the control loop has has effectively a low pass pole and a pole at zero and a net phase shift of almost -180°. This would cause poor transient response in the control loop. The output might exhibit squegging. Oliver[10] showed that slight compression of the gain by the amplifier mitigates the envelope transfer function so that most oscillators showed good transient response, except in the rare case where non-linearity in the tubes canceled each other producing an unusually linear amplifier.

## Volt, voltage, EMF and Potential

My 2 cents worth.
• Volt is about the unit of measurement and should be a separate article.
Everything else is tied together by the equation E = -∇(φ) -σA/σt, where φ is the electrodynamic retarted electric scalar potential and A is the retarded magnetic vector potential.
• Voltage is, to me, a circuit quantity. It’s what a voltmeter would read between two points in a circuit. If the leads to the voltmeter went through a region where σA/σt ≠ 0, then the voltmeter reading would depend on the path of the leads to the voltmeter and the voltage would be ill defined.
• Electromotive force, I usually associate with a path integral of the electric field (as defined above) tangential to a definite path. Usually the path follows a conductor. Typical paths may include a battery, generator, transformer, loop antenna, microphone, transducer. EMF for a specific path is unique. The path may be implicit. In a particular usage, there may be no mention of the path integral, but it is there never-the-less. If the beginning point and ending point of the path integral were in a continuous region where σA/σt = 0 then the voltage measured between those points by a voltmeter connected to those points and entirely in the region would read a unique value and EMF would numerically be the same as the voltage.
• Potential has more than one meaning. The article will have to deal with that. For me, potential is a field quantity, although potential is used as a synonym for voltage in both DC and AC circuits.
- The electrodynamic retarded scalar electric potential, which is used above in the equation for E. It is well defined everywhere except for points occupied by point charges. The difference in potential between two points is also well defined but may be physically meaningless. In general, the potential difference between two points is not the voltage between those two points.
- The electrostatic potential that applies to electrostatics and DC circuits or any situation where σA/σt = 0. In this case, the difference in potential between two points is the same as the voltage between those two points and EMF evaluated along any path is also equal to the same voltage.

## Signal Flow Graph solution of linear equations

The topology of a signal flow graph has a one to one relationship with a system of linear equations of the following from:

{\displaystyle {\begin{aligned}x_{\mathrm {j} }&=\sum _{\mathrm {k} =1}^{\mathrm {N} }t_{\mathrm {jk} }x_{\mathrm {k} }(\mathrm {where} )t_{\mathrm {jk} }&=(\mathrm {t_{\mathrm {jk} }transmission\ from\ k\ to\ j} )\end{aligned}}}

{\displaystyle {\begin{aligned}x_{\mathrm {k} }&=\sum _{\mathrm {k} =1}^{\mathrm {N} }(M_{\mathrm {kj} })y_{\mathrm {j} }\end{aligned}}}
where Mkj = Mason's gain formula for the path from input yj to variable xk.

For N equations with N unknowns where each yj is a known value and each xj is an unknown value, there is equation for each known of the following form.

{\displaystyle {\begin{aligned}\sum _{k=1}^{\mathrm {N} }c_{kj}x_{k}&=y_{j}\end{aligned}}}

These equations can be rewritten as

{\displaystyle {\begin{aligned}\sum _{k=1}^{N}c_{kj}x_{k}-y_{j}&=0\end{aligned}}}

and further rewritten as

{\displaystyle {\begin{aligned}\sum _{k=1}^{N}c_{kj}x_{k}+x_{j}-y_{j}&=x_{j}\end{aligned}}}

and finally rewritten as

{\displaystyle {\begin{aligned}\sum _{k=1}^{N}(c_{kj}+\delta _{ij})x_{k}-y_{j}&=x_{j}\end{aligned}}}
where δij = Kronecker delta

These equations obviously are in the form specified in the Elements of signal flow graphs section.

[Elements of signal flow graphs]

User:Constant314#Elements of signal flow graphs

## Regarding Signal flow graphs

_________________________________________________________

{\displaystyle {\begin{aligned}x_{j}&=\sum _{k=1}^{N}t_{kj}x_{k}\end{aligned}}}

with the understanding that all the equations have this interpretation

{\displaystyle {\begin{aligned}(\mathrm {effect\ at\ j} )&=\sum _{k=1}^{N}(\mathrm {transmission\ from\ k\ to\ j} )(\mathrm {cause\ at\ k} )\end{aligned}}}

A signal-flow graph (SFG), also known as a Mason graph, is a directed graph that may be regarded as type of block diagram that is a designed to represent cause and effect relationships in linear systems. The SFG for this purpose was introduced by Samuel Jefferson Mason,[11] in 1953. SFG's are most commonly used to represent signal flow physical systems and their controllers. Among their other uses are the representation of signal flow in digital filters, state variable filters and some other types of analog filters. The equations that model such a systems are constrained[12] in such a way that they must be linear algebraic equations that may be represented by a directed graph. The topology of the directed graph has a one-to-one relationship with the constrained system of equations. The SFG consists of nodes indicated by dots and weighted directional branches indicated by arrows. The nodes are the variables of the equations and the branch weights are the coefficients. Signals may only traverse a branch in the direction indicated by its arrow. The elements of a SFG can only represent the operations of multiplication by a coefficient and addition, which are sufficient to represent the constrained equations. When a signal traverses a branch in its indicated direction, the signal is multiplied the weight of the branch. When two or more branches direct into the same node, their outputs are added.

Kou on cause and effect [13]

## Regarding Skin Effect

_________________________________________________________

Fink, Donald G.; Beaty, H. Wayne (2000), Standard Handbook for Electrical Engineers (14th ed.), McGraw-Hill, ISBN 0-07-022005-0

regarding counter emf at center of the wire [14]

regarding inductance at the center of the wire [15]

Bedell[16]

_________________________________________________________

The text was “ is equal to the rate of change of the magnetic flux through the circuit.”

I am changing that to “is equal to the rate of change of the magnetic flux enclosed[17][18] by the circuit.”

I have provided two in-line citations with virtually the same wording (the sources say path instead of circuit).

The reason for this change is there are two interpretations of flux through the circuit

• The correct one which is the flux through a surface bounded by the circuit.

• The incorrect one meaning lines of flux pushed through the conductors, which is not consistent which the equations.

If anyone doubts this look at the equations that follow that text.

${\displaystyle {\mathcal {E}}=-{{d\Phi _{\mathrm {B} }} \over dt}\ }$,

which says that the emf is equal to the time rate of change of The Flux.

and

${\displaystyle \Phi _{\mathrm {B} }=\iint \limits _{\Sigma (t)}\mathbf {B} (\mathbf {r} ,t)\cdot d\mathbf {A} \ ,}$

which says that The Flux is the integral of the B flux density over a 2 dimensional surface bounded by the closed path, ${\displaystyle \Sigma }$, which is the path of the circuit.

[16]

[18]

[19]

If you use a definition of circuit like a roughly circular line, route, or movement that starts and finishes at the same place. That is, a mathematical definition, then path and circuit are similar. But in EE, circuit has a different definition. Gah4 (talk) 06:28, 22 September 2016 (UTC)
Good morning Gah4. Thanks for your comments. I use my user page to prepare comments to be pasted into other pages, so they may not make any sense unless seen in the context of the target page. Also, I may edit them on the target page, but I don't come back here an echo the edits. So, what you find here may be incorrect, incomplete or obsolete and I don't make any attempt to maintain it. In this particular case, circuit means whatever it means in the target article, which was probably Faraday's law of induction. Comments and suggestions are welcome on that article's talk page. Cheers. Constant314 (talk) 14:27, 22 September 2016 (UTC)
That is what I suspected. Your talk page is on my watch list from a previous discussion, and so I happened to see it. The one on wire gauge is interesting. I never saw the formula listed before, and didn't even know that there was one. I wasn't looking in too much detail, though. Sometimes I am not ready for a real talk page discussion. Thanks, Gah4 (talk) 14:48, 22 September 2016 (UTC)

## Quantities and units

_________________________________________________________

Electromagnetic units are part of a system of electrical units based primarily upon the magnetic properties of electric currents, the fundamental SI unit being the ampere. The units are:

In the electromagnetic cgs system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity (relative permeability) whose value in a vacuum is unity. As a consequence, the square of the speed of light appears explicitly in some of the equations interrelating quantities in this system.

SI electromagnetism units
Symbol[20] Name of Quantity Derived Units Unit Base Units
I electric current ampere (SI base unit) A A (= W/V = C/s)
Q electric charge coulomb C A⋅s
U, ΔV, Δφ; E potential difference; electromotive force volt V kg⋅m2⋅s−3⋅A−1 (= J/C)
R; Z; X electric resistance; impedance; reactance ohm Ω kg⋅m2⋅s−3⋅A−2 (= V/A)
ρ resistivity ohm metre Ω⋅m kg⋅m3⋅s−3⋅A−2
P electric power watt W kg⋅m2⋅s−3 (= V⋅A)
C capacitance farad F kg−1⋅m−2⋅s4⋅A2 (= C/V)
E electric field strength volt per metre V/m kg⋅m⋅s−3⋅A−1 (= N/C)
D electric displacement field coulomb per square metre C/m2 A⋅s⋅m−2
ε permittivity farad per metre F/m kg−1⋅m−3⋅s4⋅A2
χe electric susceptibility (dimensionless)
G; Y; B conductance; admittance; susceptance siemens S kg−1⋅m−2⋅s3⋅A2 (= Ω−1)
κ, γ, σ conductivity siemens per metre S/m kg−1⋅m−3⋅s3⋅A2
B magnetic flux density, magnetic induction tesla T kg⋅s−2⋅A−1 (= Wb/m2 = N⋅A−1⋅m−1)
Φ magnetic flux weber Wb kg⋅m2⋅s−2⋅A−1 (= V⋅s)
H magnetic field strength ampere per metre A/m A⋅m−1
L, M inductance henry H kg⋅m2⋅s−2⋅A−2 (= Wb/A = V⋅s/A)
μ permeability henry per metre H/m kg⋅m⋅s−2⋅A−2
χ magnetic susceptibility (dimensionless)

## Hidden Content

________________________________________________________________________________________________________________

________________________________________________________________________________________________________________

Testing

Talk:Gyroscope#Another animation

These work.

Electric current#Conventions

near-field

Causality#Engineering
Causality#Physics
Fermi energy#Related quantities
Wikipedia:Causality#Fields#Science#Engineering
Wikipedia:Make technical articles understandable
Wikipedia:Make technical articles understandable#Rules of thumb
Wikipedia:Make technical articles understandable#Write one level down

These do not work. They link to the top of the article.

Causality#Fields#Science
Wikipedia:Make technical articles understandable#Rules of thumb_Write one level down
Wikipedia:Make technical articles understandable#(Rules of thumb#Write one level down)
[[Wikipedia:Make technical articles understandable#{Rules of thumb#Write one level down}]]
Wikipedia:Make technical articles understandable#Rules of thumb(Write one level down)
[[Wikipedia:Make technical articles understandable#Rules of thumb[Write one level down]]]
[[Wikipedia:Make technical articles understandable[Rules of thumb#Write one level down]]]
Wikipedia:Make technical articles understandable#Rules of thumb##Write one level down
Wikipedia:Make technical articles understandable#Rules of thumb.Write one level down
Wikipedia:Make technical articles understandable#Rules of thumb/Write one level down
Wikipedia:Make technical articles understandable#Rules of thumb\Write one level down

## Ideal Transformer Citations

________________________________________________________________________________________________________________

begin testing

"The approximate analysis of a circuit containing an iron-core transformer may be achieved very simply by replacing that transformer by an ideal transformer" [21]

An ideal transformer is defined as Vs - a Vp, Is = Ip/a Reitz, Milford & Christy (1993, p. 323)

An ideal transformer is defined as Vs - a Vp, Is = Ip/a [22]

An ideal transformer is defined as Vs - a Vp, Is = Ip/a Reitz, Milford & Christy 1993, p. 323

end testing

## References

________________________________________________________________________________________________________________

Hayt, William; Kemmerly, Jack E. (1993), Engineering Circuit Analysis (5th ed.), McGraw-Hill, ISBN 007027410X

Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993), Foundations of Electromagnetic Theory (4th ed.), Addison-Wesley, ISBN 0201526247

________________________________________________________________________________________________________________

Talk:Telegrapher's equations#Solutions of the Telegrapher's Equations as Circuit Components

this talk page comment

## Fix up equations from Telegrapher's Equations

________________________________________________________________________________________________________________ Before:

${\displaystyle {\frac {\partial ^{2}}{{\partial x}^{2}}}V=LC{\frac {\partial ^{2}}{{\partial t}^{2}}}V+(RC+GL){\frac {\partial }{\partial t}}V+GRV}$
${\displaystyle {\frac {\partial ^{2}}{{\partial x}^{2}}}I=LC{\frac {\partial ^{2}}{{\partial t}^{2}}}I+(RC+GL){\frac {\partial }{\partial t}}I+GRI}$

After:

${\displaystyle {\frac {\partial ^{2}}{{\partial x}^{2}}}V={\mathsf {LC}}{\frac {\partial ^{2}}{{\partial t}^{2}}}V+{\mathsf {(RC+GL)}}{\frac {\partial }{\partial t}}V+{\mathsf {(GR)}}V}$
${\displaystyle {\frac {\partial ^{2}}{{\partial x}^{2}}}I={\mathsf {(LC)}}{\frac {\partial ^{2}}{{\partial t}^{2}}}I+{\mathsf {(RC+GL)}}{\frac {\partial }{\partial t}}I+{\mathsf {GR}}I}$

## equation with alt text

________________________________________________________________________________________________________________

${\displaystyle Z_{\mathsf {S}}=Z_{\mathsf {L}}^{*}\,}$
${\displaystyle Z_{\mathrm {load} }=Z_{\mathrm {source} }^{*}\,}$
${\displaystyle Z_{\mathsf {load}}=Z_{\mathsf {source}}^{*}\,}$
${\displaystyle Z_{\text{load}}=Z_{\text{source}}^{*}\,}$

## FIR Filter Definition

________________________________________________________________________________________________________________

For an N'th order discrete-time FIR filter, each value of the output sequence is a weighted sum of the (N+1) most recent input values:

{\displaystyle {\begin{aligned}y[n]&=h_{0}x[n]+h_{1}x[n-1]+\cdots +h_{N}x[n-N]\\&=\sum _{i=0}^{N}h_{i}\cdot x[n-i],\end{aligned}}}

where:

• ${\displaystyle \scriptstyle x[n]}$ is the input signal,
• ${\displaystyle \scriptstyle y[n]}$ is the output signal,
• ${\displaystyle \scriptstyle \{h_{i}\}}$, is the impulse response of the filter.
• ${\displaystyle \scriptstyle N}$ is the filter order; an ${\displaystyle \scriptstyle N}$th-order filter has ${\displaystyle \scriptstyle (N\,+\,1)}$ terms on the right-hand side.

This summation is also known as a discrete convolution.

### Direct Form

A discrete-time FIR filter of order N. The top part is an N-stage delay line with N + 1 taps. Each unit delay is a z−1 operator in Z-transform notation.

For a direct form discrete-time FIR filter, as shown in the figure, the formula for the output can be written down by inspection:

{\displaystyle {\begin{aligned}y[n]&=b_{0}x[n]+b_{1}x[n-1]+\cdots +b_{N}x[n-N]\\&=\sum _{i=0}^{N}b_{i}\cdot x[n-i],\end{aligned}}}

where:

• ${\displaystyle x[n]}$ is the input signal,
• ${\displaystyle y[n]}$ is the output signal,
• ${\displaystyle \{b_{i}\}}$, a set of constants (coefficients) that define the filter.
• ${\displaystyle N}$ is the filter order; an ${\displaystyle \scriptstyle N}$th-order filter has ${\displaystyle (N\,+\,1)}$ terms on the right-hand side.

The ${\displaystyle x[n-i]}$ in the direct form are commonly referred to as taps, based on the structure of a tapped delay line that in many implementations provides the delayed inputs to the multiplication operations. One may speak of a 5th order/6-tap filter, for instance.

In the direct form FIR filter, the set of coefficients that define the filter are identical to the set of values that define the impulse response: ${\displaystyle \{b_{i}\}=\{h_{i}\}}$

### Other Forms

Not all FIR filters have the property that their coefficients are equal to their impulse responses. For an FIR lattice filter, as shown in the figure, the first value of the impulse response is 1 and the last value is ${\displaystyle k_{N}}$. The other values are complicated functions of all the coefficients. Lattice filters are used often preferred in adaptive situations because they converge faster.

A cascade of two or more FIR filters of any type will create a composite filter with an impulse response that is different from the coefficients of the individual filters, even if the component filters are all direct form FIR filter.

________________________________________________________________________________________________________________

Adaptive linear combiner showing the combiner and the adaption process. k = sample number, n=input variable index, x = reference inputs, d = desired input, W = set of filter coefficients, ε = error output, Σ = summation, upper box=linear combiner, lower box=adaption algorithm.
Adaptive linear combiner, compact representation. k = sample number, n=input variable index, x = reference inputs, d = desired input, ε = error output, Σ = summation.

### FIR

some symbols “rrr‘±–5 —5 τ ε ΘΦθφΚκΛλμώωΩ εk

${\displaystyle f}$
${\displaystyle \mathbf {W} _{n}=\left[w_{n}(0),\,w_{n}(1),\,...,\,w_{n}(p)\right]^{T}}$.
${\displaystyle x(n)=g_{x}(n)+u_{x}(n)+v_{x}(n)}$

For convenience, the following column vectors are defined:

${\displaystyle \mathbf {X} (n)=\sum _{k=0}^{p}w_{n}(k)x(n-k)}$,

${\displaystyle {\hat {g}}(n)=\sum _{k=0}^{p}w_{n}(k)g_{x}(n-k)}$
${\displaystyle {\hat {u}}(n)=\sum _{k=0}^{p}w_{n}(k)u_{x}(n-k)}$
${\displaystyle {\hat {v}}(n)=\sum _{k=0}^{p}w_{n}(k)v_{x}(n-k)}$
${\displaystyle {\hat {d}}(n)=\sum _{k=0}^{p}w_{n}(k)x(n-k)={\hat {g}}(n)+{\hat {u}}(n)+{\hat {v}}(n)}$

${\displaystyle \mathbf {U} (n)=\left[u_{x}(n),\,u_{x}(n-1),\,...,\,u_{x}(n-p)\right]^{T}}$
${\displaystyle \mathbf {V} (n)=\left[v_{x}(n),\,v_{x}(n-1),\,...,\,v_{x}(n-p)\right]^{T}}$

Each vector has ${\displaystyle p+1}$ components.

For convenience, the following vector dot products are defined:

${\displaystyle {\hat {g}}(n)=\mathbf {W} _{n}\cdot \mathbf {G} (n)}$
${\displaystyle {\hat {u}}(n)=\mathbf {W} _{n}\cdot \mathbf {U} (n)}$
${\displaystyle {\hat {v}}(n)=\mathbf {W} _{n}\cdot \mathbf {V} (n)}$
${\displaystyle {\hat {d}}(n)=\mathbf {W} _{n}\cdot \mathbf {X} (n)={\hat {g}}(n)+{\hat {u}}(n)+{\hat {v}}(n)}$

## The ideal transformer

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Ideal transformer circuit diagram

some statement,.[23] [l]

• Hayt, William; Kemmerly, Jack E. (1993), Engineering Circuit Analysis (5th ed.), McGraw-Hill, ISBN 007027410X

begin testing

An ideal transformer is defined as Vs - a Vp, Is = Ip/a Reitz, Milford & Christy (1993, p. 323)

An ideal transformer is defined as Vs - a Vp, Is = Ip/a [22]

An ideal transformer is defined as Vs - a Vp, Is = Ip/a Reitz, Milford & Christy 1993, p. 323

end testing

The ideal condition assumptions are:

• The windings of the transformer have negligible resistance, so RP= RS= 0, where RP represents the resistance of the primary winding and RS represents the resistance of the secondary winding. Thus, there is no copper loss in the winding, and hence no voltage drop.
• Flux is confined within the core. Therefore, it is the same flux that links both the windings.
• Permeability of the core is infinitely high which implies that zero mmf (current) is required to set up the flux and that the flux in the core due to the primary winding must be equal and opposite to the flux due to the secondary winding. There is therefore zero net flux in the core.
• The core does not incur any hysteresis or eddy current loss. Hence, no core losses.

If the secondary is an open circuit, an ideal transformer will not allow the flow of primary current.

Consider the ideal, lossless, perfectly-coupled transformer shown in the circuit diagram at right having primary and secondary windings with NP and NS turns, respectively.

The ideal transformer induces secondary voltage ES =VS as a proportion of the primary voltage VP = EP and respective winding turns as given by the equation

Eq. 1 ${\displaystyle {\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {E_{\text{P}}}{E_{\text{S}}}}={\frac {N_{\text{P}}}{N_{\text{S}}}}=a}$,

where,

- VP/VS = EP/ES = a is the voltage ratio and NP/NS = a is the winding turns ratio, the value of these ratios being respectively higher and lower than unity for step-down and step-up transformers,.[25][26][m][n]
- VP designates source impressed voltage,
- VS designates output voltage, and,
- EP & ES designate respective emf induced voltages.[o]

According to the equation shown above, when the number of turns in the primary coil is greater than the number of turns in the secondary coil, the secondary voltage must be less than the primary voltage. On the other hand, when the number of turns in the primary coil is less than the number of turns in the secondary, the secondary voltage must be greater than the primary voltage.

Any load impedance ${\displaystyle Z_{L}}$ connected to the ideal transformer's secondary winding causes current to flow without losses from primary to secondary circuits, the resulting input and output apparent power therefore being equal as given by the equation

Eq. 2 ${\displaystyle I_{P}\times V_{P}=I_{S}\times V_{S}}$.

Combining the two equations yields the following ideal transformer identity

${\displaystyle {\frac {V_{P}}{V_{S}}}={\frac {I_{S}}{I_{P}}}=a}$.

This formula is a reasonable approximation for the typical commercial transformer, with voltage ratio and winding turns ratio both being inversely proportional to the corresponding current ratio.

The load impedance ${\displaystyle Z_{L}}$ is defined in terms of secondary circuit voltage and current as follows

${\displaystyle Z_{L}={\frac {V_{L}}{I_{L}}}={\frac {V_{S}}{I_{S}}}}$.

The apparent impedance ${\displaystyle Z_{L}^{\prime }}$ of this secondary circuit load referred to the primary winding circuit is governed by a squared turns ratio multiplication factor relationship derived as follows[28][29]

${\displaystyle Z_{L}^{\prime }={\frac {V_{P}}{I_{P}}}={\frac {aV_{S}}{I_{S}/a}}=a^{2}\times {\frac {V_{S}}{I_{S}}}=a^{2}\times {Z_{L}}}$.

For an ideal transformer, the power supplied in the primary and the power dissipated by the load are both equal to the following equation:[30][31]

${\displaystyle P={\frac {V_{\text{S}}^{2}}{R_{\text{L}}}}=a^{2}{\frac {V_{\text{p}}^{2}}{R_{\text{L}}}}}$

The primary current is equal to the following equation:[30][31]

${\displaystyle I_{\text{p}}={\frac {P_{\text{R}}}{V_{p}}}=a^{2}{\frac {V_{\text{p}}}{R_{\text{L}}}}}$

The equivalent resistance on the source from the load sections and transformer is equal to the following equation:[30][31]

${\displaystyle R_{\text{eq}}={\frac {V_{\text{p}}}{I_{\text{p}}}}=a^{2}{R_{\text{L}}}}$

#### Induction law

The transformer is based on two principles: first, that an electric current can produce a magnetic field and second that a changing magnetic field within a coil of wire induces a voltage across the ends of the coil (electromagnetic induction). Changing the current in the primary coil changes the magnetic flux that is developed. The changing magnetic flux induces a voltage in the secondary coil.

Referring to the two figures here, current passing through the primary coil creates a magnetic field. The primary and secondary coils are wrapped around a core of very high magnetic permeability, usually iron,[p] so that most of the magnetic flux passes through both the primary and secondary coils. Any secondary winding connected load causes current and voltage induction from primary to secondary circuits in indicated directions.

Ideal transformer and induction law

The voltage induced across the secondary coil may be calculated from Faraday's law of induction, which states that:

${\displaystyle V_{\text{S}}=E_{\text{S}}=N_{\text{S}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}.}$

where Vs = Es is the instantaneous voltage, Ns is the number of turns in the secondary coil, and dΦ/dt is the derivative[q] of the magnetic flux Φ through one turn of the coil. If the turns of the coil are oriented perpendicularly to the magnetic field lines, the flux is the product of the magnetic flux density B and the area A through which it cuts. The area is constant, being equal to the cross-sectional area of the transformer core, whereas the magnetic field varies with time according to the excitation of the primary. Since the same magnetic flux passes through both the primary and secondary coils in an ideal transformer,[28] the instantaneous voltage across the primary winding equals

${\displaystyle V_{\text{P}}=E_{\text{P}}=N_{\text{P}}{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}.}$

Taking the ratio of the above two equations gives the same voltage ratio and turns ratio relationship shown above, that is,

${\displaystyle {\frac {V_{\text{P}}}{V_{\text{S}}}}={\frac {E_{\text{P}}}{E_{\text{S}}}}={\frac {N_{\text{P}}}{N_{\text{S}}}}=a}$.

The changing magnetic field induces an emf across each winding.[32] The primary emf, acting as it does in opposition to the primary voltage, is sometimes termed the counter emf.[33] This is in accordance with Lenz's law, which states that induction of emf always opposes development of any such change in magnetic field.

As still lossless and perfectly-coupled, the transformer still behaves as described above in the ideal transformer.

#### Polarity

Instrument transformer, with polarity dot and X1 markings on LV side terminal

A dot convention is often used in transformer circuit diagrams, nameplates or terminal markings to define the relative polarity of transformer windings. Positively-increasing instantaneous current entering the primary winding's dot end induces positive polarity voltage at the secondary winding's dot end.[34][35][36][r][s]

## Photos of Doppler 019

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Vestibule and Quonset hut housing a Transit satellite tracking station 019. 1. Triad satellite magnetometer down load antenna. 2. flag pole, 3. Utility pole in background, 4 Revolving light temperature alarm, 5 VLF antenna, 6-9 Doppler satellite tracking antennas, 10. stove pipe for heater, 11 Flood light for low visibility conditions, 12 fuel tank.

Some of the equipment inside Transit satellite tracking station 019. 1. Automatic Control Unit, 2. timer-counter, 3. Time burst detector, 4. time conversion chart, 5. satellite ephemeris, 6. tracking receiver, 7. time display, 8 Header-Tailer programmer, 9. Digitizer and main clock, 10. master oscillator, 11. strip chart recorder, 12. paper tape punch, 13. short wave receiver. Out of site: VLF receiver, refraction correction unit, backup battery system, power supplies, AC voltage regulators.

## Wien bridge oscillator

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### Analyzed from Loop Gain

In this version of the oscillator, Rb is a small incandescent lamp. Usually R1 = R2 = R and C1 = C2 = C. In normal operation, Rb self heats to the point where its resistance is Rf/2.

According to Schilling[39] the loop gain of the Wien bridge oscillator is given by

${\displaystyle T=({\frac {R_{1}/(1+sC_{1}R_{1})}{R_{1}/(1+sC_{1}R_{1})+R_{2}+1/(sC_{2})}}-{\frac {R_{b}}{R_{b}+R_{f}}})A_{0}\,}$

where ${\displaystyle A_{0}\,}$ is the frequency dependent gain of the op-amp. (Note, the component names in Schilling have been replace with the component names in the figure.)

Schilling further says that the conditon of oscillation is ${\displaystyle T=1\,}$. Which, assuming ${\displaystyle R_{1}=R_{2}=R\,}$ and ${\displaystyle C_{1}=C_{2}=C\,}$ is satisfied by

${\displaystyle \omega ={\frac {1}{RC}}\rightarrow F={\frac {1}{2\pi RC}}\,}$

and

${\displaystyle {\frac {R_{f}}{R_{b}}}={\frac {2A_{0}+3}{A_{0}-3}}\,}$ with ${\displaystyle \lim _{A_{0}\rightarrow \infty }{\frac {R_{f}}{R_{b}}}=2\,}$

## Miller effect diagrams

An ideal voltage inverting amplifier with an impedance connecting output to input.