# User:Constant314

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User:Constant314/Space Cloth

User:Constant314/Leapfrog Filter

Causality#Fields#Science#Engineering

==To Do

==

This topic should always be on the top.

Create Leapfrog Filer page. Create Space-cloth page. Add alternate explanations of skin effect. Add magnetic vector potential to the transformer page.

## 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:

\begin{align} 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{align}

\begin{align} x_\mathrm{k} &= \sum_{\mathrm{k}=1}^{\mathrm{N}} ( M_\mathrm{kj} ) y_\mathrm{j} \end{align}
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.

\begin{align} \sum_{k=1}^{\mathrm{N}} c_{kj} x_k &= y_j \end{align}

These equations can be rewritten as

\begin{align} \sum_{k=1}^{N} c_{kj} x_k - y_j &= 0 \end{align}

and further rewritten as

\begin{align} \sum_{k=1}^{N} c_{kj} x_k +x_j - y_j &= x_j \end{align}

and finally rewritten as

\begin{align} \sum_{k=1}^{N} ( c_{kj} + \delta_{ij}) x_k - y_j &= x_j \end{align}
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

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\begin{align} x_j &= \sum_{k=1}^{N} t_{kj} x_k \end{align}

with the understanding that all the equations have this interpretation

\begin{align} ( \mathrm{effect \ at \ j} ) &= \sum_{k=1}^{N} ( \mathrm{transmission \ from \ k \ to \ j } )( \mathrm{cause \ at \ k} ) \end{align}

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,[1] 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[2] 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 [3]

## Regarding Skin Effect

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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 [4]

regarding inductance at the center of the wire [5]

Bedell[6]

_________________________________________________________

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[7][8] 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.

$\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

$\Phi_\mathrm{B} = \iint\limits_{\Sigma(t)} \mathbf{B}(\mathbf{r}, t) \cdot d \mathbf{A}\ ,$

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

[6]

[8]

[9]

## Quantities and units

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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[10] 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

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Testing

These work.

Causality#Engineering
Causality#Physics

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

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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" [11]

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 [12]

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

end testing

## References

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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

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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:

$\frac{\partial^2}{{\partial x}^2} V = L C \frac{\partial^2}{{\partial t}^2} V + (R C + G L) \frac{\partial}{\partial t} V + G R V$
$\frac{\partial^2}{{\partial x}^2} I = L C \frac{\partial^2}{{\partial t}^2} I + (R C + G L) \frac{\partial}{\partial t} I + G R I$

After:

$An equation for the second partial derivative of voltage with respect to distance$
$An equation for the second partial derivative of current with respect to distance$

## equation with alt text

________________________________________________________________________________________________________________

$Z_\mathsf S = Z_\mathsf L^* \,$
$Z_\mathrm{load} = Z_\mathrm{source}^* \,$
$load impedance equals the complex conjugate of the source impedance$
$Z_\text{load} = Z_\text{source}^* \,$

## 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:

\begin{align} 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{align}

where:

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

\begin{align} 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{align}

where:

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

The $x[n-i]$ 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: $\{ b_{i} \} = \{ h_{i} \}$

### 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 $k_N$. 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.

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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

$f$
$\mathbf{W}_{n}=\left[w_{n}(0),\,w_{n}(1),\, ...,\,w_{n}(p)\right]^{T}$.
$x(n) = g_x(n) + u_x(n) + v_x(n)$

For convenience, the following column vectors are defined:

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

$\hat{g}(n)= \sum_{k=0}^p w_{n}(k)g_x(n-k)$
$\hat{u}(n)= \sum_{k=0}^p w_{n}(k)u_x(n-k)$
$\hat{v}(n)= \sum_{k=0}^p w_{n}(k)v_x(n-k)$
$\hat{d}(n)= \sum_{k=0}^p w_{n}(k)x(n-k) = \hat{g}(n) + \hat{u}(n) + \hat{v}(n)$

$\mathbf{U}(n)=\left[u_x(n),\,u_x(n-1),\,...,\,u_x(n-p)\right]^{T}$
$\mathbf{V}(n)=\left[v_x(n),\,v_x(n-1),\,...,\,v_x(n-p)\right]^{T}$

Each vector has $p + 1$ components.

For convenience, the following vector dot products are defined:

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

## The ideal transformer

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Ideal transformer circuit diagram

some statement,.[13] [a]

• 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 [12]

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

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,.[15][16][b][c]
- VP designates source impressed voltage,
- VS designates output voltage, and,
- EP & ES designate respective emf induced voltages.[d]

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 $Z_L$ 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 $I_P \times V_P = I_S \times V_S$.

Combining the two equations yields the following ideal transformer identity

$\frac{V_P}{V_S} = \frac{I_S}{I_P} = a$.

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 $Z_L$ is defined in terms of secondary circuit voltage and current as follows

$Z_L = \frac{V_L}{I_L}=\frac{V_S}{I_S}$.

The apparent impedance $Z_L^\prime$ of this secondary circuit load referred to the primary winding circuit is governed by a squared turns ratio multiplication factor relationship derived as follows[18][19]

$Z_L^\prime=\frac{V_P}{I_P}=\frac{aV_S}{I_S/a}=a^2\times\frac{V_S}{I_S}=a^2\times{Z_L}$.

For an ideal transformer, the power supplied in the primary and the power dissipated by the load are both equal to the following equation:[20][21]

$P = \frac{V^2_{\text{S}}}{R_{\text{L}}} = a^2\frac{ {V^2_{\text{p}}}}{{R_{\text{L}}}}$

The primary current is equal to the following equation:[20][21]

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

The equivalent resistance on the source from the load sections and transformer is equal to the following equation:[20][21]

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

#### 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,[e] 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:

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

where Vs = Es is the instantaneous voltage, Ns is the number of turns in the secondary coil, and dΦ/dt is the derivative[f] 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,[18] the instantaneous voltage across the primary winding equals

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

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

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

The changing magnetic field induces an emf across each winding.[22] The primary emf, acting as it does in opposition to the primary voltage, is sometimes termed the counter emf.[23] 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.[24][25][26][g][h]

## 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[29] the loop gain of the Wien bridge oscillator is given by

$T = ( \frac { R_1 / (1 + sC_1 R_1) } {R_1 / (1 + sC_1 R_1) + R_2 + 1/(sC_2)} - \frac {R_b} {R_b + R_f } )A_0 \,$

where $A_0 \,$ 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 $T = 1 \,$. Which, assuming $R_1 = R_2 = R\,$ and $C_1 = C_2 = C \,$ is satisfied by

$\omega = \frac {1} {R C} \rightarrow F = \frac {1} {2 \pi R C}\,$

and

$\frac {R_f} {R_b} = \frac {2 A_0 + 3} {A_0 - 3} \,$ with $\lim_{A_0\rightarrow \infin} \frac {R_f} {R_b} = 2 \,$

## Miller effect diagrams

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