User:Constant314

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Regarding Faraday's Law[edit]

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[1][2] 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.




[3]


[2]


[1]

[4]

Quantities and units[edit]

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[5] 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[edit]


linking to a subsection[edit]

Testing


These work.

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.

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[edit]

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" [6]

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 [7]

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

end testing

References[edit]

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 

Testing a link[edit]

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

this talk page comment

Fix up equations from Telegrapher's Equations[edit]

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[edit]

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}^* \,




Zload = Zsource*

Definition[edit]

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 Nth-order filter has \scriptstyle (N \,+\, 1) terms on the right-hand side.

This summation is also known as a discrete convolution.

Direct Form[edit]

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 Nth-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[edit]

FIR Lattice Filter.png

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.

Adaptive Filtering[edit]

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[edit]

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[edit]

Ideal transformer circuit diagram

some statement,.[8] [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 [7]

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,.[10][11][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[13][14]

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:[15][16]


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:[15][16]


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:[15][16]


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

Induction law[edit]

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,[13] 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.[17] The primary emf, acting as it does in opposition to the primary voltage, is sometimes termed the counter emf.[18] 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[edit]

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.[19][20][21][g][h]

Photos of Doppler 019[edit]

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[edit]

Analyzed from Loop Gain[edit]

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[24] 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[edit]

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


External links[edit]

One of the best usenet discussions on Poynting vector and wires and angular momentum. Some day I'm going to write it up as a dialog between the Tortoise, Achilles and some other characters.

Quanitization Noise[edit]

When a continuous signal is discretized the difference between the continuous signal and the discretized signal is an error. Strictly speaking this error is distortion since the same signal discretized repeatedly results in the same error. If a periodic signal like a sine wave is synchronously sampled and discretized then the discretized signal will exhibit harmonic distortion. However, even though it is actually distortion, it can be analyzed as noise. If the discretization is uniform and the width of the discretization interval is   \alpha \,, then the noise power, n, is   n = \frac {\alpha^2} {12} \, [25].


The interpretation of phase for mathematics vs. engineering[edit]

Consider the functions:

u(t) = 0.5\cdot \cos( 2 \pi f t ) + 0.866\cdot \sin( 2 \pi f t ) \,
v(t) = 0.866\cdot \cos( 2 \pi f t ) + 0.5\cdot \sin( 2 \pi f t ) \,


In mathematics, cos( 2 \pi f t )\, and sin( 2 \pi f t )\, are considered to be basis vectors of a two dimensional vector space. The function  v(t) \, would be represented by a point with coordinates (0.866, 0.5). The line connecting the origin to this point makes a positive angle of 30 degrees with respect to the origin. Therefore in mathematics, the phase of  v(t) \, would be 30 degrees. Likewise,  u(t) \, would be represented by (.5,0.866) and its phase would be 60 degrees. The phase difference between  u(t) \, and  v(t) \, would be 30 degrees.


In engineering, a sinusoid has a negative phase shift with respect to some other sinusoid, if the peaks of the first sinusoid occurs after the peaks of the second sinusoid. Using a trig identity,  u(t) \, and  v(t) \, can be written as:

u(t) = cos( 2 \pi f t - \pi/3 ) \, note: \pi /3 =  \, 60 degrees.
v(t) = cos( 2 \pi f t - \pi/6 ) \, note: \pi /6 =  \, 30 degrees.


The peaks of  v(t) \, and  u(t) \, occur after the peak of  cos( 2 \pi f t ) \, by 30 and 60 degrees respectively. Therefore in engineering, the phase of  v(t) \, and  u(t) \, would be -30 degrees and -60 degrees respectively. The phase difference of  u(t) \, with respect to  v(t) \, would be -30 degrees meaning that the peaks of  u(t) \, occur after the peaks of  v(t) \, by 30 degrees .

In most cases, consistantly using the changing the sign of the phase only changes the sign of the imaginary part of the computation. In othere words, the result of using one convention or the other produces results that are conjugates of each other.

One place this shows up is in the definition of the Fourier transform.


Engineering Time Convention[edit]

Here are the missing references

  x(t)= \frac {1} {2 \pi} \int_{-\infty}^{\infty}  X(\omega)e^{j \omega t} d\omega

  • from Oppenheim, Alan V.; Willsky, Alan S.; Young, Ian T. (1983), Signals and Systems (1st ed.), Prentice-Hall, ISBN 0138097313

The following use the same convention:

  • Gregg, W. David (1977), Analog & Digital Communication, John Wiley, ISBN 0471326615
  • Stein, Seymour; Jones, J. Jones (1967), Modern Communnication Principles, McGraw-Hill, page 4, equation 1-5
  • Hayt, William; Kemmerly, Jack E. (1971), Engineering Circuit Analysis (2nd ed.), McGraw-Hill, ISBN 0070273820, page 535, equation 8b.


  h(t)= \frac {1} {2 \pi} \int_{-\infty}^{\infty} e^{-j \omega t} H(\omega)d\omega

  • from Press, William H.; Teukolsky, Saul A.; Vetterling, William T. (2007), Numerical Recipes (3rd ed.), Cambridge University Press, ISBN 9780521880688

The following use the same convention:

  • Jackson, John Davd (1999), Classical Electrodynamics (3rd ed.), John-Wiley, ISBN 047130932X, page 372, equation 8.89
  • Stratton, Julius Adams (1941), Electromagnetic Theory, McGraw-Hill page 294, equation 47
  • Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993), Foundations of Electromagnetic Theory, Addison-Wesley, ISBN 0201526247, page 607, equation VI-2



I've looked in 11 references and found two ways to write down the equation for a plane wave.

(In all cases I have changed i to j for consistancy.)

The following group use a form for the plane wave that involves  -j \omega t \, such as  e^{j(\mathbf{k} \cdot \mathbf{x} - \omega t) } \,

  • Griffiths, David (1989), Introduction to Electrodynamics, Prentice-Hall, ISBN 013481374X, page 356, equation 8.60
  • Jackson, John Davd (1999), Classical Electrodynamics (3rd ed.), John-Wiley, ISBN 047130932X, page 296, equation 7.8
  • Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (1993), Foundations of Electromagnetic Theory, Addison-Wesley, ISBN 0201526247, page 416, equation 17-7


The following group use a form for the plane wave that involves  +j \omega t \, such as   e^{j(\omega t - \mathbf{k} \cdot \mathbf{x}) } \,

  • Crawford, Frank S. (1968), Waves - berkeley phisics course - volume 3, McGraw-Hill, page, page 333, equation 2-1
  • Harrington, Roger F. (1961), Time-Harmonic Electromagnetic Fields, McGraw-Hill, page 39, equation 2-13
  • Hayt, William (1989), Engineering Electromagnetics (5th ed.), McGraw-Hill, ISBN 0070274061, page 338, equation 6
  • Jordan, Edward; Balmain, Keith G. (1968), Electromagnetic Waves and Radiating Systems (2nd ed.), Prentice-Hall, page 124,
  • Marshall, Stanley V. (1987), Electromagnetic Concepts & Applications (1st ed.), Prentice-Hall, ISBN 0132490048, page 320, equation 12
  • Ramo, Simon; Whinnery, John R.; van Duzer, The odore (1965), Fields and Waves in Communication Electronics, John Wiley, page 247, equation 10
  • Sadiku, Matthew N. O. (1989), Elements of Electromagnetics (1st ed.), Saunders College Publishing, ISBN 993013846, page 432 equation 10.4
  • Kraus, John D. (1984), Electromagnetics (3rd ed.), McGraw-Hill, ISBN 0070354235, page 385, equation 29.

I think this list is enough to establish that there is agroup that uses  e^{+j \omega t} \, and a group that uses  e^{-j \omega t} \,

But which are the physicists? Jackson and Griffiths are physicists for sure, but so is Crawford. Hayt, Harrington, Marshall, Whinnery and Jordan are engineers. Kraus is an engineer but says he can use either convention.

So what is the difference? Not much, because at the end of the computation you discard the imaginary part and keep the real part. You wind up with terms of either  \cos(\omega t  - \mathbf{k} \cdot \mathbf{x})\, or  \cos(\mathbf{k} \cdot \mathbf{x} - \omega t)\, which are equal.

So I propose to change the sentence

" Mathematically, a plane wave is a wave of the following form:

u(\mathbf{x},t) = A e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}

where i is the imaginary unit, k is the wave vector, ω is the angular frequency, and A is the (complex) amplitude. This form of the plane wave uses the physics time convention; in the engineering time convention[citation needed], –j is used instead of +i in the exponent. The physical solution is found by taking the real part of this expression:

Re[u(\mathbf{x},t)] = |A| \cos (\mathbf{k}\cdot\mathbf{x} - \omega t + \arg A)

" and replacing it with

Mathematically, a plane wave is a wave of the following form:

u(\mathbf{x},t) = A e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}

where i is the imaginary unit, k is the wave vector, ω is the angular frequency, and A is the (complex) amplitude. This form of the plane wave uses the physics time convention; in the engineering time convention[citation needed], –j is used instead of +i in the exponent. The physical solution is found by taking the real part of this expression:

Re[u(\mathbf{x},t)] = |A| \cos (\mathbf{k}\cdot\mathbf{x} - \omega t + \arg A)



But, what is that quantity that is the argument of the cosine function? How are we to interpret  -\omega t \, ? Is it negative time or negative frequency? I don't think so. That only leaves  -\omega t \, as value that gets more and more negative as time increases and  \omega t \, gets more and more positve for the second group. I suppose it was a mistake to say that physicists do it one way and engineers a different way. But there are two camps and two ways and they differ by sign.


Leading and lagging[edit]

The terms leading and lagging are used by engineers to describe the phase difference of one sinusoid to another. If the peaks of one sinusoid occur after the peaks of a second sinusoid, then the first sinusoid is said to lag the second sinusoid. If the peaks of the first sinusoid occur before peaks of the second sinusoid, then the first sinusoid is said to lead the second sinusoid.
For example, sin(t) lags cos(t) by 90 degrees and cos(t) leads sin(t) by 90 degrees. Although sin(t) could be said to lead cos(t) by 270 degrees it is conventional to choose leading or lagging so that the angle is less than or equal to 180 degrees.
In electric power distribution, leading and lagging power factor always refers to current relative to voltage. Thus a leading power factor means the current is leading the voltage.

Trying out a template[edit]


Collapsing two-port to a one port[edit]

A two-port network has four variables with two of them being independent. If one of the ports is terminated by a load with no independent sources, then the load enforces a relationship between the voltage and current of that port. A degree of freedom is lost. The circuit now has only one independent parameter. The two-port becomes a one-port impedance to the remaining independent variable.

For example , consider impedance parameters


 \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \begin{bmatrix} z_{11} & z_{12} \\ z_{21} & z_{22} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \end{bmatrix}


Connecting a load, ZL onto port 2 effectively adds the constraint

 V_2 = -Z_L I_2 \,

The negative sign is because the positive direction for I2 is directed into the two-port instead of into the load. The augmented equations become

 V_1 = Z_{11} I_1  +  Z_{12} I_2 \,
 -Z_L I_2 = Z_{21} I_1  +  Z_{22} I_2 \,


The second equation can be easily solved for I2 as a function of I1 and that function can replace I2 in the first equation leaving V1 ( and V2 and I2 ) as functions of I1


 I_2 = I_1 \frac {-Z_{21}} {Z_L + Z_{22}}    \,


 V_1 = Z_{11} I_1  -  \frac {Z_{12} Z_{21}} {Z_L + Z_{22}} I_1  = (Z_{11}  -  \frac {Z_{12} Z_{21}} {Z_L + Z_{22}}) I_1 = Z_{in} I_1 \,


So, in effect, I1 sees an input impedance  Z_{in} \, and the two-port's effect on the input circuit has been effectively collapsed down to a one-port i.e. a simple two terminal impedance.

Component values for resistive pads and attenuators[edit]

This section concerns pi-pads, T-pads and L-pads made entirely from resisters and terminated on each port with a purely real resistance.

  • All impedances, currents, voltages and two-port parameters will be assumed to be purely real. For practical applications, this assumption is often close enough.
  • The pad is designed for a particular load impedance, ZLoad, and a particular source impedance, Zs.
  • The impedance seen looking into the input port will be ZS if the output port is terminated by ZLoad.
  • The impedance seen looking into the output port will be ZLoad if the input port is terminated by ZS.


Reference figures for attenuator component calculation[edit]

This circuit is used for the general case, all T-pads, all pi-pads and L-pads when the source impedance is greater than or equal to the load impedance.
The L-pad computation assumes that port 1 has the highest impedance. If the highest impedance happens to be the output port, then use this figure.
Unique resister designations for Tee, Pi and L pads.

The attenuator two-port is generally bidirectional. However in this section it will be treated as though it were one way. In general, either of the two figures above applies, but the figure on the left (which depicts the source on the left) will be tacitly assumed most of the time. In the case of the L-pad, the right figure will be used if the load impedance is greater than the source impedance.

Each resister in each type of pad discussed is given a unique designation to decrease confusion.

The L-pad component value calculation assumes that the design impedance for port 1 (on the left) is equal or higher than the design impedance for port 2.


Terms used[edit]

  • Pad will include pi-pad, T-pad, L-pad, attenuator, and two-port.
  • Two-port will include pi-pad, T-pad, L-pad, attenuator, and two-port.
  • Input port will mean the input port of the two-port.
  • Output port will mean the output port of the two-port.
  • Symmetric means a case where the source and load have equal impedance.
  • Loss means the ratio of power entering the input port of the pad divided by the power absorbed by the load.
  • Insertion Loss means the ratio of power that would be delivered to the load if the load were directly connected to the source divided by the power absorbed by the load when connected through the pad.

Symbols used[edit]

Passive, resistive pads and attenuators are bidirectional two-ports, but in this section they will be treated as unidirectional.

  • ZS = the output impedance of the source.
  • ZLoad = the input impedance of the load.
  • Zin = the impedance seen looking into the input port when ZLoad is connected to the output port. Zin is a function of the load impedance.
  • Zout = the impedance seen looking into the output port when Zs is connected to the input port. Zout is a function of the source impedance.
  • Vs = source open circuit or unloaded voltage.
  • Vin = voltage applied to the input port by the source.
  • Vout = voltage applied to the load by the output port.
  • Iin = current entering the input port from the source.
  • Iout = current entering the load from the output port.
  • Pin = Vin Iin = power entering the input port from the source.
  • Pout = Vout Iout = power absorbed by the load from the output port.
  • Pdirect = the power that would be absorbed by the load if the load were connected directly to the source.
  • Lpad = 10 log10 (Pin / Pout ) always. And if Zs = ZLoad then Lpad = 20 log10 (Vin / Vout ) also. Note, as defined, Loss ≥ 0 dB
  • Linsertion = 10 log10 (Pdirect / Pout ). And if Zs = ZLoad then Linsertion = Lpad.
  • Loss ≡ Lpad. Loss is defined to be Lpad.


Symmetric T pad resister calculation[edit]

A = 10^{-Loss/20}  \qquad R_a = R_b = Z_S \frac {1 - A} {1 + A} \qquad R_c =   \frac {Z_s^2 - R_b^2   } {2 R_b } \,
 \text{Example:  } Z_s=Z_{Load}=100 \quad Loss=\;6.0204~dB \; \implies \; A = 0.5 \qquad	R_a = R_b = 100 \frac {.5} {1.5} = 33.33 \qquad R_c = \frac {100^2 - 33.33^2 } {2 \cdot 33.33 } =133.34 \; \,

see Valkenburg p 11-3[26]

Symmetric pi pad resister calculation[edit]

A = 10^{-Loss/20}   \qquad  R_x = R_y = Z_S \frac {1 + A} {1 - A} \qquad R_z = \frac {2R_x}{\left ( \frac {R_x}{Z_0} \right ) ^2 -1} \,
 \text{Example:} \; Z_s = Z_{load} = 100 \Omega \quad Loss = 6.0204dB \, \implies \; A = 10^{-6.0204/20}   = 0.5 \quad R_x = R_y =   100 \frac {1 + 0.5} {1 - 0.5}  =   300  \Omega \quad  R_z = \frac {2 \cdot 300}{\left ( \frac {300}{100} \right ) ^2 -1}  = 75  \Omega  |; \,

see Valkenburg p 11-3[27]

L-Pad for impedance matching resister calculation[edit]

If a source and load are both resistive (i.e. Z1 and Z2 have zero or very small imaginary part) then a resistive L-pad can be used to match them to each other. As shown, either side of the L-pad can be the source or load, but the Z1 side must be the side with the higher impedance.

 
R_q = \frac {Z_m} {\sqrt {\rho - 1  } }    \qquad 
R_p = Z_m \sqrt {\rho - 1  }     \qquad   
Loss = 20 \log_{10} \left ( \sqrt{ \rho - 1 } + \sqrt{\rho } \quad \right  ) \quad \text{where} \quad 
\rho = \frac {Z_1}{Z_2}     \quad
Z_m = \sqrt{  Z_1 Z_2}  \text{   } \, see Valkenburg p 11-3[28]

Large positive numbers means loss is large. The loss is a monotonic function of the impedance ratio. Higher ratios require higher loss.


 \text{Example:} \quad Z_1=135 \Omega \quad  Z_2=100 \Omega  \; \implies \; R_q = \frac {100} {\sqrt {1 - 100/135  } } = 196.4 \Omega \qquad  R_p = \frac {135 \cdot 100} {196.4} = 68.74  \Omega \,

Converting T-pad to pi-pad[edit]

 
R_z =  \frac {R_a R_b + R_a R_c + R_b R_c}  {R_c} \qquad
R_x =  \frac {R_a R_b + R_a R_c + R_b R_c}  {R_b} \qquad
R_y =  \frac {R_a R_b + R_a R_c + R_b R_c}  {R_a}.  \qquad \text{See Hayt, page 494, problem 8. } \, [29]


 \text{Example:  } 
R_a=83.35, \quad R_b=40.57, \quad R_c=81.65 \qquad \implies \qquad 
R_z = 165.34, 
\quad R_x =  332.76,
\quad R_y =161.97
 \,


Converting pi-pad to T-pad[edit]

 R_c = \frac {R_x R_y} {R_x + R_y + R_z} \qquad 

R_a = \frac {R_z R_x} {R_x + R_y + R_z}  \qquad 

R_b = \frac {R_z R_y} {R_x + R_y + R_z}  \qquad \text{see Hayt, page 494, problem, 8.}

\, [30]


 \text{Example:  } 
R_z=100.62, \quad R_x=437.831, \quad R_y=68.935  \qquad \implies \qquad 
R_c = 49.692, 
\quad R_a =  72.532,
\quad R_b =11.421  \,

Conversion between two-ports and pads[edit]

T-pad to impedance parameters[edit]

The impedance parameters for a passive two-port are
  V_1 = Z_{11} I_1  +  Z_{12} I_2 \qquad  V_2 = Z_{21} I_1  +  Z_{22} I_2 \qquad with \qquad   Z_{12} = Z_{21}  \,
It is always possible to represent a resistive t-pad as a two-port. The representation is particularly simple using impedance parameters as follows:
 Z_{21} = R_c  \qquad   Z_{11} = R_c + R_a   \qquad  Z_{22} = R_c + R_b   \,

Impedance parameters to T-pad[edit]

The preceding equations are trivially invertible, but if the loss is not enough, some of the t-pad components will have negative resistances.
R_c = Z_{21}    \qquad R_a =  Z_{11} - Z_{21}    \qquad R_b = Z_{22} - Z_{21}   \,

Impedance parameters to pi-pad[edit]

These preceding T-pad parameters can be algebraically converted to pi-pad parameters.
 
R_z =  \frac { Z_{11}Z_{22} - Z_{21}^2 }  {Z_{21} } \qquad
R_x =  \frac { Z_{11}Z_{22} - Z_{21}^2 }  {Z_{22} - Z_{21} } \qquad
R_y =  \frac { Z_{11}Z_{22} - Z_{21}^2 }  {Z_{11} - Z_{21} } \qquad


Pi-pad to admittance parameters[edit]

The admittance parameters for a passive twp port are
  I_1 = Y_{11} V_1  +  Y_{12} V_2 \qquad  I_2 = Y_{21} V_1  +  Y_{22} V_2 \qquad  with \qquad   Y_{12} = Y_{21}  \,
It is always possible to represent a resistive pi pad as a two-port. The representation is particularly simple using admittance parameters as follows:
 Y_{21} = \frac {1} { R_z  }  \qquad   Y_{11} = \frac {1} {R_x} + \frac {1} { R_z  }  \qquad  Y_{22} = \frac {1} {R_y} + \frac {1} { R_z  }    \,

Admittance parameters to pi-pad[edit]

The preceding equations are trivially invertible, but if the loss is not enough, some of the pi-pad components will have negative resistances.
R_z = \frac {1} {Y_{21}}   \qquad       R_x = \frac {1} {Y_{11} - Y_{21} }  \qquad  R_y = \frac {1} {Y_{22} - Y_{21} }  \,



General Case[edit]

The pad can provide impedance matching and attenuation at the same time. Given ZS, ZLoad and Loss the resisters for the general case can be computed by these steps:

  • 1. Determine the impedance parameters of a two-port with the same impedances and loss.
  • 2. Convert impedance parameters to T-pad or pi-pad resister values.

Because the pad is entirely made from resisters, it must have a certain minimum loss to match source and load if they are not equal.

The minimum loss is given by

 Loss_{min} = 20 \  log_{10}  \left (  \sqrt{   \rho - 1 } + \sqrt{\rho }   \quad  \right  ) \, 
\quad \text{where} \quad \rho = \frac {\max [ Z_S, Z_{Load} ]}{\min [ Z_S, Z_{Load} ] }     \, 
[31]



If the loss in step 1 is less than the minimum loss, some of the resisters in step 2 will have negative values, which are not physical.

Determining Impedance Parameters[edit]

The equations for a passive two-port using Impedance parameters:
 V_1 = Z_{11} I_1 + Z_{12} I_2   \qquad  V_2 = Z_{21} I_1 + Z_{22} I_2 \qquad    with   \qquad   Z_{12} = Z_{21} \,


 A = 10^{-Loss/20} \qquad 
Z_{11} = Z_S \frac {1+A^2} {1-A^2} \qquad   
Z_{22} = Z_{Load} \frac {1+A^2} {1-A^2} \qquad 
Z_{21} = 2 \frac { A \sqrt { Z_S Z_{Load}}} {1-A^2}  \,

Determine pad parameters[edit]

 \text{T-pad parameters:  }  R_c = Z_{21}    \qquad R_a =  Z_{11} - Z_{21}    \qquad R_b = Z_{22} - Z_{21}   \,


  \text{Pi-pad parameters:  }  
R_z =  \frac { Z_{11}Z_{22} - Z_{21}^2 }  {Z_{21} } \qquad
R_x =  \frac { Z_{11}Z_{22} - Z_{21}^2 }  {Z_{22} - Z_{21} } \qquad
R_y =  \frac { Z_{11}Z_{22} - Z_{21}^2 }  {Z_{11} - Z_{21} } \qquad


Example[edit]

 \qquad Z_S = 75 \Omega \qquad  Z_{Load} = 50 \Omega \qquad Loss = 6.02 dB \,


 Loss_{min} = 5.72 \qquad A = 0.5 \qquad A^2 = 0.25 \,


  \text{Impedance parameters:  }
Z_{11} = 125 \Omega  \qquad  
Z_{22} = 83.34 \Omega    \qquad   
Z_{21} = 81.66\Omega  \,


 \text{T-pad resisters:  } 
R_c = 81.66    \quad 
R_a =  125 - 81.65 = 43.35   \quad 
R_b = 83.34 - 81.65 =1.68  \,


 \text{Pi-pad resisters:  }
R_z =  \frac { 125 \cdot 83.335 - 81.65^2 }  { 81.65 }  = 45.92\qquad
R_x =  \frac { 125 \cdot 83.335 - 81.65^2 }  {83.335 - 81.65 } = 2230 \qquad
R_y =  \frac { 125 \cdot 83.335 - 81.65^2 }  {125 - 81.65 } = 86.51\qquad

Pictures[edit]

Two Ports[edit]

using an L_pad for impedance matching

Two port, balanced, one-way left to right.
Two port, unbalanced, one-way left to right.


Two port, balanced, one-way right to left.
Two port, unbalanced, one-way right to left.


Two port, balanced, bi-directional.
Two port, unbalanced, bi-directional

Pads[edit]

Two port to pi pad.
Converstion between Pi pad and Tee pad.
Pi pads and O pads
Pi pad and L-pad between different impedances.
Pi pad between different impedances.

Toroidal Transformer Presentation[edit]

This figure depicts the magnetic core. Normally it is a material with a high permeability, but it could be something with the permeability of the vacuum. This example uses a toroidal form of circular cross section. However, since the form is viewed obliquely, the cross section appears to be an ellipse.
This figure depicts the primary winding using magenta colored arcs to represent the wires. The depiction shows a large space between the windings. Actually, the windings are so close together that they would look like a continuous sheet of conductor. This winding is axially symmetric and has no circumferential component. It meets all the requirements for total B field confinement.
This figure depicts the primary current using yellow arcs to represent the filaments of current. The depiction shows a large space between the filaments. Actually, the filaments are so close together that they would look like a continuous sheet of current.


This figure has the front half of the toroid removed, exposing the cross section and reveals a filament of current that circling each cross section.
In this figure, all current filaments except the filaments circulating the cross section have been hidden.
This figure depicts the B field flux that is due to the primary current. The B field contribution from the primary current is confined entirely inside the primary winding. In the left-hand cross section, blue circles indicate that the B flux is coming out of the picture from that cross section. In the right-hand cross sections, blue plus signs indicate that the B flux is going into the picture at that cross section.
In this figure, the lines of B flux due to the primary current is hidden; however, the blue dots and plus signs remain to indicate that the flux is still there.
In this figure, the A field has been replaced by the E field, which has opposite direction as A and magnitude equal to the time derivative of A. It has been assumed that the charge distribution of zero everywhere.
In this figure, a one turn secondary is represented by a brown line collinear with the axis of symmetry. If it assumed that the wire is intentionally resistive so that the secondary winding is also the load.
This figure depicts the method by which the secondary circuit is closed without violating the symmetry requirements. Everything is inside a hollow conductive sphere and the secondary is terminated at opposite ends of the sphere.
This figure depicts the secondary current caused by the E field acting on the secondary winding.
In this figure, the E field arrow along the axis of symmetry has been hidden.
This figure depicts the component of the B field due to the secondary current. The effect of the secondary adds to the B filed in all space, including the space within the primary winding. Even though the primary currents make a non-zero contribution only to the B field inside the primary winding, there is a continuous B field between the primary and secondary.
This figure depicts the Poynting vector as the cross product of the E field due to the primary current and the B field due to the secondary current.


Isolation Transformer Presentation[edit]

A simple 1:1 isolation transformer without any special features.
A simple 1:1 isolation transformer with an extra dielectric barrier between primary and secondary.
A simple 1:1 isolation transformer with an extra dielectric barrier and an electrostatic shield between primary and secondary.
A 1:1 isolation transformer with primary and secondary wound on a split bobbin.
A 1:1 isolation transformer with primary and secondary wound on separate bobbins.



Solutions of the Telegrapher's Equations as Circuit Components[edit]



Coaxial transmission line.


Signal flow gtraph for a twp onded transmision line.


Equivalent circuit of a transmission line described by the Telegrapher's equations.



Solutions of the Telegrapher's Equations as Components in the Equivalent Circuit of a Balanced Transmission Line Two-Port Implementation.


The solutions of the telegrapher's equations can be inserted directly into a ciruit as components. The circuit in the left figure implements the solutions of the telegrapher's equations [32].

The right hand circuit is derived from the left hand circuit by source transformations[33] . It also implements the solutions of the telegrapher's equations.

[34]


The solution of the telegrapher's equations can be expressed as an ABCD type Two-port network with the following defining equations [35]

V_1 = V_2 cosh ( \gamma  x) + I_2 Z sinh (\gamma x) \,
I_1 = V_2 \frac{1}{Z} sinh (\gamma x) + I_2 cosh(\gamma x) \,
The symbols: E_s, E_L, I_s, I_L, l  \, in the source book have been replaced by the symbols : V_1, V_2, I_1, I_2, x  \, in the preceeding two equations.

The ABCD type two-port gives V_1 \, and I_1 \, as functions of V_2 \, and I_2 \, .

Both of the circuits above, when solved for V_1 \, and I_1 \, as functions of V_2 \, and I_2 \, yield exactly the same equations.

In the right hand circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the forward wave and the other carries the backward wave. The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from V_1 \, to V_2 \, in the sense that V_1 \, , V_2 \, , I_1 \, and I_2 \, would be same whether this circuit or an actual transmission line was connected between V_1 \, and V_2 \, . There is no implication that there are actually amplifiers inside the transmission line.

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the right only models the differential mode.

In the left hand circuit, the voltage doublers, the difference amplifiers and impedances Z(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmision line like a coaxial cable or a microstrip line.

These are not the only possible equivalent circuits. Constant314 (talk) 01:18, 23 October 2010 (UTC)

Derivation of the Telegraphers Equations[edit]

Reflection Coefficients[edit]


Transmission Line Circuit for Reflection Coefficients.jpg


  in progress



 I_2 = (V_2)/ Z_L \,
 V_C= (V_2 +  Z_c I_2) = V_2 + (V_2) Z_c/ Z_L = V_2( 1 + Z_c/Z_L)   \,


 V_B = V_2( 1 + Z_c/Z_L)/2   \,
 V_D = V_2 - V_B  = V_2 - V_2( 1 + Z_c/Z_L)/2   = V_2( 1 - Z_c/Z_L)/2  \,
 V_E = T V_D =  V_2 T ( 1 - Z_c/Z_L)/2  \,
 V_F = 2V_E = 2T V_D =    V_2 T ( 1 - Z_c/Z_L)  \,
 V_A = V_B/T =   V_2( 1 + Z_c/Z_L)/2/T  \,
 V_1 = V_A + V_E =  V_2( 1 + Z_c/Z_L)/2/T + V_2 T ( 1 - Z_c/Z_L)/2 \,
 I_1 = (V_1 - V_F)/Z_c = ( V_2( 1 + Z_c/Z_L)/2/T + V_2 T ( 1 - Z_c/Z_L)/2 -V_2 T ( 1 - Z_c/Z_L) )  /Z_c \,


 I_1 = V_2( ( 1 + Z_c/Z_L)/T - T ( 1 - Z_c/Z_L)  )  /Z_c/2 \,
 Z_{in} = V_1/I_1 = Z_c ( 1 + T^2 \Gamma_L  ) / ( 1 - T^2 \Gamma_L   )     \,



 V_S = V_1 + Z_s I_1  = V_2( 1 + Z_c/Z_L)/2/T + V_2 T ( 1 - Z_c/Z_L)/2 + Z_s[V_2( ( 1 + Z_c/Z_L)/T - T ( 1 - Z_c/Z_L)  )  /Z_c/2]  \,
 2 T Z_L V_S   = V_2[( Z_L + Z_c) + T^2 ( Z_L - Z_c) + Z_s[( ( Z_L + Z_c) - T^2 ( Z_L - Z_c)  )  /Z_c] ] \,
 2 T Z_L Z_c V_S   = V_2[( Z_L + Z_c)Z_c + T^2 ( Z_L - Z_c)Z_c +  ( Z_L + Z_c)Z_s - T^2 ( Z_L - Z_c)Z_s    ] \,


 2 T Z_L Z_c 2V_S   = 2V_2[( Z_L + Z_c)( Z_c + Z_s)   - T^2 ( Z_L - Z_c)(Z_s-Z_c)     ] \,


 2 T Z_L V_S 2Z_c/( Z_c + Z_s)   = 2V_2[ ( Z_L + Z_c)    - T^2 ( Z_L - Z_c)(Z_s-Z_c)/( Z_c + Z_s)     ] \,


 V_2 = V_S (T/2) (1 + \Gamma_L)  ( 1  -  \Gamma_S )   / ( 1    - T^2 \Gamma_L \Gamma_S    ) \,




 \Gamma_S =   (Z_S - Z_C) / (Z_S + Z_C)       \,
 1  -  \Gamma_S =  + 2Z_C / (Z_S + Z_C)      \,
 \Gamma_L =  (Z_L - Z_C) / (Z_L + Z_C)  \,
 1 + \Gamma_L =  2Z_L / (Z_L + Z_C)   \,


 V_2 = V_S     \frac {T (1  - \Gamma_S)(1 + \Gamma_L)} { 2 ( 1 -T^2 \Gamma_S \Gamma_L)  }           \,


 Z_{in} = Z_C  \frac { (1   + T^2 \Gamma_L  ) }  {( 1 - T^2 \Gamma_L  )}        \, [36]

Calculation of potentials from source distributions[edit]

Feynman[37] and Jackson[38] give the following integral equations for calculating the electric scalar potential,  \phi \, and the magnetic vector potential,  \mathbf{A}\, at point p_1\, and time t\, from the current density distribution and charge density distribution.  \mathbf{A}\, is a 3 dimensional vector. The notation differs slightly from both sources.

 \mathbf{A}(p_1,t)= {\mu_0 \over 4 \pi} \int_{V_2}{ \mathbf{j} (p_2,t')\over r_{12}} dV \,


 \phi (p_1,t)= {1 \over 4 \pi \epsilon_0 } \int_{V_2}{ \rho (p_2,t')\over r_{12}} dV \,

where
p_1\, is the point at which the value of  \mathbf{A}\, and  \phi \, are to be calculated.
t\, is the time at which the value of  \mathbf{A}\, and  \phi\, are to be calculated.
p_2\, is a point at which the value of  \mathbf{j}\, or  \rho \, or both are non-zero.
 t' = t - { r_{12} \over c } \,  is a time earlier than t\, by { r_{12} \over c }\, which is the time it takes an effect generated at p_2\, to propagate to p_1\, at the speed of light.  t'\,  is also called retarded time.
 \mathbf{A}(p_1,t)\, is the magnetic vector potential at point p_1\, and time t \,.
 \phi(p_1,t)\, is the electric scalar potential at point p_1\, and time t\,.
 \mathbf{j}(p_2,t') \, is the current density at point p_2\, and time t'\,
 \rho(p_2,t') \, is the charge density at point p_2\, and time t'\,
 r_{12}\, is the distance from point p_1\, to point p_2\,
V_2\, is the volume of all points p_2\, where  \mathbf{j}\, or  \rho \, is non-zero at least sometimes.



 \mathbf{A}\, and  \phi \, calculated in this way will satisfy with the condition:   \nabla \cdot  \mathbf{A} + { c^2 \partial \phi \over \partial t }  = 0 \,

There are a few notable things about equation for  \mathbf{A}\, . First, the position of the source point p_2\, only enters the equation as a scalar distance from p_1\, to p_2\,. The direction from p_1\, to p_2\, does not enter into the equation. The only thing that matters about a source point is how far away it is. Second, the integrand uses retarded time. This simply reflects the fact that changes in the sources propagate at the speed of light. And third, the equation is a vector equation. In Cartesian coordinates, the equation separates into three equations thus[39]:


 \mathbf{A_x}(p_1,t)= {\mu_0 \over 4 \pi} \int_{V_2}{ \mathbf{j_x} (p_2,t')\over r_{12}} dV \, where  \mathbf{A_x}\, and  \mathbf{j_x}\, are the components of  \mathbf{A}\, and  \mathbf{j}\, in the direction of the x axis.

 \mathbf{A_y}(p_1,t)= {\mu_0 \over 4 \pi} \int_{V_2}{ \mathbf{j_y} (p_2,t')\over r_{12}} dV \,

 \mathbf{A_z}(p_1,t)= {\mu_0 \over 4 \pi} \int_{V_2}{ \mathbf{j_z} (p_2,t')\over r_{12}} dV \,

In this form it is easy to see that the component of  \mathbf{A}\, in a given direction depends only on the components of  \mathbf{j}\, that are in the same direction. If the current is carried in a long straight wire, the  \mathbf{A}\, points in the same direction as the wire.

Calculation of electric and magnetic fields from the potentials[edit]

 not posted

7.  \mathbf{B}=  \nabla \times \mathbf{A} \,  [37] [40]

8.  \mathbf{E}=  -\nabla \phi - {  \partial \mathbf{A}   \over \partial t }   \,  [37] [40]

When magnetic effects are dominent, equation 8 can be simplified to:


9.  \mathbf{E}=  - {  \partial \mathbf{A}   \over \partial t }   \,

Consider two long straight zero (or very low) resistance wires extending along the x axis. One carries a sinusoidally varying current that produces a magnetic vector potential that is directed in the same direction (along the x axis). The electric field in the second wire has the opposite direction to the magnetic vector potential. So the current in one long wire tends to produce a current of in the opposite direction in a parallel wire.


Torroidal Inductor/Transformer and Magnetic Vector Potential[edit]

Showing the development of the magnetic vector potential around a symmetric toroidal inductor.

See Feynman chapter 14[41] and 15[42] for a general discussion of magnetic vector potential. See Feynman page 15-11 [43] for a diagram of the magnetic vector potential around a long thin solenoid which also exhibits total internal confinement of the B field, at least in the infinite limit.

There is some arbitrariness in the A field which is removed by the tacit assumption that \mathbf{A} = 0</math>. This would be true under the following assumtions:

  • 1. the Coulomb gauge is used
  • 2. the Lorenz gauge is used and there is no distribution of charge, \rho = 0 \,
  • 3. the Lorenz gauge is used and zero frequency is assumed
  • 4. the Lorenz gauge is used and a non-zero frequency that is low enough to neglect  \frac{1}{c^2}\frac{\partial\phi}{\partial t} is assumed.

Number 4 wil be presumed for the rest of this section and may be referred to the "quasi-stattic condition".

Although the axially symmetric toroidal inductor with no circumferential current totally confines the B field within the windings, the A field (magnetic vector potential) is not confined. Arrow #1 in the picture depicts the vector potential on the axis of symmetry. Radial current sections a and b are equal distances from the axis but pointed in opposite directions, so they will cancel. Likewise segments c and d cancel. In fact all the radial current segments cancel. The situation for axial currents is different. The axial current on the outside of the toroid is pointed down and the axial current on the inside of the toroid is pointed up. Each axial current segment on the outside of the toroid can be matched with an equal but oppositely directed segment on the inside of the toroid. The segments on the inside are closer than the segments on the outside to the axis, therefore there is a net upward component of the A field along the axis of symmetry.

Representing the magnetic vector potential (A), magnetic flux (B), and current density (j) fields around a toroidal inductor of circular cross section. Thicker lines indicate field lines of higher average intensity. Circles in cross section of the core represent B flux coming out of the picture. Plus signs on the other cross section of the core represent B flux going into the picture. Div A = 0 has been assumed.


Since the eqiations \nabla \times \mathbf{A} = \mathbf{B} \ , and \nabla \times \mathbf{B} = \mu_0\mathbf{J} \ (assuming quasi-static conditions, i.e. \frac{\partial E}{\partial t}\rightarrow 0 ) have the same form, then the lines and contours of A relate to B like the lines and contours of B relate to j. Thus, a depiction of the A field around a loop of B flux (as would be produced in a toroidal inductor) is qualitatively the same as the B field around a loop of current.


The figure to the left is an artist's depiction of the A field. The thicker lines indicate paths of higher average intensity (shorter paths have higher intensity so that the path integral is the same). The lines are just drawn to look good and impart general look of the A field.


The E and B fields can be computed from the A and  \phi \, (scalar electric potential) fields


\mathbf{B} = \nabla \times \mathbf{A}.[37]
\mathbf{E} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t }. [37]

Stokes theorem applies[44], so that the path integral of A is equal to the enclosed B flux, just as the path integral B is equal to a constant times the enclosed current.



Brief Explanations and Justifications of the Skin Effect[edit]

All the following explanations are based on classical electromagnetics.

Coax and Skin depth.png


1. The expression for skin efffect comes directly from the solution of the wave equations. The high frequency electromagnetic wave simply attenuates rapidly as it penetrates a good conductor. As the E field vanishes then so does th curret density. See almost any of the following Griffiths[45], Harrington[46], Hayt[47], Kraus[48], Jackson[49] and Marshall[50].

2. The magnetic field circulating a filament of current within the conductor is such that it opposes current that is deeper and reinforces the current that is shallower. See Johnson[51].

3. The circular wire can be viewed as a set of concentric shells. As the diameter of the shells decrease, the inductance increases. Thus current on the inside is opposed by greater inductance than current along the outside. See Skilling[52] and Terman[53]


Effect of Skin Effect on Inductance[edit]

[54] [55]


"Skin effect ... slightly decreases the inductance." [56]


The energy stored in an inductor is  W = \frac {1} {2} L i^2

The energy stored in the magnetic field is  W_B = \int_{V} \frac{ 1}{2 \mu}  \mathbf{B} \cdot  \mathbf{B}     dv \,

  \mathbf{B} \cdot  \mathbf{B}     \,  is proportional to  i^2 \, leading to a definition of inductance as   L =  \frac {2 W_B} {i^2}  [57]

The energy stored in a series of inductors is  W = \frac {1} {2} L_1 i^2  +  \frac {1} {2} L_2 i^2   + \frac {1} {2} L_3 i^2  ...

If the volume containing the non-zero magnetic field is divided into disjoint regions (the regions themselves may be composed of unconnected regions) then the energy stored in the magnetic field is  W_B = \int_{V_1} \frac{ 1}{2 \mu}  \mathbf{B} \cdot  \mathbf{B}     dv   
 + \int_{V_2} \frac{ 1}{2 \mu}  \mathbf{B} \cdot  \mathbf{B}     dv  
 + \int_{V_3} \frac{ 1}{2 \mu}  \mathbf{B} \cdot  \mathbf{B}     dv   ...  
= W_{B1} + W_{B2} + W_{B3} + ... 
 \,

Thus each region of the volume containing the magnetic field can be assigned an inductance. For a transmission line the two regions are typically  L_{ext} \, for the field external to the conductors and  L_{int} \, for the field internal to the conductors.

At dc (zero frequency), both internal and external volumes contain magnetic field so the dc inductance is  L_{dc} = L_{ext} + L_{int} \, .

At high frequency, the region of non-zero B field inside the conductors is confined to the skin depth, giving an approximate formula  L_{\delta} \approx L_{ext} + b \delta \, where  b \, is a constant and  \delta \, is the skin depth (assumed to be the same on all conductors).

Once the frequency is high enough, inductance can be approximated by  L_f \approx L_{ext} + \frac {k} {\sqrt{f}} \, where  k \, is a constant and  f \, is the frequency.

At infinite frequency the skin depth becomes essentially zero and the inductance becomes  L_{\infty} = L_{ext} \,


[58]. gives an equation of this form for telephone twisted pair:  L(\omega) = L_{\infty} + \frac {L_{DC} - L_{\infty} }{   \sqrt[4]{1 + A (\frac{\omega}{\omega_L}) + (\frac{\omega}{\omega_L})^2}   }   \,


Chen [59] gives an equation of this form for telephone twisted pair:  L(f) = \frac {l_0 + l_{\infty}(\frac{f}{f_m})^b }{1 + (\frac{f}{f_m})^b}   \,

Most equations and discussions of transmission lines tacitly assume that  \frac { L_{int} } {L_{ext}} << 1  \,

Consider a coaxial cable[edit]

Four stages of skin effect in a coax showing the effect on inductance. Color code: black=overall insulating sheath, tan=conductor, white=dielectric, green=current into the screen, blue=current coming out of the screen, dashed blue lines with arrowheads=magnetic flux (B). The width of the dashed blue lines is intended to show relative strength of the line integral of the B field. The four stages A, B, C and D are unenergized, low frequency, middle frequency and high frequency respectively. There are three regions that may contain B flux: the center conductor, the dielectric and the outer conductor. In stage B, current covers the conductors uniformly and there is B flux in all three regions. In stage C, the flux in the dielectric is unchanged from stage B, but some of the cross sections of the center conductor and outer conductor have no current and no flux. There is less energy stored in the field for the same current so the inductance is less. In stage D the skin effect is fully developed. The current is confined to the surface of the conductors. The flux in the dielectric is unchanged from stage B but there is no current and no flux in the conductors. There is less energy stored for the same current relative to stage C so the inductance less than the inductance of stage C

Let the dimensions a, b, c be the inner conductor radius, the shield (or outer) conductor inside radius and the shield outer radius respectively.

The magnetic field inside a coaxial cable can be divided into three regions and an inductance assigned to each region. [60]

The inductance  L_{cen} \, is associated with the region with radius  r < a \, , the region inside the center conductor.

The inductance  L_{ext} \, is associated with the region with radius  a < r < b \, , the region between (and external to) the two conductors.

The inductance  L_{shd} \, is associated with the region with radius  b < r \, , the region inside the shield conductor.


 L_{total} = L_{cen} + L_{shd} + L_{ext}\,

 L_{ext} \, is not changed by the skin effect and is given by the well published formula  L = \frac{\mu}{2 \pi} ln( \frac {b}{a}  )   \,

As skin effect develops, the volume of space inside the conductors that is denied to the current and B increases until all B flux is excluded from within the conductors. In twisted pair (telephone) cables the effect causes a decrease of inductance exceeding 20%, as can be seen in the following table.

Signal Flow Graph[edit]

Examples[edit]

Circuit containing two-port[edit]

Signal flow graph of a circuit containing a two port. The forward path from input to output is shown in a different color.

The transfer function from Vin to V2 is desired.

There is only one forward path:

  • Vin to V1 to I2 to V2 with gain  G_1 = -y_{21} R_L \,

There are three loops:

  • V1 to I1 to V1 with gain  L_1 = -R_{in} y_{11} \,
  • V2 to I2 to V2 with gain  L_2 = -R_L y_{22} \,
  • V1 to I2 to V2 to I1 to V1 with gain  L_3 = y_{21} R_L y_{12} R_{in} \,


 \Delta = 1 - ( L_1 + L_2 + L_3 ) + ( L_1 L_2 ) \, note: L1 and L2 do not touch each other whereas L3 touches both of the other loops.
 \Delta_1 = 1 \, note: the forward path touches all the loops so all that is left is 1.
 G = \frac { G_1 \Delta_1 } { \Delta }   =  \frac { -y_{21} R_L } {1 + R_{in}y_{11} + R_L y_{22} - y_{21} R_L y_{12} R_{in} + R_{in}y_{11} R_L y_{22} }    \,


Digital IIR biquad filter[edit]

The signal flow graph (SFG) for a digital infinite impulse response bi-quad filter. This SFG has three forward paths and two loops.

Digital filters are often diagramed as signal flow graphs.

There are two loops
  •  L_1 = -a_1 Z^{-1} \,
  •  L_1 = -a_2 Z^{-2} \,
 \Delta = 1 - ( L_1 + L_2 ) \, Note, the two loops touch so there is no term for their product.
There are three forward paths
  •  G_0 = b_0  \,
  •  G_1 = b_1 Z^{-1} \,
  •  G_2 = b_2 Z^{-2} \,
All the forward paths touch all the loops so  \Delta_0 = \Delta_1 = \Delta_2 = 1  \,
 G = \frac { G_0 \Delta_0 +G_1 \Delta_1  + G_2 \Delta_2  } {\Delta} \,
 G = \frac { b_0 + b_1 Z^{-1} + b_2 Z^{-2} } {1 +a_1 Z^{-1} + a_2 Z^{-2} } \,


Servo[edit]

Angular position servo and signal flow graph. θC = desired angle command, θL = actual load angle, KP = position loop gain, VωC = velocity command, VωM = motor velocity sense voltage, KV = velocity loop gain, VIC = current command, VIM = current sense voltage, KC = current loop gain, VA = power amplifier output voltage, L = motor inductance, IM = motor current, RM = motor resistance, RS = current sense resistance, KM = motor torque constant (Nm/amp) , T = torque, M = momment of inertia of all rotating components α = angular acceleration, ω = angular velocity, β = mechanical damping, GM = motor back EMF constant, GT = tachometer conversion gain constant,. There is one forward path (shown in a different color) and six feedback loops. The drive shaft assumed to be stiff enough to not treat as a spring. Constants are shown in black and variables in purple.

Applying Mason's gain formula[edit]

The signal flow graph has six loops. They are:

  •  L_0 = -  \frac {\beta} {sM} \,


  •  L_1 = \frac{-1} {sL( R_M + R_S)} \,


  •  L_2 = \, \frac {-G_M K_M} {s^2 L M}


  •  L_3 = \frac {-K_C R_S} {sL} \,


  •  L_4 = \frac {-K_V K_C K_M G_T} {s^2 L M} \,


  •  L_5 = \frac {-K_P K_V K_C K_M } {s^3 L M} \,


 \Delta = 1 - (L_1+L_2+L_3+L_4+L_5+L_6) + (L_0 L_1 + L_0 L_3)\,


There is one forward path:


  •  g_0 = \frac {-K_P K_V K_C K_M } {s^3 L M} \,


The forward path touches all the loops therefore the co-factor  \Delta_0 = 1


And the gain from input to output is   \frac {\theta_L} {\theta_C} = \frac {g_0 \Delta_0} {\Delta} \,



Values of Primary Parameters for Telephone Cable[edit]

Representative parameter data for 24 gauge PIC telephone cable at 70° F

Frequency (Hz) R (ohm/Kft) L (mH/Kft) G (uS/Kft) C(nF/Kft)
1 52.50 0.1868 0.000 15.72
1k 52.51 0.1867 0.022 15.72
10k 52.64 9.1859 0.162 15.72
100k 58.41 0.1770 1.197 15.72
1M 141.30 0.1543 8.873 15.72
2M 196.03 0.1482 16.217 15.72
5M 304.62 0.1425 35.989 15.72

More extensive tables and tables for other guages, temperatures and types are available in Reeve [61] . Chen[59] gives the same data in a parameterized form that he states is useable up to 50MHz.

The variation of R and L is mainly skin effect and proximity effect .

The constancy of the capacitance is a consequence of intentional design.

The variation of G can be inferred from Terman[62] "The power factor ... tends to be independent of frequency, since the fraction of energy lost during each cycle ... is substantially independent of the number of cycles per second, over wide frequency ranges." A function of the form G(f) = G_1 ( \frac {f}{f_1})^{ge} with ge close to 1.0 would fit the statement from Terman. Chen [59] gives an equation of similar form.

G in this table can be modeled well with

 f_1  \, = 1MHz
G_1 \, = 8.873 \mu S/kft
ge = 0.87

Usually the resistive losses grow proportionately to  f^{0.5} \, and dielectric losses grow proportionately to  f^{ge} \, with ge > 0.5 so at a high enough frequency, dielectric losses will exceed resistive losses. In practice, before that point is reached, a transmission line with a better dielectric is used. The dielectric can be reduced down to air with an occasional plastic spacer.


Miller Effect on Outout Impedance[edit]

A set of four transister circuits for discussing the effect of the Miller capacitance on the output gain and output impedance. Biasing details have been suppressed

The figure depicts a common emitter transister amplifier. Biasing details are surpressed. There is a coupling capacitor Ccb that couples the output at the collector back to the input at the base. The circuit in section A is suitable for calculting the forward gain of the circuit. The circuit in section B has the same gain under the assumption that β, the transistor gain, is the same in both circuits. The circuit in section C is suitable for calculting the output impedance of the circuit. The circuit in section D has the same output impedanc under the assumption that β, the transistor gain, is the same in both circuits.


To simplify the analysis, the input resistance of the transister base is assumed to be zero. As a consequence, Vb = 0

Miller Math[edit]

The figure depicts a common emitter transister amplifier. Biasing details are surpressed. There is a coupling capacitor Ccb that couples the output at the collector back to the input at the base. To simplify the analysis, the input resistance of the transister base is assumed to be zero. As a consequence, Vb = 0.


signal flow graph has six loops. They are:

  •  L_0 = -  \frac {\beta} {sM} \,


  •  L_1 = \frac{-1} {sL( R_M + R_S)} \,


  •  L_2 = \, \frac {-G_M K_M} {s^2 L M}


  •  L_3 = \frac {-K_C R_S} {sL} \,


  •  L_4 = \frac {-K_V K_C K_M G_T} {s^2 L M} \,


  •  L_5 = \frac {-K_P K_V K_C K_M } {s^3 L M} \,


 \Delta = 1 - (L_1+L_2+L_3+L_4+L_5+L_6) + (L_0 L_1 + L_0 L_3)\,


There is one forward path:


  •  g_0 = \frac {-K_P K_V K_C K_M } {s^3 L M} \,


The forward path touches all the loops therefore the co-factor  \Delta_0 = 1


And the gain from input to output is   \frac {\theta_L} {\theta_C} = \frac {g_0 \Delta_0} {\Delta} \,



Interesting Comments[edit]

Terman[62] "The power factor ... tends to be independent of frequency, since the fraction of energy lost during each cycle ... is substantially independent of the number of cycles per second, over wide frequency ranges."

Griffiths[63], regarding the calculation of the magnitude of the B field in a toroidal inductor "determining its magnitude is ridiculously easy."

Halliday[64] regarding the calculation of the magnitude of the B field in a toroidal inductor "For a close-packed coil and no iron nearby ..." .




Harrington, Roger F.

Time-Harmonic Electromagnetic Fields, McGraw-Hill, 1961, Reprinted 1987

Page 63, "From the field theory point of view, this is equivalent to assuming that no EZ or HZ exists. Such a wave is called transverse electromagnetic, abbreviated TEM. This is not the only wave possible on a transmission line, for Maxwell's equations show that infinitely many wave types can exist. Each possible wave is called a mode, and a TEM wave is called a transmission line mode. All other waves, which must have an EZ or an HZ or both, are called higher-order modes. The higher-order modes are usually important only in the vicinity of a feed point, or in the vicinity of a discontinuity on the line."

Page 147, "We therefore conjecture that all wave functions can be expressed as superposition of plane waves".


Stratton, Julius Adams

Electromagnetic Theory, McGraw-Hill, 1941

Page 533, "The transport of energy along the cylinder takes place entirely in the external dielectric. The internal energy surges back and forth and supplies the Joule heat losses."


Weinberg, Steven

Dreams of a Final Theory, Pantheon Books, 1992

Chapter 6. Beautiful Theories:

Page 142, "Just as the electromagnetic force between two electrons is due in quantum mechanics to the exchange of photons, the force between photons and electrons is due to the exchange of electrons."

Chapter 7: Against Philosophy

Page 170, " ... Einstein's special theory of relativity in effect banished the ether and replaced it with empty space as the medium that carries electromagnetic impulses."



Hayt, William H., Jr.

Engineering Electromagnetics, McGraw-Hill, 1989

Chapter 11: The Uniform Plane Wave,

Section 5: Propagation in Good Conductors: the Skin Effect

Page 344, "Electromagnetic energy is not transmitted in the interior of a conductor; it travels in the region surrounding the conductor, while the conductor merely guides the waves. The currents established at the conductor surface propagate into the conductor in a direction perpendicular to the current density, and they are attenuated by ohmic losses. This power loss is the price exacted by the conductor for acting as a guide."




Feynman[65] regarding QED, "...you're not going to be able to understand it. ... my physics students don't understand it either. That's because I don't understand it. Nobody does."


Feynman[66] regarding The ambiguity of field energy, "... but we must say that we do not know for certain what is the actual location in space of the electromagnetic field energy."

Feynman[67] regarding Examples of energy flow, "As another example, we ask what happens in a piece of resistance wire when it is carrying a current. ... . . There is a flow of energy into the wire all around. It is, of course, equal to the energy being lost in the wire in the form of heat. So our crazy theory says that the electrons are getting their energy to generate heat because of the energy flowing into the wire from the field outside. Intuition would seem to tell us that the electrons get their energy from being pushed along the wire, so the energy should be flowing down (or up) along the wire. But the theory says that the electrons are really being pushed by an electric field, which has come from some charges very far away, and that the electrons get their energy for generating heat from these fields. The energy somehow flows from the distant charges into a wide area of space and then inward to the wire.

... that the energy is flowing into the wire from the outside, rather than along the wire. "

Feynman[68] regarding real fields, "What we really mean by a real field is this: a real field is a mathematical function we use for avoiding the idea of action at a distance."

Feynman[69] regarding real fields, "We have introduced A <magnetic vector potential> because ... it is ... a real physical field in the sense that we described above."

Feynman[69] regarding The vector potential and quantum mechanics, "In our sense then, the A-field is real. ... The B-field in the whisker acts at a distance."




Standard Handbook for Electrical Engineers, 11th Edition, Fink, Donald G. editor, McGraw-Hill

Chapter 2, Section 40,

Page 2-13, "The energies stored in the fields travel with them, and this phenomenon is the basic and sole mechanism whereby electric power transmission takes place. Thus the electrical energy transmitted by means of transmission lines flows through the space surrounding the conductors, the latter (conductors) acting merely as guides.

"The usually accepted view that the conductor current produces the magnetic field surrounding it must be displaced by the more appropriate one that the electromagnetic field surrounding the conductor produces, through a small drain on its energy supply, the current in the conductor. Although the value of the latter (current) may be used in computing the transmitted energy, one should clearly recognize that physically this current produces only a loss and in no way has a direct part in the phenomenon of power transmission."




Einstein, Albert

Ether and the Theory of Relativity, address on May 05, 1920 at University of Leyden p6 "More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it, ...

According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light"

Notes[edit]

  1. ^ "The approximate analysis of a circuit containing an iron-core transformer may be achieved very simply by replacing that transformer by an ideal transformer",[9]; establishing that the ideal transformer is a simple model useful for analyzing a circuit containing a nearly ideal transformer
  2. ^ "The turn ratio of a transformer is the ratio of the number of turns in the high-voltage winding to that in the low-voltage winding",[12] common usage having evolved over time from 'turn ratio' to 'turns ratio',
  3. ^ A step-down transformer converts a high voltage to a lower voltage while a step-up transformer converts a low voltage to a higher voltage, an isolation transformer having 1:1 turns ratio with output voltage the same as input voltage.
  4. ^ As each ideal transformer winding's impressed voltage equals its induced voltage, induced voltages are omitted for clarity in the next four equations.
  5. ^ Transformer winding coils are usually wound around ferromagnetic cores but can also be air-core wound.
  6. ^ The expression dΦ/dt, defined as the derivative of magnetic flux Φ with time t, provides a measure of rate of magnetic flux in the core and hence of emf induced in the respective winding.
  7. ^ ANSI/IEEE C57.13, ANS Requirements for Instrument Transformers, defines polarity as the 'designation of the relative instantaneous directions of the currents entering the primary terminals and leaving the secondary terminals during most of each half cycle', the word 'instantaneous' differentiating from say phasor current.[22]
  8. ^ Transformer polarity can also be identified by terminal markings H0,H1,H2... on primary terminals and X1,X2, (and Y1,Y2, Z1,Z2,Z3... if windings are available) on secondary terminals. Each letter prefix designates a different winding and each numeral designates a termination or tap on each winding. The designated terminals H1,X1, (and Y1, Z1 if available) indicate same instantaneous polarities for each winding as in the dot convention.[23]

Refs[edit]

  1. ^ a b "Faraday's Law, which states that the electromotive force around a closed path is equal to the negative of the time rate of change of magnetic flux enclosed by the path"Jordan, Edward; Balmain, Keith G. (1968). Electromagnetic Waves and Radiating Systems (2nd ed.). Prentice-Hall. p. 100. 
  2. ^ a b "The magnetic flux is that flux which passes through any and every surface whose perimeter is the closed path"Hayt, William (1989). Engineering Electromagnetics (5th ed.). McGraw-Hill. p. 312. ISBN 0-07-027406-1. 
  3. ^ Bedell, Frederick. "History of A-C Wave Form, Its Determination and Standardization". Transactions of the American Institute of Electrical Engineers 61 (12): p. 864. doi:10.1109/T-AIEE.1942.5058456. 
  4. ^ "All the concepts of Chap. 5 translate verbatim to the transmission line case.", Sophocles J. Orfanidis, Electromagnetic Waves and Antennas; Chap. 8, "Transmission Lines" [1]; Chap. 5, "Reflection and Transmission" [2]
  5. ^ International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. pp. 14–15. Electronic version.
  6. ^ Hayt & Kemmerly 1993, p. 443.
  7. ^ a b Reitz, Milford & Christy 1993, p. 323.
  8. ^ Hayt, William; Kemmerly, Jack E. (1993), Engineering Circuit Analysis (5th ed.), McGraw-Hill, ISBN 007027410X 
  9. ^ Hayt & Kemmerly , §14-5, p. 443
  10. ^ Winders, John J., Jr. (2002). Power Transformer Principles and Applications. CRC. pp. 20–21. 
  11. ^ Hameyer, Kay (2001). "Electrical Machines I: Basics, Design, Function, Operation". RWTH Aachen University Institute of Electrical Machines. p. 27.  |chapter= ignored (help)
  12. ^ Knowlton, §6-13, p. 552
  13. ^ a b Flanagan, William M. (1993). Handbook of Transformer Design & Applications (2nd ed.). McGraw-Hill. pp. 1–2. ISBN 0-07-021291-0. 
  14. ^ Tcheslavski, Gleb V. (2008). "ELEN 3441 Fundamentals of Power Engineering". Lamar University (TSU system member).  |chapter= ignored (help)
  15. ^ a b c John Avison (1989). "The World of Physics". [Thomas Nelson and Sons Ltd]. pp. 300–320. 
  16. ^ a b c Jim Breithaupt (2000). "New Understanding: Physics for Advanced Level (4th Edition)". [Stanley Thornes (Publishers) Ltd]. pp. 300–320. 
  17. ^ Heathcote, Martin (Nov 3, 1998). J & P Transformer Book (12th ed.). Newnes. pp. 2–3. ISBN 0-7506-1158-8. 
  18. ^ Calvert, James (2001). "Inside Transformers". University of Denver. Retrieved May 19, 2007. 
  19. ^ Parker, M. R; Ula, S.; Webb, W. E. (2005). Whitaker, Jerry C., ed. "The Electronics Handbook" (2nd ed.). Taylor & Francis. pp. 172, 1017. ISBN 0-8493-1889-0.  |chapter= ignored (help)
  20. ^ Kothari, D.P.; Nagrath, I.J. (2010). §3.7 'Transformer Testing' in Chapter 3 - Transformers (4th ed.). Tata McGraw-Hill. p. 73. ISBN 978-0-07-069967-0. 
  21. ^ Brenner, Egon; Javid, Mansour (1959). "Analysis of Electric Circuits". McGraw-Hill. pp. 589–590.  |chapter= ignored (help)
  22. ^ "Polarity Markings on Instrument Transformers". Retrieved 13 April 2013. 
  23. ^ "Connections - Polarity". Retrieved 13 April 2013. 
  24. ^ Schilling (1968, pp. 612-614)
  25. ^ Stein (1967, p. 196)
  26. ^ Valkenburg (1998, pp. 11_3)
  27. ^ Valkenburg (1998, pp. 11_3)
  28. ^ Valkenburg (1998, pp. 11_3-11_5)
  29. ^ Hayt (1981, p. 494)
  30. ^ Hayt (1981, p. 494)
  31. ^ Valkenburg (1998, pp. 11_3)
  32. ^ McCammon, Roy, An EE Times article about SPICE Simulation of Transmission Lines by the Telegrapher's Method, retrieved 22 Oct 2010 
  33. ^ Hayt (1971, pp. 73-77)
  34. ^ Xi Nan, Sullivan, C.R., An equivalent complex permeability model for litz-wire windings 
  35. ^ Karakash (1950, p. 44)
  36. ^ Karakash (1950, pp. 52-57)
  37. ^ a b c d e Feynman (1964, p. 15_15)
  38. ^ Jackson (1999, p. 246)
  39. ^ Kraus (1984, p. 189)
  40. ^ a b Jackson (1999, p. 239)
  41. ^ Feynman (1964, p. 14_1-14_10)
  42. ^ Feynman (1964, p. 15_1-15_16)
  43. ^ Feynman (1964, p. 15_11)
  44. ^ Purcell (1963, p. 249)
  45. ^ Griffiths (1989, pp. 369-372)
  46. ^ Harrington (1961, pp. 51-54)
  47. ^ Hayt (1981, pp. 398-405)
  48. ^ Kraus (1984, pp. 447-451)
  49. ^ Jackson (1999, pp. 219-221)
  50. ^ Marshall (1987, pp. 325-335)
  51. ^ Johnson (1950, pp. 58-61)
  52. ^ Skilling (1951, pp. 139-140)
  53. ^ Terman (1943, p. 30)
  54. ^ Skilling (1951, pp. 157-159)
  55. ^ Hayt (1981, pp. 434-439)
  56. ^ Skilling (1951, p. 139)
  57. ^ Hayt (1981, p. 330)
  58. ^ McCammon, Roy, An EE Times article about SPICE Simulation of Transmission Lines by the Telegrapher's Method, retrieved 22 Oct 2010 
  59. ^ a b c Chen (2004, p. 26)
  60. ^ Hayt (1981, p. 434)
  61. ^ Reeve (1995, p. 558)
  62. ^ a b Terman (1943, p. 112)
  63. ^ Griffiths (1989, p. 223)
  64. ^ Halliday (1962, p. 901)
  65. ^ Feynman (1985, p. 9)
  66. ^ Feynman (1964, p. 27_6)
  67. ^ Feynman (1964, p. 27_8)
  68. ^ Feynman (1964, p. 15_7)
  69. ^ a b Feynman (1964, p. 15_8)

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