Resistance, Reactance, Permeability and Permittivity

[bperet]

Since my reply to Horace triggered a new understanding of these concepts, I thought I would start a separate topic on it in the EE forum. (Click on the ↑ after "Horace wrote" to read the original topic.)

I think the full EE scalar formula is |Z| = SQRT( (X_C+X_L)² + R² )

Thus for X_L=20Ω and X_C=-20Ω and R=0Ω the scalar magnitude of impedance |Z|=0Ω.
...and for X_L=20Ω and X_C=-20Ω and R=20Ω the scalar magnitude of impedance |Z|=20Ω.

I am no longer using the linear, "graph paper" approach to impedance, but have taken it into another dimension (literally), based on a quaternion rather than complex quantity (see: Extraordinary Electric Current), in which I treat resistance as "real" (linear) and reactance as "imaginary" (angular)--but as a quaternion, not a complex quantity.

I treat quaternions differently, because I do not transform the angular velocity into a 2D, linear one, so I only need a 3-axis representation (not 4, as you'll find with regular math). As described elsewhere, linear motion is a translation along an axis, whereas angular motion is a twist of an axis.

The real axis represents the linear motion of electric current and resistance/conductance, which is analogous to the "progression of the natural reference system" as it always moves at the speed of light, as electric current does. Resistance does not change this speed... the current still flows at c, like the progression does.

The other two axes are used as poles for the angular motion of reactance, the blue is for the typical "j" axis of alternating current, and the red from Steinmetz's work on impulse and oscillating currents. So to get a typical impedance, you have a resistance (radial length) and reactance (angle)--the complex number. The three planes of rotation then form Larson's concept of "speed ranges," of which we stick with the low-speed (1-x) range of DC and AC.

Odd thing about these speed ranges is that low speed and ultra-high speed (3-x) are based in clock time--they have a displacement along the real, DC axis, but intermediate speed (2-x) does not. Intermediate speeds are always at zero on the DC axis, meaning the effect would appear instantaneous, as it occurs without a change in time.

And I just realized something... the red and blue axes must also have a resistance component (translational at the speed of light), due to the symmetry constraints of the RS. And that would infer that capacitance and inductance are the same thing, but from two, reciprocal perspectives... that means the "squished flat" graph-paper form of complex numbers may be incorrectly representing the reactance relationships--being a projection, rather than a 3D structure.

My thought here is that capacitance, having negative reactance, is not in the 1-x range (as complex quantities put it), but in the 3-x range, with only inductance in the 1-x range. Both are time-dependent angular velocities, but would be orthogonal and out of phase by 90°, which may explain the leading-lagging current of impedance.

That would infer that the "resistance" on the blue and red axes would be an indirectly observable quantity--something similar to resistance but nonlocal. If inductive is 1-x, that would make the blue axis permeability and the red axis (3-x), permittivity. A common resistance acts like a membrane, limiting flow through it. Both permeability and permittivity do the same. I will explore this further.

[SoverT]

Odd thing about these speed ranges is that low speed and ultra-high speed (3-x) are based in clock time--they have a displacement along the real, DC axis

Off topic, but what does axis have to do with clock time (or clock space)? The clock is merely a synonym for the universe taking a step, whether it's a unit step of 1/1, or an accelerated step of 47/23. The progression progresses regardless of observers looking at some arbitrary projection into a preferred reference frame

[bperet]

Off topic, but what does axis have to do with clock time (or clock space)? The clock is merely a synonym for the universe taking a step, whether it's a unit step of 1/1, or an accelerated step of 47/23. The progression progresses regardless of observers looking at some arbitrary projection into a preferred reference frame

DC current is a speed, s/t, that has a magnitude of 1/1. The DC axis projection of intermediate speed is 1/0, or infinite speed.

Let me clarify... the "clock" is not a "step," but the unit speed datum of measurement. The clock IS the progression. It can be treated as a "step" because the progression progresses in unit increments.

[Horace]

Take a look at this video and notice how the projection of a sphere onto a plane looks like the Smith Chart of impedance:
https://www.youtube.com/watch?v=d4EgbgTm0Bg

...and after differentiation - like the Biradial Matrix.

[bperet]

Take a look at this video and notice how the projection of a sphere onto a plane looks like the Smith Chart of impedance:
https://www.youtube.com/watch?v=d4EgbgTm0Bg

...and after differentiation - like the Biradial Matrix.

When you start projecting angular relations (imaginary numbers) as lines, it is common to get these types of diagrams (called "pencils" in projective geometry).

In RS2, I treat the quaternion as a scaled, 3D system--not 4D--since you cannot mix apples (real) with oranges (imaginary), which conventional math does. It is a different way to think, but when you can get to that point, it really clarifies a lot of things.

[Horace]

I know, they are different but the projection of the angular yin on our 3d linear (yang) perception is at the center of RS.
Your idea to use colors to depict this is good. I think it can be extended by mapping colors to the density of the dots projected onto the vertical line, as shown at the time index 7m 45s of that video.

[bperet]

My theoretical work uses the natural units of the Reciprocal System, just space and time. In order to do anything practical, I need to map the RS natural units to conventional units. In the case of resistance and reactance, that is Ohms. The conventional relation is V/I=R, so if a conversion can be found from natural units to volts and amperes, that would give resistance.

Reactance appears to be the analog of inertia for the "mass" of inductance and capacitance, so a conversion of natural units to Farads (capacitance) and Henries (inductance) is also needed.

Larson made an attempt in Basic Properties of Matter, using now obsolete units of ESU and gram-equivalent. Since his time, those units have been dropped from regular use and the definition of an Ampere has also changed. Secondly, the conversion Larson came up with does not seem intuitively correct... ONE natural unit of resistance being 8.8 gigaohms? That is far outside of conventional use.

This clarification of the dimensional relations is accompanied by a determination of the natural unit magnitudes of the various physical quantities. The system of units commonly utilized in dealing with electric currents was developed independently of the mechanical units on an arbitrary basis. In order to ascertain the relation between this arbitrary system and the natural system of units it is necessary to measure some one physical quantity whose magnitude can be identified in the natural system, as was done in the previous determination of the relations between the natural and conventional units of space, time, and mass. For this purpose we will use the Faraday constant, the observed relation between the quantity of electricity and the mass involved in electrolytic action. Multiplying this constant, 2.89366×10¹⁴ esu/g-equiv., by the natural unit of atomic weight, 1.65979×10^-24 g, we arrive at 4.80287×10^-10 esu as the natural unit of electrical quantity.

The magnitude of the electric current is the number of electrons per unit of time; that is, units of space per unit of time, or speed. Thus the natural unit of current could be expressed as the natural unit of speed, 2.99793×10¹⁰ cm/sec. In electrical terms it is the natural unit of quantity divided by the natural unit of time, and amounts to 3.15842×10⁶ esu/sec, or 1.05353×10^-3 amperes. The conventional unit of electrical energy, the watt-hour, is equal to 3.6×10¹⁰ ergs. The natural unit of energy, 1.49275×10^-3 ergs, is therefore equivalent to 4.14375×10^-14 watt-hours. Dividing this unit by the natural unit of time, we obtain the natural unit of power
9.8099×10¹² ergs/sec = 9.8099×10⁵ watts. A division by the natural unit of current then gives us the natural unit of electromotive force, or voltage, 9.31146×10⁸ volts. Another division by current brings us to the natural unit of resistance, 8.83834×10¹¹ ohms.

One of the things that came out of the RS2 dimensional analysis (as described in this thread with yin dimensions), is that the Farad and Henry should be reciprocals... they should not be running from 0..∞, but 1/n and n/1 -- Farads and Henries should be reciprocals, since capacitors and inductors behave in a reciprocal manner.

What also needs to be considered is the fact that current (Amperes) is based on the flow of uncharged electrons, whereas electrostatics are based on charge. Larson's values appear to be electrostatic units, not units of electric current--which are the ones that I need to "do something practical" with this research.

I've been working on this for a while, but cannot seem to get any results that make sense. Hoping that someone with better math skills than I could take a stab at doing this. Thanks.

[Horace]

I've been working on this for a while, but cannot seem to get any results that make sense. Hoping that someone with better math skills than I could take a stab at doing this. Thanks.

I believe that the attached paper will put you on track to cracking this ( especially the Annex )

[bperet]

Here is my first attempt at it, which matches Larson's values (off slightly because I used newer values for the constants). Couple interesting things showed up... the Coulomb is the MKS unit of electric charge and two values match 1 amu (atomic mass unit)... energy in Joules and voltage in Volts. That means one "natural unit" of voltage is the same as 1 amu. (Never did understand why they use voltage, a force, to express a mass.)

Code: Select all

Constants used:

Rinf     Rydberg Constant  1.097373156865e+7
   c       Speed of light  2.997924580000e+8
  Na   Avogadros constant 6.022140857740e+23
  Fc     Faraday constant  9.648533289590e+4

Calculated Values:

  us           Unit space  4.556335252708e-8
  ut            Unit time 1.519829845989e-16
  
   E               Joules 1.492418061895e-10 (electric energy = 1 amu)
   q             Coulombs 1.602176620826e-19 (electric quantity = electric charge)
   I              Amperes  1.054181575032e-3 (current)
   F               Farads 1.720007275348e-28 (capacitance)
   H              Henries  1.342949412986e-4 (inductance)
   V                Volts  9.314940952803e+8 (= 1 amu, 931.494 MeV)
   P                Watts  9.819639124957e+5 (power)
   R                 Ohms 8.836182659063e+11 (resistance)

The values for current and inductance make sense, the natural unit for current being about a milliamp and for inductance, 134 microhenries--both commonly used. However, the values for voltage and capacitance seem much too high/low. Gopi stopped by last night and we discussed this; he thinks that the way we measure capacitance is off by a c² function. Makes sense to me because inductance and capacitance should be reciprocals of each other and if you look at the units: inductance is t³/s³ but capacitance is s³/t -- missing a 1/t² dimension. May have something to do with capacitance being in the time region, where speed, s/t, becomes 1/t².

My analysis so far indicates that the values for voltage and capacitance are for static electricity--not current--since they are based on the concept of charged electrons. The Van de Graff generator recently purchased produces voltages in the 500 MeV range with virtually no current, so it is in the range of 1/2 natural unit of voltage. Got me thinking of why this works for capacitors... the field of an inductor extends outward into the surrounding space as magnetism, yet the field of a capacitor extends inward into the space of the dielectric material in the time region--meaning that the capacitor may be storing the charge as a "cosmic magnetic field"--outward into the surrounding, 3D time, just as static electricity extends outward into 3D space as dielectric lines of force.

[user737]

A common resistance acts like a membrane, limiting flow through it. Both permeability and permittivity do the same. I will explore this further.

L = Rt, inductance is resistance for a period of time (i.e. energy suddenly appears from a "potential" stored in a magnetic field -- 2D counterspatial)

In other words, inductance IS mass is resistance for a period of time (L = m = Rt)... if there was no mass, there'd be no resistance!... and oh, look mass and inductance have the same dimensions of motion: t³/s³

Conversely, C = Gt, capacitance is conductance for a period of time (i.e. energy suddenly appears from a "potential" stored as a dielectric field -- 1D counterspatial)

Conductance (G) and Resistance (R) by definition have in themselves a reciprocal relation (G = 1/R)

R = μI, resistance is permeability of current, i.e inductance
G = εI, conductance is permittivity of current, i.e. capacitance (s³/t)*

* easy to miss the true scalar reciprocal relationship of capacitance (s³/t) to inductance (t³/s³) as the cross-product does not yield 1 (unity) but rather 1/t². This is because what we call inductance... the linear "lines of force" (net push/pull of scalar motion, 1D) is on the same side of the unit speed boundary as our observation (time-space i.e. same reference frame of time to time)... the capacitance is counter-spatial... the rotational "lines of force" or the net "imaginary" (scalar motion we call magnetic, 2D) and is on the other side of the unit boundary within the Time Region (TR) of the atomic rotation and so our observance is limited to the NET rotation as we cannot observe locations in 3D time directly but we can observe how time effects space (as a NET scalar push/pull PLUS net scalar rotation, real + 1D imaginary = complex). The NET effect of three dimensions of time are reduced to the affect on one dimension in the inverse aspect -- and space is one-dimensional too, as every point in space can be related to every other point using one dimension [distance] -- the difference being the well-known 1/c² transform. Look here it is: E = mc²... energy (one-dimensional motion in time) is scaled to mass (three-dimensional motion in time) using the 1/c² transform directly (naked proof). OK, here it is again in the electrical world: P = IV (axiomatically) and V = IR (Ohm's law) → P = I²R → P = E/t = I²R → E = I²Rt → E = I²m (mass IS resistance for a period of time) → E = c²m (current IS motion at the speed of light, c) → E = mc²... looks familiar. In furtherance of demonstration, note that when crossing the unit speed boundary speed (s/t) becomes 1/t²: substitute 1/t for s (as s/t = 1; this IS motion) and the result is 1/t² (equivalent speed).

In natural units that makes μεI² = 1 or μ₀ε₀ = 1/c²

Fully reduced (taking both 1 = c and I = c as current IS a speed of +1): μ = 1/ε

Doesn't get any more obvious than that.

When the rotation extends into the 2nd rotational plane, j, you get intermediate speed (2-x) motion. Now this is interesting because unlike the low-speed plane, it has NO projection onto the real axis--it is always zero, regardless of angle, as it is orthogonal to the real axis. This makes things appear "instantaneous" (infinite speed). And two-dimensional "electric" rotation is what we call "magnetism." So your magnetic fields arise with [1 ai bj 0k]--and why electromagnetism requires electricity to produce it.

We're talking about the natural datum on the j axis... the one with no dependence on clock time... cause... effect... it's up to you. Whether the voltage "leads" current (overall inductive) or voltage "lags" current (overall capacitive) and the magnitude of the phase angle separating them in clock time simply depends on the NET reactance ("imaginary" i.e. rotational resistance) -- scalarly rotationally inward or scalarly rotationally outward -- as a function of the circuit's SUM capacitance and SUM inductance -- which are themselves governed by inverse laws of relation -- for a given switching frequency... we're talking AC here or nothing's changing. Here we see instantaneous effect in time for a given period as a phase angle (clock time!) with difference in phase angle (180-θ) expressed as a constant wavelength ("angle") in clock time!

In electric power flow, it is important to know how much current is leading or lagging because it creates the reactive power in the system, as opposed to the active (real) power. It can also play an important role in the operation of three-phase electric power systems.

PolartoRect.png (62.77 KiB) Viewed 25759 times

-- Leading and lagging current (willfully incomplete) @ Wikipedia

e^jθ has magnitude (absolute value) of 1 where j is used in place of i as 'i' has other potential meaning in EE.

This is rotation in the complex plane <1 ai 0j 0k> where θ is best expressed in radians; 2π radians is one full revolution. The sign of the exponent expresses "direction" (CW or CCW) for each of the two reciprocal aspects: induction and capacitance.

The phase angle between the two varying waveforms (current and voltage) is similar to the recursive geometry of the EM spectrum in that an unbounded turn in equivalent space (time or, if you will, counter-space) can only be represented as a bounded angle in space -- this being due to the offset in dimensionality (x±1) required to represent equivalent motion (how time effects space or how space effects time) in either inverse aspect -- in that the phase angle or what we call that "frequency" is the net shear in time-space (or space-time on the cosmic side)... where only the net linear (push/pull) and net rotational motion can be transmitted across the unit [speed] boundary (the genesis being the TR of the atom)... this IS that net rotational motion.

We know this concept as dimensional reduction: two counter-rotations (each one-dimensional) can be reduced to a single dimension of linear displaced motion through a function known as Euler's Formula (for complex numbers): e^iφ = cosφ + isinφ ... flip the sign on the exponent (opposite "direction", inward → outward or vice-versa) flips the direction in one dimension only; add them together (that IS a bi-rotation) and you get e^iφ + e^-iφ = 2×cosφ... two dimensions of motion can be reduced to a one-dimensional shear in the reciprocal aspect... unbounded bi-rotation (2D) → bounded linear [vibration] (1D). This is the basis of the concept of dimensional reduction (and its inverse, dimensional expansion) as observed in nature. For example, if combining two one-dimensional rotations results in a reduction of dimensionality from 2 to 1, then what is the result of combining two one-dimensional linear displacements (simultaneously)?... ah ha! an area (2D) or an increase (inverse decrease) combining of dimensionality from 1 to 2!

To fully describe a wave (non-local) you need three things (1) [wave] magnitude, (2) frequency, and (3) phase angle (with respect to some other "reference" wave). Magnitude is obviously the net push/pull making "frequency" the net rotation (or rotational yin's shadow masquerading as rotational yang here in the material sector). This is not a shock. We've known for a long time frequency is not energy is not color is not.... It's not a lot of things we take it for in conventional, legacy science.

Changing the excitation voltage up or down (current produced in creation of magnetic field) to the AC generator-end stator does this (varies the "lead/lag angle") in reversible AC-DC motor-generator sets. We usually set them slightly "lead" in addition to the phase angle between the AC rotor and DC stator (or vice-versa) necessary to develop the torque to keep the thing spinning (minus losses!) depending on the direction of energy transfer as most loads are inductive AC motors themselves and, well, mass IS inductance and everything's made out of mass (time, not space). Slipping a pole (too much "lead," or, if you will, too much inverse "lag") results in loss of regulation and loss of [spatial←→counterspatial] energy/speed (ah ha, motion) transfer link.

And all (electric, magnetic, kinematic motion) are at orthogonal relations to one another in 3D coordinate (extension) space. It's like a big universal dynamo where you put in one, two, or three legs of the triad and the universe gives you back the difference. Just like Faraday discovered... "dielectric" field, magnetic field, electrically-conductive medium... pick any two and they will create the missing third. Put in electric and magnetic... get motion (flow) out; put in motion and electric... get magnetic out; put in magnetic and motion... get electric out. Put in all three and you get atom building...

You'd think these guys would have figured this stuff out by now...

Choose your own direction for the "flow" of time! Fun for the whole family!

TOTAL SIDEBAR: This is why all the complexity and tapestry of thought that goes into your well-detailed explanation is reduced to the NET push/pull when it leaves you (one side of unit boundary) and is communicated to another. Only the NET effect can be felt and so emotion, or intuition (the inverse aspect of the logic of your thought) either draw that person closer to you, or push you two further apart. Consider the emotional appeal of your content even if you do not think the other party has a consideration in any regard as your message may be severely dulled otherwise. Do good for the sake of good. Like "attract" like in this speed region.