# Capacitance, Inductance, Resonant requency, Radian , and Unit Circle Question.

#### Mike M.

Joined Oct 9, 2007
104
I noticed that if C=L=0.159154943092 (Farads and Henrys), that will give a resonant frequency of 1 Hz.

What is kind of weird about that is the fact that 1 Radian is 0.159154943092 of a Unit Circle. Does anyone know why these numbers are identical?

This is especially weird since Farad=coulomb/volt and Henry=webers/amp and all those units are, as far as I know, arbitrarily composed by humans. How can such random arbitrary units be interacting in just the right way for a resonant tank circuit to be at 1 Hz, when the time-frame of a second is also a human-generated arbitrary number and how does this relate to the radian?

The units of Farads and Henrys are LINEAR and the interaction when they are at 1 RADIAN in value in a resonant tank are perfectly CIRCULAR???

Based on the units of inductance and capacitance, LC=FH=Kg-m^2/Coulomb

The Kilogram is human invented:
http://en.wikipedia.org/wiki/Kilogram

The meter is human invented:
http://en.wikipedia.org/wiki/Meter

The Coulomb is human invented:
http://en.wikipedia.org/wiki/Coulomb

So.........why do they all fit so nice and neat together when a radian is involved such as to equal a period of 1 second, which is human invented of course too:
http://en.wikipedia.org/wiki/Second

But the radian is NOT human invented but rather a property of NATURE in geometry:

#### recca02

Joined Apr 2, 2007
1,214
here is my attempt
for lc oscillator
w^2=1/LC
and frequency=w/2pi (w is angular vel)
hence
f=1/sqrt(Lc)(2pi)
for 1 Hz
L=C=1/2pi
for area of a sector A=(r^2*rad)/2
hence ratio of area for one rad to whole circle.
A=1/2pi

apart form the above all i can say is it is not a coincidence but it does prove
one thing that the units are quite wisely chosen and maths has its own way of amazing people.

#### Mike M.

Joined Oct 9, 2007
104
here is my attempt
for lc oscillator
w^2=1/LC
and frequency=w/2pi (w is angular vel)
hence
f=1/sqrt(Lc)(2pi)
for 1 Hz
L=C=1/2pi
for area of a sector A=(r^2*rad)/2
hence ratio of area for one rad to whole circle.
A=1/2pi

apart form the above all i can say is it is not a coincidence but it does prove
one thing that the units are quite wisely chosen and maths has its own way of amazing people.
That is how I derived the values of what inductance and capacitance have to be in order to make the tank oscillator work at 1 Hz.

The problem I have with it is that all of the units, with the exception of the radian, were derived by humans at different times in history and none of them were intended to be directly related to the radian which is derived from the nature of geometry. That seems just a bit odd to me. Don't you think so?

#### chulangj

Joined Oct 11, 2007
3
the solution is
the frequency is 1/(2 * pi * (sq root(L * C))

where pi = 3.14158 (or 22/7)

L is the inductance C is the capacitance

#### Mike M.

Joined Oct 9, 2007
104
the solution is
the frequency is 1/(2 * pi * (sq root(L * C))

where pi = 3.14158 (or 22/7)

L is the inductance C is the capacitance
Ok...........this is starting to make some kind of sense. If pi is introduced into the equation then the radian may just be an artifact generated by the mathematical formalism of the equation's structure and the human elements would then cancel out because that is what they are intended to do.

It is still rather odd that all 4 human-invented units work so flawlessly together with a circle and a radian when they were all developed at different times in human history, well before resonant tank circuits were discovered, and none of them seem to have anything to do with a circle or a radian in their definitions.

I guess the structure is natural and the way to numerically describe the dimentions of that structure are human. They combine together to result in a hybrid where the LC tank circuit takes on both characteristics.

#### Papabravo

Joined Feb 24, 2006
13,956
If the farad and the henry were defined in English units and the meter was replaced by feet and the kilogram by slugs then the result would be the same.

Units of mass, length, time, and charge are all fundamental. At present there are three generally used measurement systems, MKS(Meter-Kilogram-Second), the cgs(centimeter-gram-second), and the English(foot-slug-second). The values of constants change but the units are always consistent. As an example the Rydberg constant used in the ideal gas equation depends on which temerature scale you use, Kelvin or Rankine.

#### niftydog

Joined Jun 13, 2007
95
I noticed that if C=L=0.159154943092 (Farads and Henrys), that will give a resonant frequency of 1 Hz.

What is kind of weird about that is the fact that 1 Radian is 0.159154943092 of a Unit Circle. Does anyone know why these numbers are identical?
LC is just C or L squared, so the square root works out the same as L (or C). Then, 2pi is a unit circle, and the final result is the reciprocal, so that makes perfect sense.

This is especially weird since Farad=coulomb/volt and Henry=webers/amp and all those units are, as far as I know, arbitrarily composed by humans.
Are they?! They're all co-defined by each other, and in turn defined by the base units of seconds (and radians), which forms part of the definition of all these other units - either directly (2pi rads/s = 1hz = 1 cycle per second) or indirectly (farad = coulomb/volt - volt = watt/amp - watt = joules/second - joules = (kg.m^2)/s^2 etc etc etc.)

Provided the quantities remain the same then the ratios between them are mathematically related regardless of what units you're using.

#### Mike M.

Joined Oct 9, 2007
104
If you look at the links in my original post, you will notice:

On 7 April 1795, the gram was decreed in France to be equal to the absolute weight of a volume of pure water equal to a cube of one hundredth of a meter, and to the temperature of the melting ice.

------------------------------------------------

A coulomb is then equal to exactly 6.241 509 629 152 65×1018 elementary charges.

------------------------------------------------

In the eighteenth century, there were two favoured approaches to the definition of the standard unit of length. One suggested defining the metre as the length of a pendulum with a half-period of one second. The other suggested defining the metre as one ten-millionth of the length of the Earth's meridian along a quadrant, that is the distance from the equator to the north pole. In 1791, the French Academy of Sciences selected the meridional definition.

In order to establish a universally accepted foundation for the definition of the metre, measurements of this meridian more accurate than those available at that time were imperative. The Bureau des Longitudes commissioned an expedition led by Delambre and Pierre Méchain, lasting from 1792 to 1799, which measured the length of the meridian between Dunkerque and Barcelona. This portion of the meridian, which also passes through Paris, was to serve as the basis for the length of the half meridian, connecting the North Pole with the Equator.

However, in 1793, France adopted the metre based on provisional results from the expedition as its official unit of length. Although it was later determined that the first prototype metre bar was short by a fifth of a millimetre due to miscalculation of the flattening of the Earth, this length became the standard. So, the circumference of the Earth through the poles is approximately forty million metres.

------------------------------------------------

The duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom is the definition of the Second.

------------------------------------------------

Now how is it even remotely possible, given the history and definition of these units, that they can be related in any way.

If you were to just give me this information and then tell me I could use it to generate an equation that incorporates the radian/unit circle ratio and end up with something that describes the frequency of an inductive-capacitive tank circuit..........I would think you were nuts!!!!!

This is what it does though and I am just absolutely amazed that the unit system has such RANDOM CHOICES FOR THE DEFINITION OF UNITS that were developed at all different times and yet the second, ANOTHER RANDOM CHOICE, becomes unity when L&C are both identical to a radian!!!

#### niftydog

Joined Jun 13, 2007
95
As I said, the second - irrespective of how it is defined - is the fixed standard for defining the other quantities (with the exception of the kilogram.) The units are literally defined by their relationship to 1 second.

If you redefined "1 second" you would then have to redefine all of the other units to reflect the new standard and the maths would STILL work out neatly somewhere along the line.

These unit definitions simply are NOT random - they are carefully defined using other standard units.

For instance, the number you gave for 1 coulomb seems random, and it may well be for all I know, but the important thing is that the ampere was defined as 1 coulomb/sec. And it's a similar story for all the other units.

#### Mike M.

Joined Oct 9, 2007
104
For instance, the number you gave for 1 coulomb seems random, and it may well be for all I know, but the important thing is that the ampere was defined as 1 coulomb/sec. And it's a similar story for all the other units.
That is why I reduced all the units to the units on which they are based on until I hit a dead end and found no basis for them besides what only appears to be human imagination.

The coulomb, meter, and gram don't appear to have a historical definition based on the second. They are based on a number of charges passing through a point at a given potential difference, the length of the meridian between Dunkerque and Barcelona, and the absolute weight of a volume of pure water equal to a cube of one hundredth of a meter (defined above) at 32 degrees farenheit.

You say these are derived from the second and I am trusting that you are correct, but how?

#### niftydog

Joined Jun 13, 2007
95
Sorry, I gave the wrong impression. I wasn't saying that every unit was derived from the second, only those like amps, volts, farads, henrys etc. Ultimately, I don't know if I have the whole story, or whether I'm right at all!

Units like meters etc are defined differently, but in the end it doesn't matter if you're measuring in meters or "slugs" or hogsheads or whatever, these units are all co-defined and they all relate somehow (usually by a linear ratio). This fact was carefully considered by those responsible for the SI system, and as such SI units often mathematically relate very neatly.

As I say, all I can think is that these units all have a common basis for the definitions. This is exactly what makes the SI system so useful.

That's all I've got... #### Mike M.

Joined Oct 9, 2007
104
I'll do some digging and see whether I can turn up some info on how these basic units came about and try to find a connection with the unit of a second. I know most units ARE derived from the second but I would really like to find out if the ones I mentioned are or aren't and even if more units possibly exist than those I mentioned that aren't derived from the second.

It's funny how I started out with the question about a tank circuit with L & C values equal to a radian and I end up trying to figure out what units are based on.

Anyways, if I find something to support your theory then I will post it here.

It may very well be that everything IS based on the passage of time, the second. If that were the case then the original question I had to start this thread would lose all of its absurdity.

#### GS3

Joined Sep 21, 2007
408
Checking out Wikipedia I see the Farad is defined as F = (s^4 * A^2) / (m^2 * Kg) and the Henry is defined as H = (m^2 * Kg) / (s^2 * A^2)

so that 1 / (F * H) = 1 / s^2 = Hz^2

or, what is the same:

Hz = 1 / SQRT(F*H)

As simple as that. It would work out no matter the magnitude of the Kg, the m, the s or any other unit.

#### Mike M.

Joined Oct 9, 2007
104
Checking out Wikipedia I see the Farad is defined as F = (s^4 * A^2) / (m^2 * Kg) and the Henry is defined as H = (m^2 * Kg) / (s^2 * A^2)

so that 1 / (F * H) = 1 / s^2 = Hz^2

or, what is the same:

Hz = 1 / SQRT(F*H)

As simple as that. It would work out no matter the magnitude of the Kg, the m, the s or any other unit.
Well put! All the units cancel with the exception of seconds and F/H=1 when they are identical, whether that be a Radian in value or not. The Radian comes about by the structure of the equation relating it to Pi. I also see that the 2*Pi*Radian=1

Looking back on some of the previous threads, I can see that others were telling me the same thing but it didn't actually hit me until you structured it the way you did.

Sorry everyone, I must have been having a 2-day brain fart or something because it all makes perfect sense now.

#### bloguetronica

Joined Apr 27, 2007
1,372
I noticed that if C=L=0.159154943092 (Farads and Henrys), that will give a resonant frequency of 1 Hz.

What is kind of weird about that is the fact that 1 Radian is 0.159154943092 of a Unit Circle. Does anyone know why these numbers are identical?

This is especially weird since Farad=coulomb/volt and Henry=webers/amp and all those units are, as far as I know, arbitrarily composed by humans. How can such random arbitrary units be interacting in just the right way for a resonant tank circuit to be at 1 Hz, when the time-frame of a second is also a human-generated arbitrary number and how does this relate to the radian?

The units of Farads and Henrys are LINEAR and the interaction when they are at 1 RADIAN in value in a resonant tank are perfectly CIRCULAR???

Based on the units of inductance and capacitance, LC=FH=Kg-m^2/Coulomb

The Kilogram is human invented:
http://en.wikipedia.org/wiki/Kilogram

The meter is human invented:
http://en.wikipedia.org/wiki/Meter

The Coulomb is human invented:
http://en.wikipedia.org/wiki/Coulomb

So.........why do they all fit so nice and neat together when a radian is involved such as to equal a period of 1 second, which is human invented of course too:
http://en.wikipedia.org/wiki/Second

But the radian is NOT human invented but rather a property of NATURE in geometry:
Maybe because of 2*Pi, which is also used for a circle perimeter when multiplied by its radius. 159154943 is the inverse of 2*Pi. Therefore, being f = 1Hz and L = C = 1 / 2*Pi, we have:
Xl = 1 / (2*PI x f x L) = 1Ohm
Xc = 2*Pi x f x C = 1Ohm

Mystery solved.

More to add, meter, kilogram, coloumb and second are metric units. So 1L = 1dm3 = 1Kg of water. No wonder that coloumb is also made to fit, as metric unit as it is. Don't forget that metric units, unlike imperial ones, are all related to each other. This unit system is all based on the metre.

#### techroomt

Joined May 19, 2004
198
the unit circle (radius of 1.0) when placed on a cartesian coordinate graph (x and y lines) also relates directly to the sine and cosine waves very well. for example, when an angle is drawn on a unit circle, the radius is always one, which is always the hypotenuse. then, since the sine of any angle is equal to opposite/hypotenuse, and hypotenuse = 1, then the sin = the opposite, or "y" value. hence, the cosine = x value. many correlations can be made with this analogy.

#### niftydog

Joined Jun 13, 2007
95
Looking back on some of the previous threads, I can see that others were telling me the same thing but it didn't actually hit me until you structured it the way you did.
I knew it was just gonna take the right approach for you to see what we were seeing!