11/20/15

Lcoaxial_{LF} derivation

The inductance for any cylindrical geometry with circular symmetry is :

This is derived from stored energy equations.

where is the field produced by current , the total current in the center conductor. The integration is over all space containing the field.

= center conductor radius

= shield inner radius

= shield outer radius

at radius equals

where is the total current inside radius .

The integration can be divided into 3 regions: center conductor, dielectric, and shield.

For a length of 1 meter and a frequency approaching 0 :

Lcoaxial derivation

This derivation follows the same course as :

1. is 0 at high frequency and

at DC. will accurately transition between these two cases.

2. is unchanged.

3. : The derivation is the same as for DC, but the shield thickness is adjusted. Let = + Fs * Ds.

Ds is the skin depth computed as for a plane surface using the formula

Fs is a constant. Atlc2 can be used to show that this constant is about 1.7 for typical coaxial cable.

FindRatioR derivation

Resistance and Inductance ratios for solid round wires, published by the National Bureau of Standards, now NIST:

Line 1 is the x-axis. Line 1 is

where is

Line 2 is

Line 3 is

is in Hertz.

This is a curve-fitting exercise. is a proposed function for predicting the NBS resistance data. Find the constants - that will give the most accurate fit.

This data is used in determining the A.C. resistance and inductance of coaxial cable. It is not accurate for twinlead.

FindRatioL derivation

setGp derivation

The capacitance between the two conductors can be modeled as two capacitors in series :

is the capacitance of the air part

is the capacitance of the solid part

is the net of the two capacitors in series.

(You can imagine all of the air squished out of the foam, so that there is solid dielectric against one conductor and all the air against the other. Then an imaginary third plate can be added where the solid and the air meet. If the third plate follows an equipotential surface then the device will work exactly as an unmodified device, and is obviously two capacitors in series.)

Solve for :

For a second equation, if the solid were replaced with air then the velocity factor would be 1, and the net capacitance could be found from the inductance :

(L comes from )

The solid capacitance would become

which since = 1 is simply

The two capacitors in series would be described by :

Now equate the two expressions for and solve for :

Once is known, can be found from :

The effective series resistance of a capacitor is

So the effective impedance of is

DelayLineBalunZ_{un} and DelayLineBalunZ_{bal} derivations

We shall use these relationships :

(Although the diagram shows an antenna, it could be any balanced device.)

The result is derived twice, once using output functions, and once using input functions. The same result should be obtained either way.

Derive using Output functions :

Derive using Input functions :

Equate the two expressions for :

Solve the above equation for :

(I am sorry if too many steps were left out, but Mathcad solves these things effortlessly and without showing any intermediate steps.)