As coaxial cable, if the case of a **two conductor system**
separated electromagnetically from the external space,
capacitance is simple matter.
But capacitance of multi-conductor system is not so easy.

This reason is not only the structure of the cable,
but circuit strcucture using cable affects **effective capacitance**.

Therefore, the specifcation of the cable must describe the cable-specific characteristics not related to how to use.

The parameters most suitable for this purpose is the **partial capacitance**
which is defined follows in electrical engineering.
(Note 1)

Q1 = K10*V1 + K12*(V1-V2) + .... + K1n *(V1-Vn) Q2 = K20*V2 + K21*(V2-V1) + .... + K2n *(V2-Vn) ............................................... Qn = Kn0*Vn + Kn1*(Vn-V1) + .... + Kn.n-1*(Vn-Vn-1) where, Qr = charge of the r-th conductor (C) (0 <= r <= n) Vrs = potential difference between r-th conductor and s-th conductor (V) (0-th condictor is th oveall shield) Krs = partial capacitance between r-th conductor and s-th conductor (F) (0 <= r, s <= n)If there is no oveall shield, the earth (ground) is the first conductor.

**Krs** is called **mutual capacitance** or **partial capacitance**
between r-th conductor and s-th conductor.
**Kr0** is called **self capacitance** or **earth capacitance**.
These parameters are totally defind by the dielectric properties of the
insulator and the geometrical protional relationship.

For example, if the case of a shielded 2-donductor shield in Figure 1, K10 = K20 will holds by Green's reciprocity theorem.

Fig. 1 - partial capacitance of shieded 2-conductor cable

The **partial capacitance** is obtained solving lenear eqation
made by combinations of partial capacitance measurement.

For example, We can make following procedure for Fig.1.

- Measure the capacitance C1 between first conductor and overall shield connected with second conductor to overall shield.
- Measure the capacitance C2 between second conductor and overall shield connected with first conductor to overall shield.
- Measure the capacitance C12 between connected first and second conductor and overall shield.

K10 = (C1 - C2 + C12) / 2 K12 = (C1 + C2 - C12) / 2 K20 = (C2 - C1 + C12) / 2

For general multi-conductor case, see Note 1.

Fig. 2 - partial capacitance of shielded 2-conductor cable

Once the **partial capacitances** are known,
easily **effective capacitance** can be calculate by simple parallel/resial
capacirance circuit.
And this is the work of cable users.

For example, if the shielded 2-donductor cable case as a microphone cable, due to the symmetry, rewriting

K0 = K10 = K20 K1 = K12

We can get

differential mode capacitance = K1 + K0 / 2 common mode capacitance = K0 * 2 (capacitance between connected 2-condictor and shield) capacitance between one conductor connected with anothe conductor and shield = K0 + K1

The capacitance value defined in specification is the value of **20 C 1 kHz**.

For cable using material having good high-frequency characteristics
in **non-polar molecule**, such as polyethylene, these vales are not varies
with freauency and temperature.
But for the cable using PVC compound and other **polar molecule** material,
capacitance is largely depend on freauency and temperature.

On electro-magnetic thory,
relation between the potential and the charge of multi-conductor system
is defined by **coefficient of potential** or **coefficient of capacity**.

Using **coefficient of capacity**,

Q1 = C11*V1 + C21*V2 + .... + Cn1*Vn Q2 = C12*V1 + C22*V2 + .... + Cn2*Vn .................................... Qn = C1n*V1 + C2n*V2 + .... + Cnn*Vn

Compared with the **partial capacitance** definion,
we get following relation.

Ki0 = Ci1 + Ci2 + Ci3 + .... + Cin Kij = -Cij (i != k, k != 0)To interpret

- measure Cii between i-th conductor and overall shield connected with all other conductors.
- measure Cjj between j-th conductor and overall shield connected with all other conductors.
- measure Ci+j between connected i-th and j-th conductor between overall shield connected with all othe onductors.
- culculate Cij with followinf relation.
Cij = (Ci + Cj - Ci+j) / 2

As nCr is the number of combinations to choose r from n, the number of measurements is

nC1 + nC2 = n + n!/2/(n-2)!It takes lot of work if number of conductors is incresed. But in many cases, number of measurement reduces by symmetry.

Measurement of capacitance coefficient from can be performed with a minimum step in an organized manner, at first measure the capacitance coefficient, then calculate partial capacitance is the good measurement practice.

Kouichi Hirabayashi, 2013-08-27