Transmission Line Parameters: Capacitance and Conductance


The last article was about line resistance and inductance. Now we will discuss about line capacitance and conductance. We already said that leakage current flows between transmission lines and ground and also between phase conductors. Leakage current flows to ground through the surface of insulator. This leakage current depends upon the suspended particles in the air which deposit on the insulator surface. It depends on the atmospheric condition. The other leakage current flows between the phase conductors due to the occurrence of corona. This leakage current also depends upon the atmospheric condition and the extent of ionization of air between the conductors due to corona effect. Both these two are quite unpredictable and no reliable formula exist to tackle these leakage currents. Luckily these two types of leakage currents are negligibly small and their ignorance has not proved to influence much the power system analysis for line voltage and current relationships. Here we will ignore the leakage currents so we will not show the leakage resistance. Inverse of this leakage resistance is called line conductance.

Here rest of the article is about line capacitance. Like previous article on inductance here also I am not going to derive the formulas for capacitance for different line configurations rather to develop some concepts.  

As the flow of line current is associated with inductance similarly the voltage difference between two points  is associated with capacitance. Inductance is associated with magnetic field and capacitance is associated with electric field.

The voltage difference between the phase conductors gives rise to electric field between the conductors ( see Fig-A). The two conductors are just like parallel plates and the air in between the conductors is dielectric. So this arrangement of conductors gives rise to capacitance between the conductors. The value of capacitance depends on the configuration of conductors. We will discuss few configurations  and the corresponding capacitance value.  
  • Here in Fig-A is shown the single phase line conductors. In the figure is shown the cross section view of the conductors. See the Electric lines of force representing Electric field. The lines of force start from one conductor and terminate on other. In the diagram it is assumed that there is no other charged body, even the ground (which is at potential zero) is assumed to be far away and has no influence on line capacitance. In this situation,
          Let the capacitance between the two lines each of  radius r is C Farad per meter of line length. Then,

                 C =    -----------
                          ( ln is for natural logarithm )
                          k is the permittivity of air.

           Note: In this article capacitance is always per meter of line length. So the unit is F/m .

           One important thing is that here the actual radius r is used in the formula. Compare with inductance
           formula where we used the equivalent radius r' which is 0.7788 times the actual radius r.

         In the last article, inductance was found for each line individually. Here also capacitance between line to
         neutral is desired for per phase analysis of power system.
         It is important to think that the line to line capacitance is equivalent to two capacitance each of value
         2C, one between line-1 and neutral(N) and other between neutral(N) and line-2. See Fig-B.

        Note: Capacitance in series behaves similar to resistance in parallel. Also capacitance in parallel behaves
                  similar to resistance in series. When two capacitors are connected in parallel their equivalent is
                  sum of the two capacitances.

         So the line to neutral capacitance Cn is two times C.

                                                          Cn = 2pk / ln (D/r)
  • Now let us consider our favorite case of three phase circuit(see Fig-C) where the phase conductors (a, b and c) occupy the corners of equilateral triangle. The conductors are equidistant from each other. If  Cn is the capacitance from line to neutral N (per phase capacitance). Note that point N is imaginary not physical.
                                                            Cn = 2pk / ln (D/r)      

  • The general form (Fig-D) of capacitance between one phase and neutral  for a three phase line is

                                                            Cn = 2pk / ln (GMD/GMR)

          GMD is Geometric Mean Distance and GMR is Geometric Mean Radius of the particular 
          configuration. GMR used for calculation of capacitance is slightly different from GMR used for
          inductance as mentioned below. It is also assumed that the phase lines are transposed
          In actual practice in most of the cases you will find that the three phase conductors are arranged 
          horizontally or nearly vertically as per the tower design. Only in few situations you will find the   
          conductors are placed nearly equidistant from each other. Hence calculation of GMD and GMR are

         Here in Fig-D the three phase conductors are arbitrarily placed. Let the distance between the phase 
         conductors are D12, D23  and D31  . The distances are between the centers of bundled twin conductors.
         Similar to inductance, transposing the conductors the capacitance between any two phases is made
         equal. Or the capacitance between any phase and neutral point are made same. The above
         equation is actually derived considering transposed lines.

here,  GMD = ∛(D12 D23 D31)

         In fig-D ACSR twin bundled conductors are used for which GMR is calculated as below.
  • In case of bundled conductors, the GMR for commonly used bundles are as below
                                                          For twin conductor bundle
                                                                        = √(r . d)

                                                          For triple conductor bundle
                                                          GMR=[(r . d . d)(r . d . d)(r . d . d)]1/9
                                                                      = ∛(r.d2)

                                                          For quad conductor bundle
                                                          GMR = 1.09 ∜(r.d3)

          Note: In case of inductance r' is used. But here the actual radius r is used in GMR calculation
          In all the above formulas r is the actual radius of circular conductor. But usually ACSR conductors are
          used. For ACSR conductor in place of r put the value of Ds as supplied by the manufacturer. In Fig-D
          two ACSR conductors per bundle(twin) are used in each phase a, b and c.
                                So for Fig-D,      GMR =√(r . d)
  • In Fig-E is shown a three phase line (in power sector a three phase line is usually simply called as single circuit line. If the tower is carrying two numbers of such three phase lines then it is called double circuit line). The line is assumed as transposed. Here each phase conductor is comprised of four numbers of conductors(quad conductor). The conductors within a bundle are arranged in a square of side d. 

                                                          GMD = ∛(D . D . 2D)
                                                    GMR = 1.09 ∜(r.d3)
               As already said If ACSR stranded conductors are used (instead of circular one as shown) so Ds as
               per the manufacturer's data is used in place of r.  Ds is the equivalent radius of stranded conductor.
               The values of GMD and GMR are put on the above equation to find line to neutral capacitance.

  • So far we only considered one three phase circuit (single circuit). An example of double circuit will be considered exclusively  in next article for both inductance and capacitance calculation.
Earth being at zero potential influences the electric field. Some electric lines of force originating from conductors terminate on earth surface at 90 degrees. The presence of earth is tackled by considering imaginary image conductors placed below the earth, just like image of real conductors. However the influence of earth on the capacitance of line is small in comparison to the line to line capacitance. So the influence of earth is neglected in many cases. We discuss it here.

When the line parameters for all the three phase conductors are nearly equal, then the line voltages at the other end of the line are more or less balanced. Of course the balanced three phase system can be solved by considering any one phase and neutral. This is called per phase analysis. It should be remembered that here neutral does not mean the requirement of a neutral conductor for transmission. Although the above general formula for capacitance derived considering transposed lines, but it is often used for non-transposed lines to  get approximate values


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Anteneh Nebyu said...

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Saad Salman Khan said...

can anyone tell me where d square come from in triple conductor bundle....

Unknown said...

r^3/9 * d^6/9 = r^1/3 * d^2/3

shiv mittal said...

nice explanation

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