Transmission Line Parameters: Capacitance and Conductance


Capacitance

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,

                           p.k
                 C =    -----------
                          ln(D/r)
                                                  
                          ( 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
          important.

        
         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
                                                          GMR=[(r.d)(r.d)]1/4
            
                                                                        = √(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. 

                                        Here,    
                                                          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.



Transmission Line Parameters Resistance and Inductance


The transmission lines are modeled by means of the parameters resistance, inductance, capacitance and conductance. Resistance and inductance together is called transmission line impedance. Also capacitance and conductance in parallel is called admittance  Here we are not going to derive the formulas rather to develop some concepts about the transmission line parameters. It will help us understand the transmission line modelling and in analyzing the power system. In this article we will discuss about the line resistance and inductance. In the next article we will discuss about line capacitance and conductance. 

Resistance

The conductors of the transmission lines have small resistance. For short lines, resistance plays an important role. As the line current increases so do the ohmic loss (I2R loss). When the current exceed a certain value the heat generated due to ohmic loss starts to melt the conductor and the conductor becomes longer that results in more sag. The current at which this condition of conductor is irreversible is called thermal limit of conductor. Short overhead lines should be operated well within this limit.

The resistance R of a conductor of length 'l'  and cross section 'a' is given by the formula

                         l
                 R =  ρ ----
                         a

Here  ρ is the resistivity of the conductor material which is a constant.

Transmission lines usually use ACSR conductors with spirally twisted strands. So the actual length of the conductor is about 2 % more than the ACSR conductor length. So from the above formula, the resistance of the line is proportionately 2% more than the conductor length. Another important factor is that when the frequency of current increases the current density increases towards the surface of conductor  and current density at the center of conductor is less. That means more current flows towards the surface of conductor and less towards the center. This is well known skin effect. Even at power frequency (60/50 Hz) due to this skin effect  the effective cross sectional area of conductor is less. Again from the above equation it is clear that the conductor resistance is more for higher frequency. So AC resistance of conductor is more than the DC resistance. Temperature is another factor that influences the resistance of conductor. The resistance varies linearly with temperature. The manufacturers specify the resistance of the conductor and one should use the manufacturers data.


Inductance

For medium and long distance lines the line inductance (reactance) is more dominant than resistance. The value of current that flows in a conductor is associated with another parameter, inductance. We know that a magnetic field is associated with a current carrying conductor. In AC transmission line this current varies sinusoidally, so the associated magnetic field which is proportional to the current also varies sinusoidally. This varying magnetic field induces an emf (or induced voltage) in the conductor. This emf(or voltage) opposes the current flow in the line. This emf is equivalently shown by a parameter known as inductance. The inductance value depends upon the relative configuration between the conductor and magnetic field. Inductance in simple language is the flux linking with the conductor divided by the current flowing in the conductor. In the calculation of inductance the flux inside and outside of the conductor are both taken care of. The inductance so obtained is total inductance. Now onwards if not exclusively mentioned then inductance means total inductance due to conductor internal and external flux linkages. The symbol L is used universally to represent inductance.  L is measured in Henry (H). It is usually expressed in smaller unit, milli Henry(mH). Manufactures usually specify inductance value per kilometer or mile.
     
      It should be noted that, in all the formulas below inductance L is in Henry per unit length and not simply Henry. Here few cases are depicted.
  • For a single phase line see the fig-A. The conductor inductance is

                                           L = 2 * 10-7 ln ( D/r1)

                                     
          Here D is the distance between the centers of conductors.


                                             r1= r* e-(1/4) = 0.7788 r1
                                        
              r1 is the actual radius of the conductor.

For a single phase line the return path also has inductance say L'. If the return conductor is of radius r2, then

                              L' = 2 * 10-7 ln ( D/r2)
                                    
Therefore the total inductance of single phase circuit is Lt = L+L'
rearranging  we get
                                    

                                             L= 4 * 10 -7 ln [D / √ (r1'. r2')]


                                    


  • For three phase circuit whose three circular conductors are at the corners of equilateral triangle(Fig-B(i)) then the above formula for single phase case is applied here. In this case inductance per phase L is as below:
                          If the Denominator is renamed as Ds, then

                               L = 2 * 10-7 ln ( D / Ds )
              
                          Here Ds = r'
                        
                          As already said r' is 0.7788 times the actual radius(r) of conductor.


  • For three phase circuit whose three circular conductors are arbitrarily placed (Fig-B(ii)) and the conductors are transposed then,

                               L = 2 * 10-7 ln [ ∛(D1 . D2 . D3) / Ds ]

                                                   
Beginning from the single phase line, it is observed that all the three equations for inductance of a phase                    conductor are similar. Remember that this formula for three phase line is not valid for non-transposed lines.

 Observing the formula for single phase and three phase lines we can generalize the formula for inductance of  a phase line as in the form

                       L = 2 * 10-7 ln ( D / Ds )

                                  Where
                                  D = Geometric Mean Distance (GMD)
                                  Ds= Geometric Mean Radius (GMR)

In single phase case GMD is simply the distance between the centers of two conductors.
In three phase case for conductors equidistant from each other GMD is the distance between any two phase conductors.
In the three phase case, for line conductors arbitrarily placed GMD = ∛(D1D2D3 )

In all the three cases D = r'.
  • From above we can conclude that GMD is like equivalent distance between conductors. When two or more conductors per phase are used as in bundled conductors then GMD is required to be computed. Here distances from each conductor in one phase to each conductor in other phase is calculated. If for example in a single phase line there are 4 conductors in one phase and 3 conductors in other phase (Fig-C) then we will have 12 numbers of distances between the conductors. I have shown four distances only. 
                                     GMD = [D1 . D2 ........ D121/12
                                    
                          so here GMD is the 12th root of product of 12 numbers of distances.  


  • GMR is calculated for each phase separately. Each of the phases may have different GMR values depending upon the conductor size and arrangement.  GMR is to be calculated when each phase is comprised of more than one conductor per phase as in the example above. For GMR calculation when two or more conductors per phase are used, first  product of all the groups (one group for each conductor)are found where each group is product of possible distances from one conductor to other conductors including r' of that conductor.  In the above example case GMR for line with 3 conductors per phase is
                                    GMR = [(r1'.D12.D13)(r2'.D23.D21)(r3'.D31.D32)]1/9
                  
                           It should be noted that D12 = D21,   D13 = D31 and D23 = D32

          For three conductors per phase (triple conductor)
                                              GMR = ∛(Ds *d 2)
      
          For four conductors per phase (quad conductor)
                                              GMR = 1.09 ∜(Ds *d 3)          


        
How to calculate GMD of three phase line with bundled conductors? For an example see Fig-E where three phase bundles (triple conductror) are placed horizontally on  transmission towers. In this case the distance between the conductors (D) is taken as distance between the centers of bundled conductors.

                                        So,  GMD =  ∛(D.D.2D)

You can also calculate considering the distance from each bundled conductors  of one phase to other conductors of two other phases. But the GMD calculated does not vary significantly from our simple form above. This is due to the fact that D is quite larger than d.


  • For ACSR conductors GMR is specified by the manufacturer.If this GMR is called Ds. For example if two such ACSR conductors(twin conductor) are used in a bundle for each phase. The GMR of the phase conductor arrangement is calculated imagining that the supplied GMR (or Ds) as the equivalent radius of ACSR conductor. 
          Hence if d is the distance between the centers of the two ACSR conductors, similar to the formulas in
          Fig-D,

                                             GMR= [(Ds.d).(Ds.d)]1/4 =√(Ds . d)

  • We will discuss Inductance and capacitance for double circuit after discussing line capacitance in next article.                     

Usually it is not always possible to arrange the phase lines equilaterally on the towers. To make the inductance and capacitance of all the three phases nearly equal, the conductors are transposed. Which means the conductors exchange the position after 1/3 rd of line length. By transposing the inductance and capacitance of all the three phase lines are made nearly equal. This helps balancing the three phase voltages at the receiving end of the line. Although the above formulas are derived considering transposition, the same formulas are also used for non-transposed cases to get approximate values.

Transmission line capacitance and conductance