For transmission lines of length up to 80 km and transmitting power at relatively low voltage; (voltage below 20 kV), the capacitance is too small that its effects can be neglected. However, the effects of shunt capacitance become more and more pronounced with the increase in length and operating voltage of the line.
Since medium transmission lines have sufficient length exceeding 80 km and usually operate at voltages exceeding 20 kV, the capacitive current is appreciable and, therefore, line capacitance is to be taken into account. The capacitive current is always flowing in the line while the supply end switches are closed even though the receiving end of the line may be open-circuited.
The magnitude of capacitive current flowing at any point along the line is that required to charge the section of the line between the given point and the receiving end, hence it has a maximum value at the sending end and diminishes at a practically uniform rate down to zero at the receiving end.
Actually the capacitance of the line is uniformly distributed over its entire length, as illustrated in Fig. 5.4. However, in order to make the calculations simple, the capacitance of the system is assumed to be divided up, and ‘lumped’ in the form of capacitors shunted across the line at one or more points—the more the points the closer the approximation.
The most common methods of representation, called the localised capacitance methods, are:
(i) End condenser method, nominal-T method (or middle condenser method), and
(ii) Nominal π-method (or split-condenser method).
The above methods of solution of transmission lines are only of academic interest as far as long lines are concerned because these involve considerable error owing to the actual distributed capacitance having been assumed lumped. However, for lines up to 200 km at 50 Hz, the error involved by the use of these methods will be of the order of 10 per cent for sending-end voltage calculations which is an acceptable approximation.
ADVERTISEMENTS:
It is interesting to note that the end condenser method and the nominal T-method over compensates (giving sending-end voltages which are too low), while the nominal-π method under compensates. Because of the approximations involved in T and π-methods, they are often referred to as nominal-T and nominal-π methods.
1. End Condenser Method:
In this method the capacitance of the line is assumed to be lumped at the load-end, as shown in Fig. 5.5. This method overestimates the effect of capacitance.
Let the load current per phase be IR lagging behind the load end phase voltage VR by and angle ɸR. Let resistance, and reactance per phase of the line be R and X ohms respectively. Capacitance per phase of line is C farads.
ADVERTISEMENTS:
Taking receiving-end phase voltage as reference phasor we have VR = VR (1 + j 0)
Receiving-end or load-end current, IR = IR (cos ɸR – j sin ɸR)
Capacitive current, IC = j VR ω C = j 2 π f C VR
Sending-end current, IS = IR + IC = IR (cos ɸR – j sin ɸR) + j 2 π f C VR
ADVERTISEMENTS:
Voltage drop in line (per phase) = ISZ = IS (R + j X)
= [IR (cos ɸR – j sin ɸR) + j 2 π f C VR] (R + j X)
= IR R cos ɸR + IRX sin ɸR – 2 π f C VR X + j [IR X cos ɸR — IR R sin ɸR + 2 π f C VR R]
= IR [R cos ɸR + X sin ɸR) + j (X cos ɸR – R sin ɸR)] – 2 π f C VR (X – j R)
ADVERTISEMENTS:
Phase voltage at the sending end, VS = VR + ISZ … (5.23)
Phasor diagram is shown in Fig. 5.6 where horizontal line OA is the voltage to neutral at the receiving end; OD is the load current IR that lags behind VR by ɸR; DF is the capacitive current IC that leads the receiving-end voltage VR by 90°; OF is the sending-end current IS that flows through the line and is the phasor sum of receiving-end current IR and capacitive current IC; AB is the resistive voltage drop ISR that is in phase with IS; BC is the reactive voltage drop ISX that leads the IS by 90°; OC is the sending-end voltage VS that is the phasor sum of VR, IS R and IS X (or VR and IS Z); and ɸS, the phase angle between the sending-end voltage VS and sending-end current IS the sending-end phase angle that determines the sending-end power factor.
2. Nominal T-Method (or Middle Condenser Method):
In this method the whole of the line capacitance is assumed to be concentrated at the middle point of the line and half the line resistance and reactance (i.e. R/2 and X/2) to be lumped on either side, as shown in Fig. 5.7. Thus in this arrangement, at the terminals of the condenser the voltage has a value V’ intermediate between the sending-end voltage VS and the receiving-end voltage VR.
A capacitive current IC leading V’ by 90° accordingly flows over first half of the line from the sending-end. The current in the receiving-end half of the line is IR, and in the sending-end half of the line is IS which is the phasor sum of the receiving-end (or load) current IR and charging or capacitance current IC.
Taking receiving-end phase voltage (voltage to neutral) as the reference phasor we have,
In the phasor diagram shown in Fig. 5.8, horizontal line OA is the voltage to neutral at the receiving-end; OB is the current in the receiving-end half of the line (i.e. IR) drawn ɸR degrees behind OA; ɸR is the load power factor angle; AC drawn parallel to OB is the ohmic voltage drop in the receiving-end half of the line, CD drawn 90° ahead of OB is the reactive voltage drop in the receiving-end half of the line; OD is the voltage across the capacitor; BF drawn 90° ahead of OD is the charging or capacitive current IC; OF (the phasor sum of OB and BF) is the sending-end current IS; DG drawn parallel to OF is the ohmic voltage drop in the sending-end half of the line; GH drawn 90° ahead of OF is the reactive voltage drop in the sending end half of the line; OH is the sending-end voltage; and ɸS (the angle between OH and OF) is the phase angle at the sending-end.
3. Nominal π-Method (or Split-Condenser Method):
In this method the capacitance of each line conductor is assumed to be divided into two halves, one half being shunted between line conductor and neutral at the receiving-end and the other half at the sending end, as shown in Fig. 5.9. The current flowing in the line at any point in between the two capacitors is IL which is phasor sum of the load current IR and the current ICR drawn by the receiving-end capacitor. Knowing the line current IL, the voltage at the sending-end can easily be determined by the fundamental impedance method used for short lines.
The sending-end voltage is, therefore, more readily calculated by this method than by nominal-T method which needs the calculation of two separate halves of the line. Though the capacitance at the sending-end has no effect on the voltage drop in the line but if the sending-end current IS and power factor cos ɸS are to be determined, the current drawn by the sending-end capacitor, ICS is to be added vectorially to that in the line.
Taking receiving-end phase voltage (voltage to neutral) as the reference phasor we have:
Sending-end current, IS = IL + ICS … (5.31)
In phasor diagram shown in Fig. 5.10 horizontal line OA is the voltage to neutral at the receiving end VR; OB is the load current IR drawn ɸR degrees behind OA; ɸR is the load power factor angle. The capacitive current due to capacitance localised at the receiving end, ICR will lead the phasor VR by 90° and will be equal to ½ ω VR in magnitude. Let it be represented by phasor BC. The line current IL will be the phasor sum of IR and ICR and is represented by phasor OC.
The voltage drop in the line is equal to phasor sum of resistive drop IL R in phase with phasor IL represented by AD and reactive drop IL X in quadrature with IL represented by DF. The phase voltage at the sending-end VS, the phasor sum of receiving-end phase voltage VR and the line drop is represented by phasor OF. The sending end current IS, which is phasor sum of IL and ICS (ICS is equal to ½ ωCVS, in magnitude and leads VS by 90°), is represented by phasor OG.