Name: Ziad Ezzat Mohammed Mohammed
ID:19015724
Modeling transmission lines
Abstract
In order to obtain sending and receiving ends currents, voltages and powers of the transmission
lines, it’s required to solve the two ports model of the lines and define ABCD parameters.
Since the three phase system is balance, it’s enough to obtain ABCD, Vs, Is, Vr and Ir for one
phase and apply the results for the remaining ones.
ABCD parameters give relation between sending and receiving voltages and currents as follows:
Characteristics:
A=D (non-dimensional) (1)
A2 -B*C=1 (2)
B has the units of impedance (ohm)
C has the units of admittance (mho)
Short lines
From the model:
Is=Ir (5) (series connection)
By applying KVL on the circuit:
Vs = Vr + Z*Ir (6)
Since Vs= A*Vr+B*Ir and Is= C*Vr+D*Ir (3) and (4)
From 1, 3,4,5,6
A=D=1, B=Z, C=0
Medium lines with ∏ representation
Y/2
From the model:
Is=Icd+Iab+Ir (6) KCL on nodes c and a
By applying KVLon the left circuit
Icd=Vs*Y/2 (7) where Y= J*W*C
By applying KVLon the right circuit
Iab=Vr*Y/2 (8)
By applying KVLon the outside circuit and KCL on node a
Vs=(Iab+Ir)*Z + Vr (9)
From 7,8: Vs=(Vr*Y/2 + Ir)Z+Vr=Vr(1+Z*Y/2) +Ir*Z (10)
Since Vs= A*Vr+B*Ir (3) , A=D (1)
A=D=1+Z*Y/2 , B=Z
since A2 -B*C=1 (2)
(1+ZY/2)2 -Z*C=1
1+ZY+(ZY/2)2 -ZC=1
ZC= ZY+(ZY/2)2
C=Y(1+ZY/4)
A=D=1+Z*Y/2
B=Z
C=Y(1+ZY/4)
Y/2
Medium lines with T representation
From the figure
Is=Iab+Ir (11)
By applying KVL on the right circuit
Iab/Y=Ir*Z/2+Vr
Iab=Ir*Y*Z/2+Vr*Y (12)
By substituting in 11 by 12
Is= Ir*Y*Z/2+Vr*Y+Ir=Ir(YZ/2+1)+Vr*Y (13)
Since Is= C*Vr+D*Ir (4) and A=D (1)
D=A=1+YZ/2
C=Y
since A2 -B*C=1 (2)
(1+ZY/2)2 -B*Y=1
1+ZY+(ZY/2)2 -B*Y=1
D=A=1+YZ/2
2
B*Y= ZY+(ZY/2)
B=Z(1+ZY/4)
B=Z(1+ZY/4)
C=Y