Lesson 5:
“CAPACITANCE” OF TRANSMISSION LINE
OUTLINE:
Lesson 5:
ON A TRANSMISSION LINE
I. The Representation of Transmission Lines
II. Two Port Network Analysis
III.The Short Transmission Line (STL)
IV.The Medium Transmission Line (MTL)
V. The Long Transmission Line (LTL)
“V-I RELATIONSHIP” OF TRANSMISSION LINE
B. REPRESENTATION OF LINES BASED ON LENGTH
TR ANS M IS S ION
LINE (TL)
TRANSMISSION LINE
TRANSMISSION LINE
RECEIVING END
SENDING END
A. INTRODUCTION
KILOMETER
(KM)
MILES
(MI)
80 below
50 below
80-240
50-150
Above 240
Above 150
***What values we usually looked for?
 V&I
 S,P,Q, pf
 Line losses
 Efficiency of the
line
 %VD & %VR
NOTE: R,L,C of transmission lines are called the
of TL and they are considered
once installed.
means NEGLECTED only
“V-I RELATIONSHIP” OF TRANSMISSION LINE
VSN  f VRN , I R 
I S  f VRN , I R 
TWO PORT NETWORK ANALYSIS
 used in analyzing transmission line.
 it includes:
port and
port.
VSN  A VRN  B I R
I S  C VRN  D I R
O
U
T
P
U
T
I
N
P
U
T
In a transmission line,
I
N VS /VSN
P
U IS
T
In a transmission line the GENERAL EQUATION is,
VR /VRN
TRANSMISSION
LINES
(RLC CONSTANTS)
IR
VSN
O
U
T
P
U
T
IS
V & I at
“SENDING “
end

A
C
B VRN
D IR
“ABCD CONSTANT”
*dependent on the type of line
V & I at
“RECEIVING “
end
“V-I RELATIONSHIP” OF TRANSMISSION LINE
Constants
 Generalized circuit constants of
transmission line using 2-port network.
1. Using Circuit Analysis
CONSTANT
2. Using Two-Port Network Analysis
I
N VSN
P
U IS
T
VRN
TRANSMISSION
LINES
(RLC CONSTANTS)
VSN  A VRN  B I R
I S  C VRN  D I R
VSN
IS

A
C
B VRN
D IR
IR
UNIT
Unitless
O
U
T
P
U
T
Ohm
Mho
Unitless
TAKE NOTE:
z= series impedance per unit length per phase
y= shunt admittance per unit length per phase to neutral
I = length of line
Z= z(l) = total series impedance per phase
Y= y(l) = total shunt admittance per phase to neutral
“V-I RELATIONSHIP” OF TRANSMISSION LINE
In Matrix Form,
A. Single Phase, Line-to-Line Analysis
VS
IS

1
0
2 Z L VR
IR
1
Constants
CONSTANT
FORMULA
1
2ZL
VS  1 VR
I S  0  VR


2 Z L  I R
1 I R
0
1
“V-I RELATIONSHIP” OF TRANSMISSION LINE
In Matrix Form,
B. Single Phase, Line-to-Neutral Analysis
VSN
IS
NOTE:

1
0
VS= 2VSN
Z L VRN
1 IR
VR= 2VRN
Constants
CONSTANT
FORMULA
1
VSN  1 VRN
I S  0  VRN


Z L 
1
ZL
IR
IR
0
1
“V-I RELATIONSHIP” OF TRANSMISSION LINE
In Matrix Form,
C. Three Phase, Line-to-Line Analysis
VS
IS

1
0
Z L 1  1-120 VR
IR
1
Constants (line ab)
CONSTANT
FORMULA
1
(1-1cis120)*Z
L
VS  1 VR
I S  0  VR


1  1-120Z L  I R
1 I R
0
1
“V-I RELATIONSHIP” OF TRANSMISSION LINE
In Matrix Form,
D. Three Phase, Line-to-Neutral Analysis
VSN
IS

1
0
Z L VRN
1 IR
Constants
CONSTANT
FORMULA
1
ZL
VSN  1 VRN
I S  0  VRN


Z L 
1
0
IR
IR
1
“V-I RELATIONSHIP” OF TRANSMISSION LINE
A. NOMINAL π MTL, Line-to-Neutral Analysis
VSN
Z LYC 

 1 
 VRN  Z L  I R
2 


Z LYC2
I S   YC 
4


Z Y 

 VRN  1  L C  I R
2 


In Matrix Form,
Constants
Z Y
A  1 L C
2
Z LYC2
C  YC 
4
VSN
B  ZL
D  1
IS
Z LYC
2
Z LYC
1
2

Z LYC2
YC 
4
ZL
VRN
Z LYC
1
2
IR
“V-I RELATIONSHIP” OF TRANSMISSION LINE
B. NOMINAL T MTL, Line-to-Neutral Analysis
L
L
VSN
Z Y

 1  L C
2

IS 
YC  VRN

Z L2YC 

 I R
 VRN   Z L 
4 


Z LYC 

 1 
 IR
2 

In Matrix Form,
Constants
Z Y
A  1 L C
2
C  YC
Z L2YC
B  ZL 
4
Z Y
D  1 L C
2
VSN
IS
Z Y
1 L C
2

YC
Z L2YC
ZL 
4 VRN
Z LYC
IR
1
2
“V-I RELATIONSHIP” OF TRANSMISSION LINE
SHORT TRANSMISSION LINE and MEDIUM TRANSMISSION LINE
1.
A 100 mile, three phase transmission line delivers 55MVA at 0.8 pf lagging
to the load at 132kV. The line is composed of “DRAKE” conductors with flat
horizontal spacing of 11.9ft between adjacent conductors. Assume a wire
temperature of 50°C. Determine the sending voltage, sending current,
sending power and line losses using:
a. “SHORT” line representation
b. “MEDIUM (NOMINAL π)” line representation
c. “MEDIUM (NOMINAL T)” line representation
“V-I RELATIONSHIP” OF TRANSMISSION LINE
where
A. LONG TRANSMISSION LINE:
SOLUTION TO DIFFERENTIAL EQUATIONS
z= series impedance per unit length per phase
yc= shunt admittance per unit length per phase to neutral
x = length of line from “RECEIVING” end
TAKE NOTE:
x= Independent Variable
I, V = Dependent Variables
z, yc= constant parameters
V  C1e
C


yc z x
yc z x

yc z x

yc z x
 C2 e
C1e
C2 e
I

z / yc
z / yc
“V-I RELATIONSHIP” OF TRANSMISSION LINE
Thus,
A. LONG TRANSMISSION LINE:
SOLUTION TO DIFFERENTIAL EQUATIONS
when
x=0
V= VRN
I=IR
VRN
C1 

2
z / yc
IR
2
VRN
C2 

2
z / yc
IR
2
VRN
V 

 2
 
z / yc
I R e
2

VRN


 2
 
z / yc
I R e
2

 VRN
1  
I 
 I R e
2 
 2 z / yc
 VRN
1  

 I R e
2 
 2 z / yc
yc z x
yc z x
yc z x
yc z x
“V-I RELATIONSHIP” OF TRANSMISSION LINE
Let:
B. LONG TRANSMISSION LINE:
INTERPRETATION OF THE EQUATION
ZC 
z
yc
 
yc z
    j
From
Where:
VRN
V 


 2
V
  RN 

 2
z / yc
2
 
I R e


 
z / yc
I R e
2



 
VRN
1
I  

I R e
2


 2 z / yc


 
VRN
1


I R e
2


 2 z / yc

yc z x
α= Attenuation Constant (nepers per unit length)
β= Phase constant (radians per unit length)
Thus
yc z x
yc z x
yc z x
ZC
V

V   RN 
I R  e x
2
 2

ZC
V

  RN 
I R  e  x
2
 2

V

1
I   RN 
I R  e x
2
 2Z C

V

1
  RN 
I R  e 
2
 2Z C

x
“V-I RELATIONSHIP” OF TRANSMISSION LINE
Constants
A  cosh x 
C. LONG TRANSMISSION LINE:
HTPERBOLIC FORM OF THE EQUATIONS
B  Z c sinh x 
C
V  cosh x  VRN  Z c sinh x  I R
I
 sinh x  

 VRN  cosh x  I R
 Zc

cosh x 
 sinh x 
I
Zc
D  cosh x 
RECALL: Note in “RADIANS”
cosh x   cosh x  j x 
In Matrix Form,
V
sinh x 
Zc
Z c sinh x 
cosh x 
cosh x   cosh x  cos x   j sinh x sin  x 
VRN
IR
sinh x   sinh x  j x 
sinh x   sinh x  cos x   j cosh x sin  x 
“V-I RELATIONSHIP” OF TRANSMISSION LINE
LONG TRANSMISSION LINE
A three phase 60Hz transmission line is 250 mile long. The voltages at the
sending end is 220kV. The parameters of the line are r= 0.2Ω/mi, xL=0.8Ω/mi
and yC=5.3μmho/mi.
a. Find the following at no load
1.
2.
3.
IS and VR
I and V at the middle of the line
I and V at 50 miles from the sending end
b. Find the following if the load on the line is 80MW at 220kV with unity power
factor
1.
2.
3.
IS , IR and VS
I and V at the middle of the line
I and V at 50 miles from the sending end
Lesson 5: