In this video, the characteristic impedance of a two conductor transmission line is been derived and calculated for a general case by applying KVL and KCL and also by considering time harmonic dependence as well as linear homogeneous differential equations. Thereafter, two special cases are also been explained by considering the general case as the reference to it.

1. LECTURE-2
-
CHARACTERISTIC
IMPEDANCE (Z0)
OF A
TRANSMISSION
LINEELECTROMAGNETIC
FIELDS &
TRANSMISSION LINES

2. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
A two-conductor transmission line supports a TEM wave; that is, the
electric and magnetic fields on the line are transverse to the direction of
wave propagation.
An important property of TEM waves is that the fields E and H are
uniquely related to voltage V and current I, respectively:
 dlEV .
 dlHI .
We will use circuit quantities V and I in solving the transmission line
problem instead of solving field quantities E and H (i.e., solving
Maxwell's equations and boundary conditions).

3. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
Let us examine an incremental portion of length z of a two-conductor
transmission line.
We intend to find an equivalent circuit for this line and derive the line
equations.
We expect the equivalent circuit of a portion of the line to be as in
Figure 1.
The model in Figure 1 is in terms of the line parameters R, L, G, and
C, and may represent any of the two-conductor lines.
The model is called the L-type equivalent circuit; there are other
possible types.
In the model of Figure 1, we assume that the wave propagates along
the +z-direction, from the generator to the load.

4. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
Figure 1 L-type equivalent circuit model of a differential length z of a
two-conductor transmission line

5. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
By applying Kirchhoff's voltage law to the outer loop of the circuit in
Figure 1, we obtain
),(
),(
.),(.),( tzzV
t
tzI
zLtzIzRtzV 



- (1)
t
tzI
LtzIR
z
tzVtzzV




 ),(
.),(.
),(),(
Taking the limit of the above equation as z0, which leads to
z
tzI
LtzIR
z
tzV






),(
.),(.
),(

6. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
By applying Kirchhoff's current law to the main node of the circuit in
Figure 1, we obtain
ItzzItzI  ),(),(
- (2)
t
tzzV
CtzzVG
z
tzItzzI




 ),(
.),(.
),(),(
Taking the limit of the above equation as z0, which leads to
z
tzV
CtzVG
z
tzI






),(
.),(.
),(
t
tzzV
zCtzzVzGtzzItzI



),(
),(.),(),(

7. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
If we assume harmonic time dependence, so that
]).(Re[),( tj
S ezVtzV 
 - (3)
In the differential equations, (5) and (6), VS and IS are coupled. To
separate them, we need to take the second derivative of VS in eq. (5)
S
S
ILjR
dz
dV
).( 
]).(Re[),( tj
S ezItzI 
 - (4)
Where VS(Z) and IS(Z) are the phasor forms of V(z,t) and I(z,t),
respectively, so therefore equation (1) and (2) can be written as follows:-
S
S
VCjG
dz
dI
).( 
- (5)
- (6)
dz
dI
LjR
dz
Vd SS
).(2
2

S
S
VCjGLjR
dz
Vd
)).((2
2
 
Substitute eq. (6) in the above equation, we obtain
S
S
V
dz
Vd 2
2
2
(or)
02
2
2
 S
S
V
dz
Vd

where,
)).(( CjGLjRj   and
- (7)

8. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
Similarly, we need to take the second derivative of IS in eq. (6)
dz
dV
CjG
dz
Id SS
).(2
2

S
S
ILjRCjG
dz
Id
)).((2
2
 
Substitute eq. (5) in the above equation, we obtain
S
S
I
dz
Id 2
2
2
(or)
02
2
2
 S
S
I
dz
Id

where, )).(( CjGLjRj  
and - (8)
The wavelength  and the wave velocity u are respectively given as



2




fu 

9. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
The solutions of the linear homogeneous differential eqn. (7) and (8)
are given by
zz
S eVeVzV  
 ..)( 00
where V0
+, V0
-, I0
+, and I0
- are the wave amplitudes; the + and - signs,
respectively, denote wave traveling along ‘+z’ and ‘-z’ directions.
zz
S eIeIzI  
 ..)( 00
- (9)
- (10)
Differentiate eq. (9) with respect to ‘z’, we have
dzeVdzeV
dz
zdV zzS
).().(
)(
00

 
 ).().( 00
zzS
eVeV
dz
dV 
 

]..[ 00
zzS
eVeV
dz
dV 
 


Substituting eq. (5) in the above equation, we have
]..[)( 00
zz
S eVeVILjR 
 

+Z
+Z
-Z
-Z

10. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)












zz
zz
eIeI
eVeVLjR




..
..)(
00
00
Here, ‘+z’ direction it taken, so V0
- and I0
- are considered to be zero










z
z
eI
eVLjR




.
.)(
0
0










0
0)(
I
VLjR


- (11)
Differentiate eq. (10) with respect to ‘z’, we have
dzeIdzeI
dz
zdI zzS
).().(
)(
00

 
 ).().( 00
zzS
eIeI
dz
dI 
 

]..[ 00
zzS
eIeI
dz
dI 
 


Substituting eq. (6) in the above equation, we have
]..[)( 00
zz
S eIeIVCjG 
 






 

 
S
zz
I
eVeVLjR 

 ..)( 00 

11. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)












zz
zz
eVeV
eIeICjG




..
..)(
00
00
Here, ‘-z’ direction it taken, V0
+ and I0
+ are considered to be zero.










z
z
eV
eICjG




.
.)(
0
0







 

0
0
)( I
V
CjG 

- (12)





 

 
S
zz
V
eIeICjG 

 ..)( 00 










0
0)(
V
ICjG


The CHARACTERISTIC IMPEDANCE (Z0) of the line is the ratio of
positively traveling voltage wave to current wave at any point on the
line.

 )(
0
0
0
LjR
I
V
Z







 

, according to eq. (11)

12. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
)(0
0
0
CjGI
V
Z









 

)).((
)(
0
CjGLjR
LjR
Z





since, )).(( CjGLjRj  
Also, , according to eq. (12)
)(
)).((
0
CjG
CjGLjR
Z





since, )).(( CjGLjRj  
000
)(
)(
jXR
CjG
LjR
Z 





Finally,
Finally, - (13)
000
)(
)(
jXR
CjG
LjR
Z 






13. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
The derivation which we have done till yet is valid only for general
case, we may now consider two special cases, one is lossless
transmission line and the other is distortionless line.
CASE: I - Lossless Transmission Line (R=G=0)
A transmission line is said lo be lossless if the conductors of the line
are perfect (σC ) and the dielectric medium separating them is lossless
(σC  0).
For such a line, (R=G=0)
Recalling the equation of ‘’, i.e.;
)).(( CjGLjRj  
)).(()0).(0( CjLjCjLjj  
LCjLCjLCjj   ))()( 222
LCjj   0
0 LC and

14. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
Recalling the equation of velocity ‘u’, i.e.;





f
LCLC
u 
1
At last recalling the equation of characteristic ‘Z0’, i.e.;
000
)(
)(
jXR
CjG
LjR
Z 





000
)0(
)0(
jXR
C
L
Cj
Lj
Cj
Lj
Z 







000 0 jXRj
C
L
Z 
C
L
R 0
00 Xand

15. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
CASE: II - Distortionless Transmission Line (R/L = G/C)
A distortionless line is one in which the attenuation constant ‘α’ is
frequency independent while the phase constant ‘β’ is linearly
dependent on frequency.
For such a line, (R/L = G/C)
Recalling the equation of ‘’ once again, i.e.;
)).(( CjGLjRj  
)1).(1()1().1(
G
Cj
R
Lj
RG
G
Cj
G
R
Lj
Rj

 
)1).(1(
G
Cj
R
Lj
RGj

 
Since, (R/L = G/C), it can also be written as (L/R = C/G)
2
)1()1).(1(
G
Cj
RG
G
Cj
G
Cj
RGj

 

16. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)







G
Cj
RGj

 1.
G
C
RGjRG
G
Cj
RGRGj .. 

 
L
R
G
jRG
R
L
RGjRGj ..  
LCjRGCLjRGj  
LCjRGj  
RG LC and
Recalling the equation of velocity ‘u’, i.e.;





f
LCLC
u 
1
)..(. LL
L
C
jRGL
L
C
jRGj  

17. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
At last recalling the equation of characteristic ‘Z0’, i.e.;
000
)(
)(
jXR
CjG
LjR
Z 





000
)1(
)1(
)1(
)1(
jXR
G
R
G
Cj
G
G
Cj
R
G
Cj
G
R
Lj
R
Z 










000 0 jXRj
G
R
Z 
C
L
G
R
R 0
00 Xand

18. DERIVATION OF CHARACTERISTIC IMPEDANCE (Z0)
NOTE:-
The phase velocity is independent of frequency because the phase
constant ‘β’ Linearly depends on frequency. We have shape distortion
of signals unless ‘α’ and ‘u’ are independent of frequency.
‘u’ and ‘Z0’ remain the same as for lossless lines.
A lossless line is also a distortionless line, but a distortionless line is
not necessarily lossless. Although lossless lines are desirable in power
transmission, telephone lines are required to be distortionless.
CASES
PROPAGATION CONSTANT
(γ = α + jβ)
CHARACTERISTIC
IMPEDANCE
(Z0 = R0 + jX0)
GENERAL
CASE
CASE –I
(LOSSLESS)
CASE-II
(DISTORTIONLESS)
)).(( CjGLjR  
)(
)(
0
CjG
LjR
Z





LCj  0
LCjRG  
00 j
C
L
Z 
00 j
C
L
Z 

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