Transcript transmission line equations

TRANSMISSION LINE EQUATIONS
• Transmission line is often schematically represented as a
two-wire line.
i(z,t)
z
V(z,t)
Δz
Figure 1: Voltage and current definitions.
The transmission line always have at least two conductors.
Figure 1 can be modeled as a lumped-element circuit, as
shown in Figure 2.
TRANSMISSION LINE EQUATIONS
• Supposed a T-Line is used to connect a source to a
load, as shown below
At position x along the line, there exist a time-varying
voltage v(x,t), and current, i (x,t)
TRANSMISSION LINE EQUATIONS
• For a small section between x and x+Δx equivalent
circuit for this section can be represented by the
distributed elements
Applying Kirchoff’s voltage and current law to this
equivalent circuit, 2 equations are produced
TRANSMISSION LINE EQUATIONS
i ( x, t )
 ( x  x, t )   ( x, t )   ( x, t )  ( Rx)i ( x, t )  ( Lx)
t
( x  x, t )
i ( x  x, t )  i ( x, t )  i ( x, t )  (Gx) ( x  x, t )  (Cx)
t
• Dividing the above eqs by Δx, and taking the limits as Δx
-> 0;
( x, t )
i ( x, t )
  Ri ( x, t )  L
..................( 2.18)
x
t
i ( x, t )
( x, t )
 G ( x, t )  C
................( 2.19)
x
t
TRANSMISSION LINE EQUATIONS
Differentiating eq 2.18 wrt x and eq2.19 wrt t;
  ( x, t )
i ( x, t )
 i ( x, t )
 R
L
...................(2.20)
2
x
x
xt
 2 i ( x, t )
( x, t )
 2 ( x, t )
 G
C
................(2.21)
2
2
x
t
t
2
2
TRANSMISSION LINE EQUATIONS
• By substituting eq2.19, eq2.21 into eq2.20, one can
2

i
eliminate i x and
(xt ) . If only the steady state
sinusoidally time varying solution is desired, phasor
notation can be used to simplify these equations
• Here, v and i can be expressed as:
 ( x, t )  ReV ( x)e jt ...................(2.22)


i( x, t )  Re I ( x)e jt .....................(2.23)
Where:
Re [ ]is the real part
ω is the angular frequency
TRANSMISSION LINE EQUATIONS
• A final equation can be written as:
d 2V ( x)
2


V ( x)  0.................................................(2.24)
2
dx
** Note that eq2.24 is a wave eq, and γ is the wave
propagation constant given by
1
2
  [( R  jL)(G  jC )]    j ......................(2.25)
Where:
α = attenuation constant (Np/m)
β = phase constant (rad/m)
TRANSMISSION LINE EQUATIONS
• The general solution to eq2.24 is
V ( x)  V  e x  V  ex .............................................(2.26)
• Eq2.26 gives the solution for voltage along the
transmission line.
• The voltage is a summation of forward wave (V+e-γx) and
reflected waves (V-e+γx) propagating in the +x and –x
directions, respectively
TRANSMISSION LINE EQUATIONS
• The current I(x) can be found from eq2.18 in the freq
domain:
I ( x)  I  e x  I e x .............................................(2.27)
• Where;

I 

I 

R  jL

R  jL
V
V
TRANSMISSION LINE EQUATIONS
• The characteristic impedance of the line is defined by;
V
V
R  jL  R  jL 

Z0     
 
I
I

 G  jC 


1
2
• For a lossless line, R=G=0, and we have;
  j  j LC
Z0 
p 
L
C

1
 f g 

LC
Where:
λg = guided wavelength
β = propagation constant
TRANSMISSION LINE
EQUATIONS
• If a coaxial line with inner and outer radii of a and b,
respectively, as shown below:
55.63 r
C
ln b
a
 
b
L  200 ln  
a
b
L  200 ln  
a
Where:
λg = guided wavelength
β = propagation constant
0.3495 r f GHz tan 
G
ln b
a
1 1
R  10  
a b
f GHz
