Dx

1
I(xDx) I(x) DI
I(x)
LDx
RDx
V(xDx) V(x) DV
CDx
GDx
V(x)
xDx
x

• Dx

2

• Letting ?x tend to zero

Hence
Equations 3 4 are called the "Telegrapher's
Equations" and are important because they govern
the variation of I and V as a function of
distance along the line (x) and time (t).
3

• But we can substitute for ?I/?t from one of the
  Telegrapher's Equations Equation (4)

Rearranging
Using similar steps it can also be shown that
The above two equations govern the voltage and
current along a general transmission line.
4

• Consider the special case of a lossless
  transmission line, i.e. one for which R 0 and G
  0. The previous equations reduce to

Note the similarity between
these two equations
and . . . . . . the wave equations
obtained from Maxwell's
Equations in Prof. Murray's section
Equations 9 and 10 describe voltage and current
WAVES travelling down the transmission line.
5

• Solution to the wave equations for a lossless
  transmission line.

Any function, F, which has tx/v or t-x/v as the
variable is a solution to these equations, v
being a constant with the dimensions of velocity
(i.e. units of m/s)
Note that F is a function of x AND t. It is not
surprising that F can be any function since we
can propagate sine waves, square waves,
triangular waves and waves of arbitrary shape
down e.g. a coaxial cable.
To verify that F(t x/v) is a solution we need
to put F into the wave equation and check that
6

Therefore F(t-x/v) is a solution of the wave
equations for the lossless transmission line
(Equations 9 and 10) if 1/v2 LC (or v
1/vLC) Similarly you can show that F(tx/v) is a
solution if 1/v2 LC (or v -1/vLC)

We will see later that the "" and "-" signs in
the solution show which direction the wave is
moving in. "-" indicates the wave is moving to
the right (ve x direction) "" indicates the
wave is moving to the left (-ve x direction)
.
7

• SYLLABUS
• Part 1 - Introduction and Basics
• Lecture Topics
• 1. General definition
• Practical definition
• Types of transmission line TE, TM, TEM
  modes
• TEM wave equation - equivalent circuit
  approach
• 2. The "Telegrapher's Equations"
• Solution for lossless transmission lines
  F(tx/v)
• Simplest case of F(tx/v)
• 3. Direction of travel of cos/sin (?t ?x)
  waves
• Phase velocity of a wave on a transmission
  line
• General transmission line attenuation

8

• Example 2.2 - Effect of a medium on the
• wavelength of an EM wave.
• A 3GHz EM wave is incident on a sheet of
  polystyrene ( er 2.7) with a hole. How
  thick should the sheet be in order that the wave
  passing through the sheet and the wave through
  the hole are in phase when they emerge from the
  sheet?

x0 xd
E?
E?
9
2.3 Properties of a wave on a transmission line.

• A wave of frequency 4.5 MHz and phase constant
  0.123 rad/m propagates down a lossless
  transmission line of length 500 m. Find
• (i) the wavelength
• (ii) the velocity of the wave
• (iii) the phase difference between the phasor
  voltages at the two ends of the line
• (iv) the time required for a reference point on
  the wave to travel down the line
• (v) the relative permittivity of the dielectric
  in the line

10

Direction of travel of cos/sin(wt bx) waves
This figure shows how the function VV1 sin (wt
bx) changes with time a point, P, of constant
constant phase moves to the LEFT with time.
P
l
T period t 0 (a) tT/4 (b) t T/2 (c)
P
P
VV1 sin (wt bx)
11

Direction of travel of cos/sin(wt bx) waves
This figure shows how the function VV1 sin (wt -
bx) changes with time a point, P, of constant
constant phase moves to the RIGHT with time.
P
l
V 1 0 -1
T period t 0 (a) tT/4 (b) t T/2 (c)
p 2p 3p
4p bx
P
V 1 0 -1
p 2p 3p
4p bx
P
V 1 0 -1
p 2p 3p
4p bx
VV1 sin (wt - bx)
12
Phase Velocity of a Wave on a Transmission Line

The PHASE VELOCITY of a wave is defined as the
velocity of a point of constant phase. For a
point of constant phase, V is constant, hence
wt bx constant To find
the velocity of this constant phase point we must
obtain ?x/?t, so differentiate the above w.r.t
time, t w b ?x/?t
0 \ PHASE VELOCITY ?x/?t w/b v For a
lossless transmission line with inductance per
unit length L and capacitance per unit length C
Example 2.1 shows that the velocity of a TEM wave
on a transmission line is the same as that of an
EM wave in the dielectric medium of the line.
13
General Transmission Line R g 0, G g 0
Until now we have dealt with the simplest case,
i.e. that of a lossless line (R0 and G0). We
know that for this type of line the voltage is
governed by the wave equation with
solution V V1e jwt e ?x For a line which isn't
lossless we saw that the voltage is governed by
Equation 7
By analogy with the solution for the wave
equation, for the general line let's take as a
trial solution to Equation 7 V V1e jwt e gx
V1e(jwtgx)
(14) But take ? to be complex rather than real
like ?.
14

V V1 e jwt e jbx Lossless Line (? is
real) V V1 e jwt egx General Line (g
is complex)
Here we are assuming that the time variation is
as before but the spatial variation is different
and as yet unknown since g is unknown. To check
that our trial function is a solution we need to
put it in Equation 7 and check that L.H.S.
R.H.S. Differentiating the trial function
(Equation 14) w.r.t. time
15
Substituting for ?V/?t and ?2V/?t2 in 7
If V V1 e(jwtgx) then
Comparing (16) and (17), we see that for our
trial function to be a solution
?2V (R jwL)(G jwC)V
gt g2 (R jwL)(G jwC)
(18)
Thus V1 e(jwtgx) is a solution to Equation 7 if
g (R jwL)(G jwC)1/2 (19)

16
The sign again indicates the direction the wave
is travelling in the solution with e-gx
corresponds to a forward travelling wave (x
direction) the solution with egx corresponds to
a backward travelling wave (-x
direction) The general solution is
V V1 e jwt e -gx V2 e jwt e
gx where V1 and V2 are independent arbitrary
amplitudes which depend on the circumstances.

17
Physically, this corresponds to two waves moving
simultaneously in opposite directions on the
transmission line V V1 e jwt e -gx V2 e
jwt e gx
- Backward (Reflected) voltage wave
- Forward voltage wave
x
-x
For the lossless case put R G 0 in Equation
19 g (R jwL)(G
jwC)1/2 (19) g (j2w2LC)1/2
jw(LC)1/2 jw/v since v 1/(LC)1/2
jb
since w/b v
18
Thus for the lossless case egx e jbx
i.e. g is
purely imaginary (? j?). In general, g is
complex and has both real and imaginary parts
g (RjwL)(GjwC)1/2 (a
jb) Hence egx e(ajb)x
eaxejbx The factor eax operates on the
amplitude of the wave, decreasing it
exponentially. a is termed the
ATTENUATION CONSTANT eax gives the amplitude
attenuation as the wave travels ejbx gives the
phase change over distance x
g is termed the PROPAGATION CONSTANT
19
For a forward travelling wave
V V1 e-ax e j(wt-bx)
amplitude factor phase factor
time
variation

20

• Example 3.1 - Phase velocity of a wave

Derive an expression for the phase velocity of a
wave on a general transmission line.
Voltage V1
x
21
Summary
q Direction of travel of cos/sin(wtbx) waves
the sign gives the DIRECTION OF TRAVEL for
the waves indicates the wave is
travelling to the left (i.e. in -x
direction) - indicates the wave is
travelling to the right (i.e. in x
direction) q The PHASE VELOCITY of a wave on a
transmission line is defined as the velocity of a
point of constant phase.
Phase velocity v fl w/b 1/(LC)1/2 for
a lossless line
22

q Voltage on a general transmission line R g 0,
G g 0
V V1e(jwtgx)
where g (RjwL)(GjwC)1/2 (a
jb) g is termed the PROPAGATION CONSTANT a is
termed the ATTENUATION CONSTANT b is the PHASE
CONSTANT V V1e(jwt-gx) V1e-axe
j(wt-bx) GENERAL SOLUTION is
V V1ejwte-gx V2ejwtegx
forward reflected
wave wave