W.C.Chew
ECE 350 Lecture Notes
5. Transmission Lines
σ= +
σ= -
GENERAL STRIP LINE COAXIAL
Examples of Tr ansmission lines
Symbol of a Tr ansmission Line
Symbol of a Transmission Line
Another place where wa v ephenomenon is often encountered is on trans-
mission lines. A transmission line consists of t wo parallel conductors of ar-
bitrary cross-sections that can carry t wo opp osite currents or t wo opp osite
c harges. A transmission line has capacitances between the t wo conductors,
and the conductors hav e inductances to them. We can c haracterize this ca-
1
pacitance b y a line capacitance C which has the unit of farad m , and the
1
inductance b y a line inductance L, which has the unit of henry m . Hence
a transmission line can be approximated b y a lump ed element equivalent as 1
shown
L∆ZLVV+∆ZL∆VV+2∆Z ∆V
I II+∆II+2∆I
C∆Z C∆ZC∆Z Ð∆I
Z
We can derive the voltage equation between no des (1) and (2) to get
@I
; (1) V (V +V )= Lz
@t
or
@I
V = Lz : (2)
@t
Similarly, the current relation at no de (3) says that
@V @ (V +V )
' C z : (3) I = C z
@t @t
In the limit when we let our discrete or lump ed element mo del become very
small, or z ! 0, we have
@V @I
= L ; (4)
@z @t
and
@V @I
= C : (5)
@z @t
The ab ove are known as the telegrapher's equations. Wave equations can b e
easily derived from the ab ove
2 2
@ V @ V
LC =0; (6)
2 2
@z @t
and
2 2
@ I @ I
LC =0: (7)
2 2
@z @t
Comparing with Equation (3.17), we deduce that the velocity of the current
and voltage waves on a transmission line is
1
p
v = : (8)
LC 2
The solution to (6) maybe of the form
V (z; t)= f (z vt): (9)
Substituting into (4), we have
@I
0
L = f (z vt) (10)
@t
or
1
I (z; t)= f (z vt): (11)
Lv
Hence,
r
V (z; t) L
= Lv = (12)
I (z; t) C
for a forward going wave. The quantity
r
L
Z = (13)
0
C
is the characteristic impedance of a transmission line.
Lossy Transmission Line
Often time, a transmission line has loss to it. For example, the conductor
has a nite conductivity and hence is a little resistive. The insulation b etween
the conductors mayhave current leakage, thus not forming an ideal capacitor.
A more appropriate lump ed element mo del is as follows.
R∆ZL∆ZR∆ZL∆ZR∆ZL∆Z
G∆ZC∆ZG∆ZC∆ZG∆ZC∆Z
The ab ove circuit is more easily treated using phasor techniques. If we
have applied phasor technique to (4) and (5), we would have obtained
~
dV
~
I; (14) = j!L
dz
~
dI
~
= j!C V: (15)
dz 3
Note that j!L is the series imp edance per unit length of the lossless line
while j!C is the shunt admittance p er unit length of the lossless line. In the
lossy line case, the series imp edance per unit length b ecomes
Z = j!L + R (16)
while the shunt admittance per unit length b ecomes
Y = j!C + G (17)
where R and G are line resistance and line conductance resp ectively. The
telegraphers equations b ecome
~
dV
~
= Z I; (18)
dz
~
dI
~
= Y V; (19)
dz
and the corresp onding Helmholtz wave equations are
2
~
d V
~
ZY V =0; (20)
2
dz
2
~
d I
~
ZY I =0: (21)
2
dz
Similarly, the characteristic imp edance, is
s s
r
j!L j!L + R Z
) Z = = : (22) Z =
0 0
j!C j!C + G Y
Equations (20) and (21) are of the same form as (4.22) or
2
~
d V
2
~
V =0; (23)
2
dz
2
~
d I
2
~
I =0; (24)
2
dz
where
p
p
= ZY = (j!L + R )(j!C + G)= + j : (25)
The general solution is of the form (4.23). For example,
z + z
~
V (z )=V e + V e
+
z j z z +j z
= V e + V e : (26)
+
j +j
+
If V = jV j e ; V = jV j e , then the real time representation of
+ +
V is
j!t
~
V (z; t)= z z = jV j e cos (!t z + )+jV j e cos (!t + z + ): (27) + 1 2 4 The rst term corresp onds to a decaying wave moving in the p ositive z - direction while the second term corresp onds to a wave decaying and moving z in the negative z -direction. Hence, e corresp onds to a p ositive going wave, + z while e corresp onds to a negative going wave. If the transmission line is lossless, i.e., R = G =0, then, the attenuation constant = 0, and the propagation constant becomes = j . In this case, there is no attenuation, and (26) b ecomes j z +j z ~ V (z )=V e + V e ; (28) + and (27) becomes V (z; t)= jV j cos(!t z + )+ jV j cos(!t + z + ): (29) + 1 2 The wave propagates without attenuation or without decay in this case. The velo city of propagation is v = != . Furthermore, we can derive the current that corresp onds to the voltage in (26) using Equation (18). Hence ~ 1 dV Z + Z ~ I = = V e V e : (30) + Z dz Z Z But r Y 1 = = (31) Z Z Z 0 where Z is the characteristic imp edance given by Equation (22). Hence, 0 V V + Z Z Z Z ~ I = e e = I e + I e ; (32) + Z Z 0 0 where V V + = Z ; = Z : (33) 0 0 I I + 5