Chapter 5 Transmission Lines
51 Characteristics of Transmission Lines
Transmission line: It has two conductors carrying current to support
an EM wave, which is TEM or quasiTEM mode. For the TEM mode,
, , and .
The current and the EM wave have different characteristics. An EM wave
propagates into different dielectric media, the partial reflection and
the partial transmission will occur. And it obeys the following rules.
Snell’s law: and θiθr
The reflection coefficient: Γ and the transmission
coefficient: τ
for perpendicular polarization (TE)
for parallel polarization (TM)
In case of normal incidence, , where η1 and η2
.
Equivalentcircuit model of transmission line section:
, , ,
Transmission line equations: In higherfrequency range, the
transmission line model is utilized to analyze EM power flow.
Set v(z,t)Re[V(z)ejωt], i(z,t)Re[I(z)ejωt]
where γα+jβ
Characteristic impedance: Z0
Note:
1.
International Standard Impedance of a Transmission Line is Z050Ω.
2.
In transmissionline equivalentcircuit model, G≠1/R.
Eg. The following characteristics have been measured on a lossy
transmission line at 100 MHz: Z050Ω, α0.01dB/m1.15×103Np/m, β0.8π(rad/m).
Determine R, L, G, and C for the line.
(Sol.) , 1.15×103+j0.8π
, ,
,
Eg. A dc generator of voltage and internal resistance is connected to
a lossy transmission line characterized by a resistance per unit
length R and a conductance per unit length G. (a) Write the governing
voltage and current transmissionline equations. (b) Find the general
solutions for V(z) and I(z).
(Sol.) (a)
(b)
Lossless line (RG0):
Lowloss line (R2: , where
Case 2: t/h＞0.005. In this case, we obtain Weff firstly.
For W/h :
For W/h :
And then we substitute Weff into W in the expressions in Case 1.
Assuming not the quasiTEM mode:
, where , (h in cm)
and , G0.6+0.009Z0, (f in GHz)
The frequency below which dispersion may be neglected is given by
, where h must be expressed in cm.
Attenuation constant: ααd +αc
For a dielectric with low losses: ( )
For a dielectric with high losses: ( )
For W/h → ∞: , where
For W/h :
For ＜W/h 2: , where
and
For W/h 2:
Eg. A highfrequency test circuit with microstrip lines.
52 Wave Characteristics of Finite Transmission Line
Eg. Show that the input impedance is Zi .
(Proof) ,
Let zl, V(l)VL, I(l)IL
, Zi
Lossless case (α0, γjβ, Z0R0, tanh(γl)jtanβl): Zi
Note: In the highfrequency circuit, the input current Ii :
the value in the lowfrequency case. And the highfrequency Ii is
dependent on the length l, the characteristic impedance Z0, the
propagation constant γ of the transmission line, and the load
impedance ZL. But the lowfrequency Ii is only dependent on Z0 and ZL.
Eg. A 2m lossless airspaced transmission line having a characteristic
impedance 50Ω is terminated with an impedance 40+j30(Ω) at an
operating frequency of 200MHz. Find the input impedance.
(Sol.) , , ,
Eg. A transmission line of characteristic impedance 50Ω is to be
matched to a load ZL40+j10(Ω) through a length l’ of another
transmission line of characteristic impedance R0’. Find the required l’
and R0’ for matching.
(Sol.)
Eg. Prove that a maximum power is transferred from a voltage source
with an internal impedance Zg to a load impedance ZL over a lossless
transmission line when ZiZg*, where Zi is the impedance looking into
the loaded line. What is the maximum power transfer efficiency?
(Proof) ,
When and , , ∴
In this case, , ,
Transmission lines as circuit elements:
Consider a general case: Zi
1. Opencircuit termination (ZL→∞): ZiZioZ0coth(γl)
2. Shortcircuit termination (ZL 0): ZiZisZ0tanh(γl)
∴ Z0 , γ
3. Quarterwave section in a lossless case (lλ/4, βlπ/2):
4.
Halfwave section in a lossless case (lλ/2, βlπ):
Eg. The opencircuit and shortcircuit impedances measured at the
input terminals of an airspaced transmission line 4m long are 250∠50°Ω
and 360∠20°Ω, respectively. (a) Determine Z0, α, and β of the line.
(b) Determine R, L, G, and C.
(Sol.) (a) ,
(b) ,
,
Eg. Measurements on a 0.6m lossless coaxial cable at 100kHz show a
capacitance of 54pF when the cable is opencircuited and an inductance
of 0.30μH when it is shortcircuited. Determine Z0 and the dielectric
constant of its insulating medium.
(Sol.) (a) ,
Lossless 74.5Ω,
General expressions for V(z) and I(z) on the transmission lines:
Let Γ , z’lz
For a lossless line, V(z)
Eg. A 100MHz generator with Vg10∠0° (V) and internal resistance 50Ω
is connected to a lossless 50Ω air line that is 3.6m long and
terminated in a 25+j25(Ω) load. Find (a) V(z) at a location z from the
generator, (b) Vi at the input terminals and VL at the load, (c) the
voltage standingwave radio on the line, and (d) the average power
delivered to the load.
(Sol.) , , , , ,
, ,
,
(a)
(b)
(c)
(d) ,
Eg. A sinusoidal voltage generator Vg110sin(ωt) and internal
impedance Zg50Ω is connected to a quarterwave lossless line having a
characteristic impedance Z050Ω that is terminated in a purely
reactive load ZLj50Ω. (a) Obtain the voltage and current phasor
expressions V(z’) and I(z’). (b) Write the instantaneous voltage and
current expressions V(z’,t) and I(z’,t).
(Sol.) (a)
(c) ,
,
(b)
Eg. A sinusoidal voltage generator with Vg0.1∠0° (V) and internal
impedance ZgZ0 is connected to a lossless transmission line having a
characteristic impedance Z050Ω. The line is l meters long and is
terminated in a load resistance ZL25Ω. Find (a) Vi, Ii, VL, and IL;
(b) the standingwave radio on the line; and (c) the average power
delivered to the load.
(Sol.) (a) , ,
(b) , (c)
Eg. Consider a lossless transmission line of characteristic impedance
R0. A timeharmonic voltage source of an amplitude Vg and an internal
impedance RgR0 is connected to the input terminals of the line, which
is terminated with a load impedance ZLRL+jXL. Let Pinc be the average
incident power associated with the wave traveling in the +z direction.
(a) Find the expression for Pinc in terms of Vg and R0. (b) Find the
expression for the average power PL delivered to the load in terms of
Vg and the reflection coefficient Γ. (c) Express the ratio PL/Pinc in
terms of the standingwave ratio S.
(Sol.) , ,
(a)
(b)
(c)
Case 1 For a pure resistive load: ZLRL
, 1. Γ0 S1 when ZL Z0 (matched load)
2. Γ1 S∞ when ZL0 (shortcircuit), 3. Γ1 S∞ when
ZL∞ (opencircuit)
occurs at
occurs at
If
If
If
Eg. The standingwave radio S on a transmission line is an easily
measurable quality. Show how the value of a terminating resistance on
a lossless line of known characteristic impedance R0 can be determined
by measuring S.
(Sol.) If , , occurs at and
occurs at .
, , , , or .
If , , occurs at , and occurs
at .
, , , . or
Case 2 For a lossless transmission line, and arbitrary load:
ZL , zm’+lmλ/2
Find ZL?
1. , 2. At θΓ2βzm’π, V(z’) is a minimum.
3. ZLRL+jXL
Eg. Consider a lossless transmission line. (a) Determine the line’s
characteristic resistance so that it will have a minimum possible
standingwave ratio for a load impedance 40+j30(Ω). (b) Find this
minimum standingwave radio and the corresponding voltage reflection
coefficient. (c) Find the location of the voltage minimum nearest to
the load.
(Sol.)
, ,
,
Eg. SWR on a lossless 50Ω terminated line terminated in an unknown
load impedance is 3. The distance between successive minimum is 20cm.
And the first minimum is located at 5cm from the load. Determine Γ, ZL,
lm, and Rm.
(Sol.)
0.15m
,
Eg. A lossy transmission line with characteristic impedance Z0 is
terminated in an arbitrary load impedance ZL. (a) Express the
standingwave radio S on the line in terms of Z0 and ZL. (b) Find the
impedance looking toward the load at the location of a voltage
maximum. (c) Find the impedance looking toward the load at a location
of a voltage minimum.
(Sol.) (a)
(b)
(c)
53 Introduction to Smith Chart
,
: rcircle, : xcircle
Several salient properties of the rcircles:
1.
The centers of all rcircles lie on the Γraxis.
2.
The r0 circle, having a unity radius and centered at the origin,
is the largest.
3.
The rcircles become progressively smaller as r increases from 0
toward ∞, ending at the (Γr1, Γi0) point for opencircuit.
4.
All rcircles pass through the (Γr1, Γi0) point.
Salient properties of the xcircles:
1.
The centers of all xcircles lie on the Γr1 line, those for x>0
(inductive reactance) lie above the Γr–axis, and those for x<0
(capacitive reactance) lie below the Γr–axis.
2.
The x0 circle becomes the Γr–axis.
3.
The xcircle becomes progressively smaller as |x| increases from 0
toward ∞, ending at the (Γr1, Γi0) point for opencircuit.
4.
All xcircles pass through the (Γr1, Γi0) point.
Summary
1.
All |Γ|–circles are centered at the origin, and their radii vary
uniformly from 0 to 1.
2.
The angle, measured from the positive real axis, of the line drawn
from the origin through the point representing zL equals θΓ.
3.
The value of the rcircle passing through the intersection of the
|Γ|–circle and the positivereal axis equals the standingwave
radio S.
Application of Smith Chart in lossless transmission line:
, when
keep |Γ| constant and subtract (rotate in the clockwise direction) an
angle from θΓ. This will locate the point for |Γ|ejφ, which
determine Zi.
Increasing z’ wavelength toward generator in the clockwise
direction
A change of half a wavelength in the line length A
change of in φ.
Eg. Use the Smith chart to find the input impedance of a section of a
50Ω lossless transmission line that is 0.1 wavelength long and is
terminated in a shortcircuit.
(Sol.) Given , ,
1.
Enter the Smith chart at the intersection of r0 and x0 (point
on the extreme left of chart; see Fig.)
2.
Move along the perimeter of the chart by 0.1 “wavelengths
toward generator’’ in a clockwise direction to P1.
At P1, read r0 and , or , .
Eg. A lossless transmission line of length 0.434λ and characteristic
impedance 100Ω is terminated in an impedance 260+j180(Ω). Find (a) the
voltage reflection coefficient, (b) the standingwave radio, (c) the
input impedance, and (d) the location of a voltage maximum on the
line.
(Sol.) (a) Given l0.434λ, R0100Ω, ZL260+j180
1.
Enter the Smith chart at zLZL/R02.6+j1.8 (point P2 in Fig.)
2.
With the center at the origin, draw a circle of radius .
( 1)
3.
Draw the straight line and extend it to P2’ on the
periphery. Read 0.22 on “wavelengths toward generator” scale.
, .
a.
The circle intersects with the positivereal axis
at rS4.
b.
To find the input impedance:
1. Move P2’ at 0.220 by a total of 0.434 “wavelengths toward
generator,” first to 0.500 and then further to 0.154 to P3’.
2. Join O and P3’ by a straight line which intersects the
circle at P3.
3. Read r0.69 and x1.2 at P3. .
d.
In going from P2 to P3, the circle intersects the
positivereal axis at PM, where the voltage is a maximum.
Thus a voltage maximum appears at (0.2500.220) or 0.030
from the load.
Application of Smith Chart in lossy transmission line
∴ We can not simply move close the |Γ|circle; auxiliary calculation
is necessary for the e2αz’ factor.
Eg. The input impedance of a shortcircuited lossy transmission line
of length 2m and characteristic impedance 75Ω (approximately real) is
45+j225(Ω). (a) Find α and β of the line. (b) Determine the input
impedance if the shortcircuit is replaced by a load impedance ZL
67.5j45(Ω).
(Sol.) (a) Enter in the chart as P1 in Fig.
Draw a straight line from the origin O through P1 to P1’.
Measure ,
Record that the arc is 0.20 “wavelengths toward generator”.
, . .
(b) To find the input impedance for:
1. Enter on the Smith chart as P2.
2. Draw a straight line from O through P2 to P2’ where the
“wavelengths toward generator” reading is 0.364.
3. Draw a –circle centered at O with radius .
4. Move P2’ along the perimeter by 0.2 “wavelengths toward generator”
to P3’ at 0.364+0.200.564 or 0.064.
5. Joint P3’ and O by a straight line, intersecting the
–circle at P3.
6. Mark on line a point Pi such that .
7. At Pi, read .
54 Transmissionline Impedance Matching
Impedance matching by λ/4transformer: R0’
Eg. A signal generator is to feed equal power through a lossless air
transmission line of characteristic impedance 50Ω to two separate
resistive loads, 64Ω and 25Ω. Quarterwave transformers are used to
match the loads to the 50Ω line. (a) Determine the required
characteristic impedances of the quarterwave lines. (b) Find the
standingwave radios on the matching line sections.
(Sol.) (a) .
,
(b) Matching section No. 1:
,
Matching section No. 2:
,
Application of Smith Chart in obtaining admittance:
, , where
Eg. Find the input admittance of an opencircuited line of
characteristic impedance 300Ω and length 0.04λ.
(Sol.) 1. For an opencircuited line we start from the point Poc on
the extreme right of the impedance Smith chart, at 0.25 in Fig.
2. Move along the perimeter of the chart by 0.04 “wavelengths toward
generator” to P3 (at 0.29).
3. Draw a straight line from P3 through O, intersecting at P3’ on the
opposite side.
4. Read at P3’: , .
Application of Smith Chart in singlestub matching:
, where yBR0YB, ysR0Ys
∵ 1+jbs yB, ∴ ysjbs and lB is required to cancel the imaginary
part.
Using the Smith chart as an admittance chart, we proceed as yL follows
for singlestub matching:
1.
Enter the point representing the normalized load admittance.
2.
Draw the |Γ|circle for yL, which will intersect the g1 circle at
two points. At these points, yB11+jbB1 and yB21+jbB2. Both are
possible solutions.
3.
Determine loadsection lengths d1 and d2 from the angles between
the point representing yL and the points representing yB1 and yB2.
Determine stub length lB1 and lB2 from the angles between the
shortcircuit point on the extreme right of the chart to the points
representing –jbB1 and –jbB2, respectively.
Eg. SingleStub Matching:
Eg. A 50Ω transmission line is connected to a load impedance ZL 35j47.5(Ω).
Find the position and length of a shortcircuited stub required to
match the line.
(Sol.) Given , ,
1. Enter on the Smith chart as . Draw a –circle
centered at O with radius .
2. Draw a straight line from through O to on the
perimeter, intersecting the –circle at , which
represents . Note 0.109 at on the “wavelengths toward
generator” scale.
3. Two points of intersection of the –circle with the g1
circle.
At : . At : ;
4. Solutions for the position of the stubs:
For (from to ):
For (from to ):
For (from to , which represents ):
For (from to , which represents ):
55 Introduction to Sparameters
Sparameters: for analyzing the highfrequency circuits.
Define ,
,
, where , , , and .
New Sparameters obtained by shifting reference planes:
, , ,
, where
and
Tparameters: , where
and