THE SMITH CHART AND ITS APPLICATIONS TRANSMISSION LINES
1
1. TRANSMISSION LINES
To understand the Smith Chart, the theory of electric waves along a uniform transmission line
is first considered. A uniform transmission line can be defined as a line which has identical
dimensions and electrical properties in all planes transverse to the direction of propagation.
Unlike the waveguide, the transmission line consists of two conductors separated by a dielectric.
If the two conductors are identical and placed along side each other then a twin line is formed.
These lines are usually used at the lower frequencies. Examples of twin lines are the familiar
TV twin-lead and the open two-wire line. If one conductor is placed inside the hollowed tube
of the other, then a coaxial line is formed. Coaxial lines are used for frequencies up to 30 GHz.
At frequencies from 3 to 300 GHz, hollow waveguides are used. Transmission lines provide a
method of transmitting electrical energy between two points in space, similar to antennas.
Their behavior is normally described in terms of the current and voltage waves which
propagate along it. The performance of a transmission line is normally described in terms of
the secondary coefficients, as discussed below.
1.1 DISTRIBUTED CONSTANTS AND TRAVELLING WAVES
In a coaxial line the centre conductor may be held in place by dielectric spacers or by a
continuous solid dielectric fill which supports the inner conductor keeping it central to the outer
conductor. A coaxial cable is self-shielded and has no external field except possibly near
terminations. For this reason it is widely used throughout the radio frequency range and well
within the microwave region; which is the name given to the radio spectrum at wavelengths
below half a metre.
Twin lines, used as open wire lines, are normally balanced with respect to ground, but as a
coaxial cable is unsymmetrical it cannot be balanced with respect to ground.
Figure 1(a) Open two-wire line Figure 1(b) Coaxial cable
When the transmission-line is analyzed by means of electromagnetic field theory, it is found
that the principal-mode of transmission of the electromagnetic wave is not the only one that
can exist on a set of parallel conductors. Our discussion will consider only the principal-mode
in which the electric and the magnetic fields are perpendicular to each other and to the direction
of the conductors. The principal-mode wave is called the transverse electromagnetic, or TEM,
wave, and is the only kind that can exist on a transmission line at the lower frequencies. When
the frequencies become so high that the wavelength is comparable with the distance between
conductors, other types of waves, of the kind utilized in hollow waveguides become possible.
Except in very special cases, these “higher modes” are considered undesirable on the
transmission systems that we are considering; therefore, whenever possible, the spacing
between conductors is kept much smaller than a quarter wavelength. Another reason for a
small spacing is that, when the distance between the wires of an unshielded line approaches a
quarter wavelength, the line acts as an antenna and radiates a considerable portion of the energy
THE SMITH CHART AND ITS APPLICATIONS TRANSMISSION LINES
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that it carries. In this chapter, we will assume a very small spacing and shall neglect radiation
losses altogether.
1.2 DISTRIBUTED CONSTANTS OF THE LINE
Transmission lines are most easily analyzed by an extension of lumped-constant theory. The
same theory will apply to the lines shown in Figure 1 above.
The most important constants of the line are its distributed inductance and capacitance. When a
current flows in the conductors of a transmission line, a magnetic flux is set up around the
conductors. Any change in this flux will induce a voltage (L di/dt). The inductance of the
transmission-line conductors is smoothly distributed throughout their length. The distributed
inductance, representing the net effect of all the line conductors and is associated with the
magnetic flux linking these conductors, is given the symbol L′ and is expressed in henries per
unit length.
Between the conductors of the line there exists a uniformly distributed capacitance C′ which is
associated with the charge on the conductors. This will be measured in farads per unit length of
the line. The distributed inductance and capacitance are illustrated in Figure 2. When a line is
viewed in his way, it is not hard to see that the voltage and current can vary from point to
point on the line, and that resonance may exist under certain conditions.
Figure 2 Schematic representation of the distributed inductance and
capacitance of a transmission line
In addition to inductance and capacitance, the conductors also have a resistance R′ ohms per
unit length. This includes the effect of all the conductors. Finally, the insulation of the line may
allow some current to leak from one conductor to the other. This is denoted by a conductance
G′, measured in siemens per unit length of line. The quantity R′ represents the imperfection of
the conductor and is related to its dimensions and conductivity, while G′ represents the
imperfection of the insulator and is related to the loss tangent of the insulating material between
the conductors. Note however, that G′ does not represent the reciprocal of R′. When solid
insulation is used at very high frequencies, the dielectric loss may be considerable. This has the
same effect on the line as true ohmic leakage and forms the major contribution to G′ at these
frequencies.
Even though the line constants are uniformly distributed along the line, we can gain a rough
idea of their effect by imagining they line to be made up of short sections of length ∆z, as
shown in Figure 3. If L′ is the series inductance per unit length (H/m), the inductance of a short
section will be L′∆z henrys. Similarly, the resistance of the section will be R′∆z ohms, the
capacitance will be C′∆z farads, and the leakage conductance will be G′∆z siemens.
Although the inductance and resistance are shown lumped in one conductor in Figure 3, they
actually represent the net effect of both conductors in the short section ∆z. As the section
lengths ∆z are made smaller and approach zero length, the “lumpy” line of Figure 3 will
approach the actual smooth line and since ∆z can be made small compared to the operating
wavelength, an individual section of line can be analysed using lumped circuit theory. As ∆z
may approach zero, the results of the derivation to follow are valid at all frequencies.
THE SMITH CHART AND ITS APPLICATIONS TRANSMISSION LINES
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Figure 3 Lumped circuit representation of a uniform transmission line
1.2.1 BALANCED TO UNBALANCED LINE CONVERSION
As may have been noticed when considering Figure 2 and Figure 3, the schematic
representation of the distributed inductance and capacitance of a balanced transmission line has
been changed to the lumped circuit representation of an unbalanced uniform line. This section
considers how this transition is justified.
Figure 4 shows how the transmission line of Figure 2 can be represented by lumped circuit
representation of impedances, where Z1 = (jωL′ + R′)∆z and Z2 = 1/(jωC′ + G′)∆z, where ∆z is
an infinitesimal section of the line. As the line is balanced half of the distributed inductance
and resistance, which could be measured, is placed in each of the upper and lower arms.
Figure 4 Lumped circuit representation of a balanced transmission line
However, because in the following analysis, we want to deal with T-networks, the balanced
lumped circuit model can be divided into a number of cascaded T-sections, each of length ∆z,
and each of which are contained within the dotted region shown in Figure 4. To do this each of
the Z1/2 impedances must be split into two Z1/4 series impedances as shown in Figure 5. This
permits each of the Z1/4 series impedances to be balanced around each of the Z2 branch
impedances. The dotted region shown in figure 5 also represents the infinitesimal length ∆z.
The remaining section of the transmission line at either end of the section is shown replaced by
the impedance Zo which, as will be discussed in more detail below, is known as the
“characteristic impedance” of the line and is normally taken as being resistive.
THE SMITH CHART AND ITS APPLICATIONS TRANSMISSION LINES
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Figure 5 Balanced lumped circuit representation of a transmission line
If the balanced circuit is considered to work into a resistive impedance Zo then, the lower arm
Z1/4 series impedances can be moved into the upper arm and added to the existing Z1/4 series
impedances to give each upper arm an impedance of Z1/2 and provide the unbalanced Tsection
required for analysis. This is shown in Figure 6.
Figure 6 Unbalanced lumped circuit representation of a transmission line
1.3 PRIMARY COEFFICIENTS OF A LINE
1.3.1 RESISTANCE
The resistance R′ of a line is the sum of the resistances of the two conductors comprising a pair.
At zero frequency the resistance of a pair per unit length is merely the d.c. resistance R′ but at
frequencies greater than a few kilohertz a phenomenon known as skin effect comes into play.
This effect ensures that current flows only in a thin layer or “skin” at the surface of the
conductors. The thickness of this layer reduces as the frequency is increased and this means
that the effective cross-sectional area of the conductor is reduced. Since resistance is equal to
ρl/A (resistivity x length / cross-sectional area), the a.c. resistance of a conductor will increase
with increase in frequency. While the skin effect is developing, the relationship between a.c.
resistance and the frequency is rather complicated., but once skin effect is fully developed
(around 12 kHz) the ac resistance becomes directly proportional to the square root of the
frequency, that is:
Rac = K1 .√f
where K1 is a constant.
THE SMITH CHART AND ITS APPLICATIONS TRANSMISSION LINES
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Figure 7(a) below shows how the resistance of a line varies with the frequency. Initially, little
variation from the d.c. value is observed but at higher frequencies the shape of the graph is
determined by Rac = K1 .√f
Figure 7 (a) Variation of line resistance with frequency
(b) Variation of line leakance with frequency
1.3.2 INDUCTANCE
The inductance of a line L′, usually expressed in henrys/kilometre, depends upon the
dimensions and the spacing of its two conductors. The equations for the inductance of both twowire
and coaxial pairs are given below. The permeability of free space µo, is given as 4π x
10-7 H/m.
1.3.2.1 TWO-WIRE OR TWIN
L′ =
µ
π
µ
π
o
e
d r
r 4
+
−
log
( )
H/m ≈
µ
π
µ
π
o
e
d
r 4
+ log H/m when d >> r or
01 0 92 10 . . log + µr
d
r µH/m, where “d” is the spacing between the two conductors, each of
radius “r”, µ is the permeability (H/m), µ µ µ r o = , is the relative permeability and the
component µo/4π arises because of the flux linkages inside the conductors themselves.
1.3.2.2 COAXIAL
L′ =
µ
π
µ
π
o
e
w
r 8 2
+ log H/m = 01 0 46 10 . . log + µr
w
r µH/m where the component µo/8π
arises due to the interior flux linkages inside the two conductors, and where ”w” is the radius of
the outer conductor and “r” is the radius of the inner conductor. Neither the dimensions of a line
nor the absolute permeability are functions of frequency and consequently the inductance of a
line is not a frequency-dependent parameter.
1.3.3 CAPACITANCE
The capacitance C′, usually expressed in farads/kilometre, for both two-wire and coaxial pairs
are given below: The permittivity is given by ε = εo εr where εo is the permittivity of free-space
and is given by 8.854187818 x 10-12 F/m or approximately, by 10-9/36π F/m, and εr is the
relative permittivity or dielectric constant, which is a dimensionless quantity.
1.3.3.1 TWO-WIRE OR TWIN
C′ =
πε
log ( )
e
d r
r
− F/m ≈
πε
loge d
r
F/m, when d >> r, or
12 08
10
.
log
ε r
d
r
pF/m
where “d” is the spacing between the two conductors, each of radius “r”.
Skin effect developed
where R f ac ∝
Frequency
Conductor
resistance
R
Rdc
0
Leakance
G
(a) (b)
Frequency
G f ∝............... |
/2 
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发表于 2007-10-13 23:55:37
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