74 Radio Engineering for Wireless Communication and Sensor Applications
•In an antenna array a mismatched element causes deterioration of the overall antenna performance due to phase and amplitude errors.
A standing wave in a line can be measured and displayed using a slotted line, which is usually made of a rectangular metal waveguide or a coaxial line. In the case of a rectangular waveguide, there is a narrow slot in the middle of the wide wall, as shown in Figure 4.3; this slot does not disturb the fields of the waveguide because the surface currents of the TE10 mode do not cross the centerline of the wide wall. In the slot there is a movable probe, into which a voltage proportional to the electric field is induced. This voltage is then measured with a square-law diode detector and displayed with a proper device. By moving the probe, the maximum and minimum are found and their ratio gives the VSWR . The impedance at the standing wave minimum is Z 0 /VSWR . Then the impedance at any position z can be calculated using (4.12). In practice, nowadays the impedance is measured using a network analyzer. For more information concerning measurement techniques, see [1].
4.2 Smith Chart
The Smith chart is a useful tool for displaying impedances measured versus frequency or for solving a matching problem in a circuit design. The Smith chart clearly shows the connection between the reflection coefficient and
Figure 4.3 A slotted line made of a rectangular waveguide.
Impedance Matching |
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impedance, and also displays readily how the input impedance changes when moving along the line.
If the load is passive, the absolute value of the voltage reflection coefficient is never more than 1. Then any complex reflection coefficient of a passive load can be presented in the polar form within a unity circle. All possible normalized impedances of passive loads can be presented within this unity circle. This is the great idea of the Smith chart, presented by P. Smith in 1939 [2].
The normalized input impedance at z = −l can be presented as
z (−l ) = |
Z (−l ) |
= r + jx |
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Z 0 |
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The corresponding voltage reflection coefficient is
r(−l ) = rL e −2jbl = u + jv
According to (4.10) we have
z (−l ) = |
1 + r L e −2jbl |
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1 |
− rL e −2jbl |
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and after substituting (4.17) and (4.18) into this we obtain
1 + (u + jv ) r + jx = 1 − (u + jv )
(4.17)
(4.18)
(4.19)
(4.20)
We can form the two following equations by separating (4.20) into real and imaginary parts:
r = |
1 − (u 2 + v 2 ) |
(4.21) |
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(1 |
− u )2 |
+ v 2 |
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x = |
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2v |
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(4.22) |
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(1 |
− u )2 |
+ v 2 |
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76 Radio Engineering for Wireless Communication and Sensor Applications
These can be solved for two equations of circles as
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r |
2 |
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1 |
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Su − |
D + v |
2 = |
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(4.23) |
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1 + r |
(1 + r )2 |
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(u − 1)2 + Sv − |
1 |
D2 |
= |
1 |
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(4.24) |
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x |
x 2 |
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Graphically presented, these equations form the Smith chart shown in Figure 4.4.
Figure 4.4 Smith chart.
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At the center of the Smith chart the normalized impedance is z = 1; that is, the load is matched to the line ( r = 0). At the top of the Smith chart there is a point representing a short circuit, z = 0 or r = −1, and at the bottom there is a point representing an open circuit, z = ∞ or r = 1. Points elsewhere on the unity circle perimeter represent pure imaginary impedances X| r | = 1C. All pure real impedances are on the vertical diameter, and from that to the left there are the capacitive impedances and to the right the inductive impedances.
Figure 4.5 shows how an impedance is related to the corresponding voltage reflection coefficient. Point A represents a normalized load impedance, z L = 0.5 + j 0.5. The magnitude of the reflection coefficient is the distance of point A from the center of the chart, point O, or | r| = 0.45 (remember that the radius of the chart is 1). The phase of the reflection coefficient is the angle between the directions from point O to the point z = ∞ and to point A measured counterclockwise, in this case r = 117°.
When moving along a lossless line, the absolute value of the reflection coefficient is constant and the phase changes 360° per one half-wavelength. Therefore the impedance locus following this move is along a circle, the
Figure 4.5 Using the Smith chart: The relation between an impedance and the corresponding voltage reflection coefficient, movement along a lossless transmission line (A → B), and the relation between an impedance (point A) and the corresponding admittance (point A′).