MICROVE DEVICES
3. Further Transmission Lines Characteristics
3.1. Impedance Matching
3.3 Impedance Matching
Impedance matching is very desirable with radio frequency (RF) transmission lines. Stand ing waves lead to increased losses and frequently cause the transmitter to malfunction. A line terminated in its characteristic impedance has a standing-wave ratio of unity and trans mits a given power without reflection. Also, transmission efficiency is optimum where there is no reflected power. A "flat" line is non-resonant; that is, its input impedance al ways remains at the same value Zo when the frequency changes.
Matching a transmission line has a special meaning, one differing from that used in circuit theory to indicate equal impedance seen looking both directions from a given terminal pair for maximum power transfer. In circuit theory, maximum power transfer requires the load impedance to be equal to the complex conjugate of the generator. This condition is some times referred to as a conjugate match. In transmission-line problems matching means simply terminating the line in its characteristic impedance.
A common application of RF transmission lines is the one in which there is a feeder con nection between a transmitter and an antenna. Usually the input impedance to the antenna itself is not equal to the characteristic impedance of the line. Furthermore, the output im pedance of the transmitter may not be equal to the Zo of the line. Matching devices are necessary to flatten the line. A complete matched transmission-line system is shown in figure 3.3.1.
For a low-loss or lossless transmission line at radio frequency, the characteristic impedance Zo of the line is resistive. At every point the impedances looking in opposite directions are
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conjugate. If Zo is real, it is its own conjugate. Matching can be tried first on the load side to flatten the line; then adjustment may be made on the transmitter side to provide maximum power transfer.
3.3.1 Single Stub Matching
Although single-lumped inductors or capacitors can match the transmission line, it is more common to use the susceptive properties of short-circuited sections of transmission lines. Short-circuited sections are preferable to open-circuited ones because a good short circuit is easier to obtain than a good open circuit.
For a lossless line with Yg = Y0, maximum power transfer requires Y11 = Y0, where Y11 is the total admittance of the line and stub looking to the right at point 1 as shown in the figure 3.3.2.
The stub must be located at that point on the line where the real part of the admittance, looking toward the load, is Y0. In a normalized unit Y11 must be in the form
y11 = yd ±ys = 1
if the stub has the same characteristic impedance as that of the line. Otherwise Y11 = Yd ±Ys = Y0
The stub length is then adjusted so that its susceptance just cancels out the susceptance of the line at the junction.
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Figure 3.3.2: Single-stub matching.
3.3.2 Double Stub Matching
Since single-stub matching is sometimes impractical because the stub cannot be placed physically in the ideal location, double-stub matching is needed. Double-stub devices consist of two short-circuited stubs connected in parallel with a fixed length between them. The length of the fixed section is usually one-eighth, three-eighths, or five-eighths of a wavelength.
The stub that is nearest the load is used to adjust the susceptance and is located at a fixed wavelength from the constant conductance unity circle (g = 1) on an appropriate constant-standing-wave-ratio circle. Then the admittance of the line at the second stub as shown in figure 3.3.3 is
y22 = yd2 ±ys2 = 1
Y22 = Yd2 ±Ys = Y0
In these two equations it is assumed that the stubs and the main line have the same charac teristic admittance. If the positions and lengths of the stubs are chosen properly, there will be no standing wave on the line to the left of the second stub measured from the load.