4
Impedance Matching
In Chapter 3 we considered homogeneous transmission lines and waveguides in which a wave propagates only in the z direction. In a homogeneous line the characteristic impedance is independent of z , and accordingly the ratio of the electric and magnetic field as well as the ratio of the voltage and current (in a TEM line) is constant.
If there is a discontinuity in the line disturbing the fields, the impedance changes and a reflection occurs. The discontinuity may be a change in the line dimensions or a terminating load, the impedance of which is different from that of the line. This mismatch of impedances may cause serious problems. Elimination of the reflection—that is, matching the load to the line—is a frequent and important task in radio engineering.
In this chapter we first consider the fundamental concepts needed in impedance matching: the reflection coefficient, input impedance, standing wave, and the Smith chart. Then we consider different methods of impedance matching, such as matching with lumped elements, with tuning stubs, with quarter-wave transformers, and with a resistive circuit.
4.1 Reflection from a Mismatched Load
In the following analysis we will use the transmission line model and use voltages and currents. In principle we could, of course, use electric and magnetic fields, but the advantage of using voltages and currents is that the characteristic impedance of a line is always (by definition) the ratio of the
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70 Radio Engineering for Wireless Communication and Sensor Applications
voltage and current, but not always directly the ratio of the electric and magnetic field [see (3.51)]. In impedance matching it is the characteristic impedance of the line that matters, not the wave impedance.
Let us consider a situation shown in Figure 4.1 in which the line is terminated at z = 0 with a load. The characteristic impedance of the line is Z 0 and the impedance of the load is Z L . Let us assume that there is a voltage wave propagating toward the load, V +e −gz, and the corresponding current wave is I +e −gz (Z 0 = V +/I + ). Then at the input of the load, normally
one part of the propagating wave power is reflected while the other part is |
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absorbed to the load. The reflected voltage wave is V |
−e +gz, and the corre- |
sponding reflected current wave is I −e +gz (Z 0 = V −/I |
− ). V +, I +, V −, and |
I − are complex amplitudes. At z = 0 the voltages of the line and those of |
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the load must be equal, and the same applies of course to the currents:
V + + V − = VL |
(4.1) |
I + − I − = IL |
(4.2) |
Note that the directions of positive I + and I − are defined to be opposites. As Z L = VL /IL , we can present (4.2) as
V |
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V |
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VL |
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− |
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(4.3) |
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Z L |
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Z 0 |
Z 0 |
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The voltage reflection coefficient of the load is defined as
Figure 4.1 A line terminated with a load.
Impedance Matching |
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r L = |
V |
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V |
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If we eliminate V − from (4.1) and (4.3), and then solve for rL , we obtain
rL = |
Z L − Z 0 |
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z L − 1 |
(4.5) |
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Z L + Z 0 |
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z L + 1 |
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where z L = Z L /Z 0 is the normalized load impedance (do not confuse the normalized impedances with the z coordinate). The voltage transmission coefficient is
tL = |
VL |
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+ r L = |
2Z L |
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V |
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Z L + Z 0 |
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If the load impedance and the characteristic impedance of the line are equal, there will be no reflected wave but all power will be absorbed into the load. In such a case the load is matched to the line.
If the load is mismatched or Z L ≠ Z 0 , there will be a wave propagating in both directions, that is, the voltage and current as a function of z are
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V (z ) = V +e −gz + V −e +gz |
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(4.7) |
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I (z ) = I |
+e −gz − I −e +gz = |
V + |
e −gz − |
V − |
e +gz |
(4.8) |
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Z 0 |
Z 0 |
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Let us assume that the length of the line is l . In the following we consider the load impedance seen through this line. At z = −l the voltage reflection coefficient is
V −e −gl r(−l ) = V +e +gl
Now the input impedance at z = −l is
V (−l ) V +e +gl + V −e −gl Z (−l ) = I (−l ) = Z 0 V +e +gl − V −e −gl
=r L e −2gl
=1 + rL e −2gl
Z 0 1 − rL e −2gl
(4.9)
1+ r(−l )
=Z 0 1 − r(−l ) (4.10)
72 Radio Engineering for Wireless Communication and Sensor Applications
When substituting (4.5) into (4.10) and taking into account that e x = sinh x + cosh x , we obtain
Z L + Z 0 |
tanh gl |
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Z (−l ) = Z 0 |
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(4.11) |
Z 0 + Z L |
tanh gl |
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In practice we use low-loss lines, which means that the attenuation (or damping) constant is small, often negligible, and then we can assume that g = jb. In such a case the input impedance is
Z L + jZ 0 tan bl |
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Z (−l ) = Z 0 |
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(4.12) |
Z 0 + jZ L tan bl |
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In a lossless case the line voltage as a function of z is |
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V (z ) = V +e −jbz (1 + rL e 2jbz ) |
(4.13) |
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The voltage (or the field strength) repeats itself periodically at every halfwavelength, so there is a standing wave in the line, as shown in Figure 4.2. At the maximum, the voltages of the forward and reflected waves are in
the same |
phase, and therefore the total amplitude |
is Vmax = |
| V + | + | V |
− | . At the minimum, the voltages are in an opposite phase, and |
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therefore the total amplitude Vmin is the difference | V + | − |
| V − | . On the |
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other hand, the current has a minimum where the voltage has a maximum, and vice versa. The voltage standing wave ratio is defined as
VSWR = |
Vmax |
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| V + | + | V − | |
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1 + | r |
L |
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Vmin |
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1 − | r L | |
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Figure 4.2 Standing wave pattern (rhs) and its phasor presentation (lhs).
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VSWR is equal to 1 when the load is matched, and 1 < VSWR ≤ ∞ when the load is mismatched. The input impedance of the line is Z 0 × VSWR at the maximum and Z 0 /VSWR at the minimum, that is, it is real in both cases.
There are many problems caused by the load mismatch:
• Part of the power is not absorbed by the load. Power is proportional
to the square of the voltage (or field strength); therefore the power |
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reflection coefficient is | r L |2. If the forward propagating power is |
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P, then the reflected power is |
| rL |2 × P and the power absorbed |
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by the load is X1 − | r L |2 C × P. The power loss due to reflection |
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or the reflection loss L refl is (in decibels) |
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L refl = 10 log |
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1 − | rL |2 |
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The return loss L retn describes how much smaller the reflected power is compared to the incident power P, and is defined (in decibels) as
L retn = 10 log |
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•Due to reflection, the field strength may be doubled at the standing wave maximum, and therefore danger of electrical breakdown increases in high-power applications such as radar.
•The standing wave increases the conductor loss in the line. The loss is proportional to the square of the current. At the current maximum the loss increase is higher than the decrease at the current minimum compared to a matched case.
•A mismatch at the input of a sensitive receiver deteriorates the signal- to-noise ratio of the receiver.
•If the line is long, the input impedance fluctuates rapidly versus frequency. This is a disadvantage for active devices such as amplifiers because their performance depends on the feeding impedance. For example, the gain of an amplifier may change greatly if its input load impedance (feed impedance) changes.
•In digital radio systems the reflected pulses may cause symbol errors.