- Electromagnetics I
- Ch 3
- Loc 3.22
Single-Reactance Matching
An impedance matching structure can be designed using a section of transmission line combined with a discrete reactance, such as a capacitor or an inductor. In the strategy presented here, the transmission line is used to transform the real part of the load impedance or admittance to the desired value, and then the reactance is used to modify the imaginary part to the desired value. (Note the difference between this approach and the quarter-wave technique described in Section 3.19. In that approach, the first transmission line is used to zero the imaginary part.) There are two versions of this strategy, which we will now consider separately.
The first version is shown in Figure 3.22.1. The purpose of the transmission line is to transform the load impedance
into a new impedance
for which
. This can be done by solving the equation (from Section 3.15)
for
, using a numerical search, or using the Smith chart. The characteristic impedance
and phase propagation constant
of the transmission line are independent variables and can be selected for convenience. Normally, the smallest value of
that satisfies Equation 3.22.1 is desired. This value will be
because the real part of
is periodic in
with period
.
After matching the real component of the impedance in this manner, the imaginary component of
may then be transformed to the desired value (
) by attaching a reactance
in series with the transmission line input, yielding
. Therefore, we choose
The sign of
determines whether this reactance is a capacitor (
) or inductor (
), and the value of this component is determined from
and the design frequency.
Exercise
#1 Single reactance in series.
Design a match consisting of a transmission line in series with a single capacitor or inductor that matches a source impedance of
to a load impedance of
at 1.5 GHz. The characteristic impedance and phase velocity of the transmission line are
and 0.6c respectively.
Solution:
From the problem statement:
and
are the source and load impedances respectively at
GHz. The characteristic impedance and phase velocity of the transmission line are
and
respectively.
The reflection coefficient
(i.e.,
with respect to the characteristic impedance of the transmission line) is
The length
of the primary line (that is, the one that connects the two ports of the matching structure) is determined using the equation:
where here
. So a more-specific form of the equation that can be solved for
(as a step toward finding
) is:
By trial and error (or using the Smith chart if you prefer) we find
rad for the primary line, yielding
for the input impedance after attaching the primary line.
We may now solve for
as follows: Since
(Section 3.8), we find
Therefore
.
The impedance of the series reactance should be
to cancel the imaginary part of
. Since the sign of this impedance is negative, it must be a capacitor. The reactance of a capacitor is
, so it must be true that
Thus, we find the series reactance is a capacitor of value
.
The second version of the single-reactance strategy is shown in Figure 3.22.2. The difference in this scheme is that the reactance is attached in parallel. In this case, it is easier to work the problem using admittance (i.e., reciprocal impedance) as opposed to impedance; this is because the admittance of parallel reactances is simply the sum of the associated admittances; i.e.,
where
,
, and
is the discrete parallel susceptance; i.e., the imaginary part of the discrete parallel admittance.
So, the procedure is as follows. The transmission line is used to transform
into a new admittance
for which
. First, we note that
where
is characteristic admittance. Again, the characteristic impedance
and phase propagation constant
of the transmission line are independent variables and can be selected for convenience. In the present problem, we aim to solve the equation
for the smallest value of
, using a numerical search or using the Smith chart. After matching the real component of the admittances in this manner, the imaginary component of the resulting admittance may then be transformed to the desired value by attaching the susceptance
in parallel with the transmission line input. Since we desire
in parallel with
to be
, the desired value is
The sign of
determines whether this is a capacitor (
) or inductor (
), and the value of this component is determined from
and the design frequency.
In the following example, we address the same problem raised in Exercise 1, now using the parallel reactance approach:
Exercise
#2 Single reactance in parallel.
Design a match consisting of a transmission line in parallel with a single capacitor or inductor that matches a source impedance of
to a load impedance of
at 1.5 GHz. The characteristic impedance and phase velocity of the transmission line are
and 0.6c respectively.
Solution:
From the problem statement:
and
are the source and load impedances respectively at
GHz. The characteristic impedance and phase velocity of the transmission line are
and
respectively.
The reflection coefficient
(i.e.,
with respect to the characteristic impedance of the transmission line) is
The length
of the primary line (that is, the one that connects the two ports of the matching structure) is the solution to:
where here
mho and
mho. So the equation to be solved for
(as a step toward finding
) is:
By trial and error (or the Smith chart) we find
rad for the primary line, yielding
mho for the input admittance after attaching the primary line.
We may now solve for
as follows: Since
(Section 3.8), we find
Therefore,
.
The admittance of the parallel reactance should be
mho to cancel the imaginary part of
. The associated impedance is
. Since the sign of this impedance is negative, it must be a capacitor. The reactance of a capacitor is
, so it must be true that
Thus, we find the parallel reactance is a capacitor of value
.
Comparing this result to the result from the series reactance method (Exercise 1), we see that the necessary length of transmission line is much shorter, which is normally a compelling advantage. The tradeoff is that the parallel capacitance is much smaller and an accurate value may be more difficult to achieve.
Additional Reading
- “Smith chart” on Wikipedia.
Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0
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