Basics of impedance (K) inverters for microwave filter design
Introduction
Conceptually, most common microwave bandpass filters are constructed out of only three basic building blocks: transmission line segments, resonators, and impedance (K) inverters. In this tutorial we will discuss impedance inverters. While the basic concepts behind impedance inverters are covered in a selection of popular microwave engineering textbooks [1,2], there are, it seems to me, certain concepts and identities which are not as well known or explained. In this tutorial I will attempt to 1. fill in this gap, 2. collect my knowledge of impedance inverters in one convenient place for my own reference, and 3. hopefully give you a better understanding of microwave filters in the process.
Quarter-wave transformer = one implementation of an impedance inverter
I first encountered impedance inverters when learning transmission line theory. Figure 1 shows a transmission line with characteristic impedance Z0, propagation constant β, and length l terminated in a load impedance ZL. At a distance z=-l from the load, the input impedance seen looking toward the load is given by the well known equation:
We see from this equation that if the line is a
quarter-wavelength long l =λ/4, then the tangent term in Eq. (1) become
,
and the input impedance becomes:
Note that Eq. (2) holds for any integer multiple of a quarter wavelength; namely, l=λ/4+nλ/4, for n=1,2,3, . Such a line is called a quarter-wave transformer, because it transforms the load impedance, in an inverse manner, depending on the characteristic impedance of the line.
Figure 1 Uniform transmission line with characteristic impedance Z0, propagation constant β, and length l terminated in a load impedance ZL.
Generalized impedance inverter
A quarter-wave transformer is not the only circuit element that forms the inverse of a load impedance. In fact, there are an infinite number of circuit implementations which act as so-called impedance inverters. The concept of the ideal impedance inverter is shown in Figure 2. The impedance inverter of impedance K is a symmetrical two-port which, looking into its input port, inverts and scales to impedance K any impedance ZL connected to its output port. As you can see, its circuit symbol is a generic two-port network block labeled with its impedance K and reference angle θ. The reference angle θ can be only one of two possible values, ±900, which tells you the phase shift across the impedance inverter. Often times you will see this angle omitted from the circuit symbol. In such cases, the sign of impedance K is implicitly associated with that of the angle θ, e.g. a positive K might mean θ =+900, while a negative K means θ=-900, or visa versa, depending on your convention. In simple ladder networks the sign of K does not matter. For instance, from Figure 2 we see that the input impedance depends on K squared. However, in coupled resonant filters with cross couplings, the sign of K does matter, since signals of opposite phase can cancel out to produce transmission zeros.
Figure 2 Definition of impedance inverter.
One application of the impedance inverter is in the construction of simple matching networks. From Figure 2 we see that an impedance inverter terminated in a load ZL can be perfectly matched to any line impedance Z0 on the condition that the impedance K is made equal to the geometric mean of ZL and Z0, i.e.
This property is exploited in microwave filter design to normalize input/output terminations to any desired level.
Equivalent circuits for impedance and admittance inverters
Ideally, the impedance K is invariant with respect to frequency, temperature, and anything else, i.e., an ideal impedance inverter would work perfectly from DC to gamma ray frequencies through a nuclear winter. This means an ideal K inverter does not really exist in practice. However, many simple lumped element circuits do approximate the behavior of impedance inverters over relatively wide bandwidths. Table 1 shows some useful equivalent circuits, along with those for the corresponding admittance inverter. An admittance inverter with admittance J is equal to an impedance inverter with impedance K = 1/J. Admittance inverters are used when it is more convenient to use admittance rather than impedance; this is usually the case when dealing with circuit elements in parallel. It is worth mentioning that the equivalent circuits shown in Table 1 form only a subset of an infinite number of equivalent circuits for impedance and admittance inverters. In fact, any linear, passive, reciprocal, two port network can be converted into an impedance inverter by connecting each of its two ports to transmission lines with lengths that satisfy a few simple design equations [3].
Table I. Some useful equivalent circuits for impedance and admittance inverters*
* The equivalent circuits shown here form only a subset of an infinite number of equivalent circuits for impedance inverters. See Ref [3].
Useful transformations for microwave bandpass and bandstop filter design
In the operation of RF/microwave filters, the main roles of impedance/admittance inverters are to scale impedance levels, couple energy, and convert series to parallel resonances and vice versa. These roles are made possible by a few useful equivalent circuit transformations. Below are shown some these transformations, which I compiled with lossless bandpass/bandstop filters in mind. Some more generalized transformations are given in Refs [1,2].
Trasformation 1. Convert a parallel LC circuit into a series LC circuit:
Figure 3 Illustration of transformation 1
Trasformation 2. Convert a series LC circuit into a parallel LC circuit:
Figure 4 Illustration of transformation 2
Trasformation 3. Scale the impedance of a series LC circuit:
Figure 5 Illustration of transformation 3
Trasformation 4. Scale the impedance of a parallel LC circuit:
Figure 6 Illustration of transformation 4
Another equivalence transformation you should know
Figure 7 shows another useful equivalence transformation. Here we start with an initial ladder of N lumped element LC circuits connected in tandem by impedance inverters. The LC networks are indexed 1, 2, , N, and their resonant frequencies are denoted ω1,ω2, , ωN. The impedance (K) values of the impedance inverters are denoted K0,1, K1,2, , KN,N+1. The filters source (generator) and load impedances are denoted Rg and RL, respectively. As shown in Figure 7, this network is electrically equivalent to another ladder network with different source/load impedances, lumped LC values, and K values (these variables are denoted with a prime symbol in the figure to indicate they are different). Note that this equivalence works because the resonant frequencies of the new LC circuits are the same as that in the original ladder network (ω1,ω2, , ωN).
Figure 7 A ladder network and its equivalent circuit with different lumped LC values and source/load impedances. This transformation is made possible by properly scaling the impedance inverters as shown.
Example: So how does this all fit into microwave filter design?
Suppose you are designing a bandpass filter, which is supposed to yield the amplitude response, |S21| and |S11|, shown in Figure 8. This amplitude response, coincidently, corresponds to that of a 0.5 dB equal-ripple (Chebyshev) filter of third-order (there are three bumps or ripples in the passband) with a center frequency of 1 GHz over a 10% bandwidth. In old school microwave filter design it is standard practice to start by designing (or looking up in tables) a prototype filter circuit which yields the desired response. It turns out that a prototype circuit for this response is the ladder network shown in Figure 9. If you need verification, try simulating the S-parameters of the circuit in Figure 9 using a circuit simulator like Agilent ADS or Ansoft Designer (you can download the free student version of Designer from Ansofts website), and you will find that its amplitude response over frequency exactly matches that shown in Figure 8. Here we denote Zinput on the schematic diagram as the input impedance seen by the generator.
Figure 8 Ideal amplitude response, |S21| and |S11|, for our example.
Figure 9 Third-order prototype bandpass filter circuit, implemented as a ladder network, which yields the response shown in Figure 8.
Everything looks good so far. So, what exactly is the problem here? For starters, its very unlikely that our actual (physical) load and source impedances will be 1 Ω. Let us suppose then that our bandpass filter is required to operate in, say, a 50 Ω system. Unfortunately, you cant just replace the 1 Ω resistors with 50 Ω resistors in Figure 9 and expect the same response. That would cause a huge mismatch. The prototype bandpass circuit as shown yields the correct response only when the main filter body sees 1 Ω impedances at both the generator and load ends. What are we to do?
Lets start by working specifically with the 1 Ω load termination. Fortunately, as we already said, so long
as the main filter body of the bandpass circuit
sees a 1 Ω impedance at both the generator and
load ends, the overall circuit will yield the same amplitude
response. What we are going to do then is replace the 1 Ω load
resistor with a subcircuit consisting of a K inverter
cascaded with a 50 Ω load as shown in Figure 10. In order that this subcircuit presents a 1 Ω load to the filter, we
utilize Eq. (3) and choose the value 
=
7.07107 Ω. In a way, we are tricking the main filter body
into seeing a 1 Ω load (via our handy K inverter) even though we really
have a 50 Ω load. This is actually a very common sort of thing to do
in microwave engineering. In fact, the entire subject of impedance
matching and tuning might be better thought of as impedance tricking
but I
digress
Note that in this circuit the generator still sees an input impedance Zinput
(the same input impedance as in Figure 9).
Figure 10 Bandpass filter with a 50 Ω load coupled to the circuit through an impedance inverter.
We emphasize again that the amplitude response, |S21| and |S11|, of the circuit shown in Figure 10 is equal to that of Figure 9. It is worth mentioning, however, that the phase of S21 is drastically different between the two circuits. This is because the impedance inverter shifts the phase of S21 by 90 degrees note the j term in the ABCD matrix of the impedance inverter (see Figure 2). Luckily, in the world of filter design, we rarely (if ever) are interested in the absolute phase of S21 and S11. It is the amplitude which counts (as well as group delay variation, which is proportional to the derivative of the phase).
We can now play the same sort of trick on the generator
side. This involves replacing the original generator with a subcircuit consisting of an impedance inverter with
impedance 
cascaded
with a new generator as shown in Figure 11 (this new generator itself consists
of some internal voltage source Vg and an impedance of 50
Ω).
Figure 11 Bandpass filter with 50 Ω source and load impedances coupled to the circuit through K inverters.
We did it! The bandpass filter circuit shown in Figure 11 yields the same amplitude response as the original prototype circuit shown in Figure 9 but with the required 50 Ω generator and termination impedances. In case you are wondering, we dont need to specify the values of the voltage sources Vg and Vg in Figures 10 through 11. Nor do we even need to care how they are related to each other. This is because we are only interested in the amplitude response of the filer (|S21| and |S11|). Remember that the scattering parameters are linear transfer functions, which by definition are independent of the specific inputs. For example, S21 is the ratio of power absorbed by the load to the input power. Here, the term input power refers to the power available from the source. So, to be sure, the specific value of input power definitely depends on the chosen value of Vg. But the ratio of the output power to the input power (i.e. S21) is independent of the value Vg. It makes no difference to the power ratio S21 whether Vg is one volt or a one million volts. The same thing holds, of course, for Vg.
Let us now analyze what we did (i.e. impedance scaling) from the perspective of the generator. As shown in Figure 9, we denote Zinput as the input impedance presented to the generator of the prototype circuit. The magnitude of the reflection coefficient, |S11|, from the point of view of the generator can then be expressed using the well-known formula:
Let us now consider the generator of our new circuit shown
in Figure 11. As shown in Figure 11, we denote 
as
the input impedance presented to the 50 Ω generator.
Now, is
related to the original input impedance Zinput
through the following impedance inverter transformation (thanks to the
impedance inverter at the input side):
With this result, we find the magnitude of the reflection coefficient, |S11|, from the point of view of the 50 Ω generator to be:
Note that the reflection coefficient amplitude |S11| is the same for both the prototype and final bandpass filter circuits (i.e. compare Eq. (4) with Eq. (6) they are equal). Furthermore, since the main body of both these filters are lossless, we can find |S21| from |S11| for either of these circuits by using the following convenient relation for a lossless network: |S21|2 + |S11|2 = 1. Now, since both filters are lossless and both have the same reflection coefficient |S11|, they therefore both have the same transmission coefficient |S21|. In other words, we have proven from the point of the view the generator this time that the final filter circuit shown in Figure 11 has the same amplitude response as our original prototype circuit shown in Figure 9.
As I mentioned before, in old school microwave filter design
it is standard practice to start with a prototype filter circuit which yields
the desired response. The next step in the design process is to apply a
series of transformations to the prototype (such as impedance scaling).
Each transformation preserves the overall amplitude response while at the same
time slowly massaging the circuit into the desired form. Take our
example above. Once we know the basic impedance scaling trick, we can take
the same prototype circuit shown in Figure 9 and modify it with appropriately
scaled impedance inverters to account for any source and load impedance we
desire. There was nothing special about choosing a 50 Ω source and a
50 Ω load in this example per say. The prototype circuit is one
size fits all. If we wanted the load impedance to be 100 Ω
instead of 50 Ω, then we would simply change the output coupling value
from to
.
I would be cheating you if I didnt mention that there is actually a completely different way to achieve impedance scaling without the use of K inverters. Let us again consider the problem of scaling the prototype network shown in Figure 9 from a 1 Ω system to a 50 Ω system (i.e. we want to scale up the generator and load impedances from 1 Ω to 50 Ω while preserving the same amplitude response). We can achieve this without using impedance inverters by simply multiplying all the inductances in Figure 9 by 50 and dividing all the capacitances by 50. So, for example, the two 2.541 nH inductors in Figure 9 would be replaced by new inductors each with an inductance of 2.541*50=111 nH. The two 9.995 pF capacitors in Figure 9 would be replaced by new capacitors each with a capacitance of 9.995/50=111 pF, etc. The disadvantage with this method is that (as far as I know) there is no simple scaling trick like this to account for load and source impedances which are different from each other. In other words, this type of trick cant handle a 50 Ω source impedance and, say, a 100 Ω load impedance. They have to be the same impedance. So impedance inverters are more versatile.
Are we done now? Not quite. Notice how the main body of the filter in Figure 11 is composed of three distinct resonant resonators a series LC resonator, a parallel LC resonator, and another series LC resonator connected in tandem (not so coincidentally, each of these has a resonant frequency f0 = ω0/2π = (LC)-1/2/2π = 1 GHz, which corresponds to the center of the passband). Everything looks good on paper, but a problem arises when we start thinking about how to physically implement this filter, i.e. going from the circuit schematic to a real world product.
Suppose for the sake of argument that we are to construct this filter in rectangular waveguide, and in such a medium we know that series LC resonators are easy to implement, while parallel LC resonators are difficult/awkward to implement (if you dont understand why this is, it makes no difference for this example just accept it as true for now). This physical insight means we must somehow get rid of the offending parallel LC resonator, while at the same time retaining the same amplitude response. How are we to do this?
The answer is easy; we are going to use transformation 1 shown in Figure 3 to convert the parallel LC resonator into a series LC resonator in tandem with a pair of impedance inverters. Incidentally, this means we are going to introduce more impedance inverters into our filter design (besides just the input/output coupling inverters we introduced previously to scale the load/source impedances). Luckily for us and humanity at large impedance inverters are easy to physically implement in most mediums because of the variety of equivalent circuits to choose from (see Table 1 and the corresponding discussion).
According to the transformation shown in Figure 3, we have the power to convert the offending parallel LC resonator with inductance Lp = 0.01455 nH and capacitance Cp = 1745.4523 pF into an equivalent circuit composed of a series LC resonator, with some new inductance Ls and capacitance Cs = CpLp/Ls, in tandem with a pair of impedance inverters of opposite sign, where K = ±(Ls/Cp)1/2. Note that we can choose Ls to be any real positive value we want. Once Ls is chosen, we then solve for Cs and K.
The question now is: what inductance Ls should we choose? The answer is to pick an inductance Ls that is easily realizable by a real world resonator. Conveniently, in waveguide it just so happens that a lossless half-wavelength (λ/2) cavity resonator can be modeled near resonance as a series LC lumped-element equivalent circuit. This equivalence is illustrated in Figure 12. The particular values of the LC lumped-elements themselves are a function of the resonant frequency and cavity dimensions. Such a cavity by itself is isolated from the outside world, but can be excited through small inductive apertures (i.e. a holes) through the cavity walls (such a hole actually models the response of a shunt inductor connected to the resonator).
Figure 12 A rectangular resonant cavity can be modeled as a series LC lumped-element circuit near resonance.
I would like to emphasize again that the electrical response of a cavity resonator depends on the cavitys dimensions and resonant frequency. This means the lumped-element LC values in its equivalent circuit model are fixed by the physical geometry of the cavity and therefore constitute a given. For the sake of argument, we shall assume that, for our particular problem at least, the given lumped-element inductance for a half-wavelength cavity corresponding to our particular geometry is Ls = 1 nH. This is then the value of inductance we must realize in the transformation, which corresponds to a capacitance Cs = CpLp/Ls = 1/{(ω0)2Ls} =25.396331 pF, and an impedance value K = (Ls/Cp)1/2 = 0.7569131 Ω . With these particular values, we arrive at the modified filter design shown in Figure 13.
Figure 13 Bandpass filter implemented using only series LC resonators. The original parallel LC resonator was removed via the transformation given in Figure 3.
We emphasize that our new design shown in Figure 13 has the same amplitude response as the original design shown in Figure 9. However, the new design contains all series LC resonators, which, as we stated earlier, are easier to implement in waveguide than parallel LC resonators. Furthermore, the LC resonator in the direct center has the necessary inductance Ls = 1 nH and capacitance Cs = 25.396331 pF, which, as we also stated earlier, can be conveniently realized with a λ/2 cavity resonator (for our particular geometry). This is a step in the right direction, but what we really want is to go all the way and transform our filter into one consisting of three identical resonators, each with inductance Ls = 1 nH and capacitance Cs = 25.396331 pF.
The two LC resonators which do not have these values can be made equivalent by applying the transformation given in Figure 5. This involves appropriately scaling the impedance values of the adjoining impedance inverters as illustrated in Figure 14. Alternatively, we could have employed the transformation shown in Figure 7 to reach the same equivalent circuit.
Figure 14 Transformation of our original filter from Figure 13 into an equivalent filter consisting of three identical resonators, each with an inductance Ls = 1 nH and capacitance Cs = 25.396331 pF.
There is one final order of business to take care of. In our equivalent circuit shown in Figure 14, one of the impedance inverters has a negative K value (K = -0.4748). While both negative and positive K values are perfectly acceptable from a theoretical point of view, it is generally preferred for all impedance inverters to be the same sign if possible from a practical point of view. For the sake of argument, let us further assume we want all the impedance inverters to have a positive sign. To accomplish this, all we need to do is drop the negative sign (i.e. change K = -0.4748 into K = +0.4748), which yields the filter shown in Figure 15. You may be wondering what gives us the right to simply drop the sign like that. Remember, the sign specifies the phase shift across the inverter as plus or minus ninety degrees. So an impedance inverter with K = -0.4748 is identical to an impedance inverter with K = +0.4748 in series with an ideal 180 degree phase shifter. In filter design we care only about the amplitude response (e.g. |S11| and |S21|) and the derivative or slope of the phase absolute phase (i.e. the group velocity). We dont care about the absolute phase in filter design.
Therefore the sign can be dropped without disturbing the things we care about (i.e. we can change K = -0.4748 into K = +0.4748 without changing the amplitude response or group delay). Beware, this little drop the sign trick only works for simple ladder networks like our filter here. There are more complicated filters out there involving cross couplings between resonators (meaning the LC resonators are also connected to nonadjacent resonator through additional K inverters). This introduces more signal paths or braches from the input to the output. If you drop the negative sign in one branch, you may have to add a negative sign to the others so no two branches subtract out their responses. This is a more advanced topic, best suited to be discussed in the framework of coupling or M matrices, which will be covered in a different tutorial.
Figure 15 The final form of our bandpass filter which consists of three identical resonators and all positive K inverters.
The circuit shown in Figure 15 is a result of massaging our original prototype by applying equivalence transformations involving impedance inverters. Our final design is in a form convenient for waveguide implementation a broad topic saved for elsewhere.
References
[1] R.E. Collin, Foundations
for Microwave Engineering 2nd Ed. ,
[2] D.M. Pozar,
Microwave Engineering 3rd Ed. ,
[3] R. Levy A generalized design technique for practical distributed reciprocal ladder networks, IEEE MTT vol 21, 1973 pp 519-526.