Lowpass ladders of series L and shunt C elements have been used throughout the history of radio engineering as filters for limiting noise bandwidth and harmonic suppression, as moderate bandwidth resistance transformers, and as structures for general impedance-matching problems^1-3,16,17.

At frequencies where inductors and capacitors no longer act like ideal lumped elements, these functions are performed by transmission-line elements assembled into systems. One such system is a cascade of electrically short transmission lines whose wave impedances have alternating high and low values. This cascade is analogous to the LC ladder in that a low-wave-impedance line imitates a shunt capacitor, and a high-wave-impedance line imitates a series inductor²,

where d is the line physical length and v is the line wave speed.

These equations will be referred to as the “Low-frequency mapping” later on. Equations (1a) and (1b) are only exact at dc. They point to one of several ways that such cascades can be designed. That is, one can start with an LC design that solves a particular problem and convert it to a short-line cascade using (1a) and (1b).

The analogy between short-line cascades and LC ladders has intrigued this author for quite some time, leading to using the behavior of one to inspire solutions using the other^4-6. Appendix A, on page 56, presents an exact high-frequency mapping for three or more elements. For both mappings, the wave impedances alternate between two fixed values, and the lengths of the lines vary according to the need. Equations (3) through equation (21) are presented in Appendix A.

In any particular transmission-line technology, there are limits to the practical values of wave impedances. For printed lines, the lower the wave impedance, the wider the line. This could ultimately lead to non-TEM modes and radiation.

On the high side, the higher the wave impedance, the narrower the line. This can lead to excessive loss and fabrication difficulty. I assume that the designer wishes to minimize the space occupied by the network, and that a small insertion loss is better than minimum insertion loss if it allows a shorter network.

There are two basic kinds of short-line cascades:

• Unit-element

  The length of all lines is the same, and the parameters for design are the wave impedances.

• Fixed wave impedances

  The parameters to be designed are the line lengths.

The main advantage of unit-element design is that a synthesis procedure is available if a desired power-gain or input impedance function is known^8,16, and closed-form design equations are available for standard filter types^8,9.

I assert that the main advantage of the fixed wave impedances structure is that it is shorter than an equivalent-performance unit-element design. This is illustrated by considering the low-frequency mapping of equation (1a) and equation (1b). If an LC ladder is converted to a unit-element design, the value of d must be chosen so that the largest values of L and C can be converted within the wave impedance limits.

Then smaller values of L or C will be converted using this same d and wave impedance values further inside the allowed range. If conversion is made to a fixed-wave-impedances cascade, the wave impedances are always at the limits, so that the d values for smaller L and C values will be less than the maximum. I support the generality of the assertion in the following examples.

In this section, I consider the design of a five-element Chebyshev filter having a passband ripple of 0.1 dB, and 50Ω terminations. The objective is to design a cascade of short lines whose wave impedances fit in the range 15Ω to 100Ω. Closed-form equation sets are available for designs based on LC prototypes⁷, and designs based on the short-line unit element^8,9.

The structure for each of the following designs is shown in Figure 1. The passband responses are shown in Figure 2, and the stopband responses are shown in Figure 3.

I note that the derivation⁸ depends on the condition that sin² (ω_C) <<1, ω_C is the corner frequency electrical length. This condition was not well satisfied in the present case. To find a design in which the wave impedances are between 15Ω and 100Ω, it was necessary to choose the electrical length as 35°. sin² (35°) is 0.33. The unit-element line length is 0.0972λ, where λ is the corner frequency wavelength.

The total length for five lines is 0.486 λ. The wave impedances are 24.506Ω, 99.768Ω, 16.529Ω, 99.768Ω, and 24.506Ω. The minimum passband gain is -0.1447 dB.

The LC prototype values, for 1 Hz, 1Ω are 0.18252, 0.21824, 0.31433, 0.21824, and 0.18252. Treating elements 1,3,5 as capacitors, the frequency response was perfect.

Using the low-frequency mapping with Z_hi = 100Ω, Z_lo = 15Ω, the line lengths are 0.054756λ, 0.10912λ, 0.094299λ, 0.10912λ, and 0.054756λ. The total length is 0.42205λ. This is about 13 percent shorter than the unit-element design. The minimum passband gain is -0.61 dB, at the corner frequency.

The five LC elements are split into two pi sections, then converted using the equations in Appendix A (page 56). The adjacent Z_lo sections in the result are joined as one line. The lengths are 0.043965λ, 0.11803λ, 0.072126λ, 0.11803λ, and 0.043965λ.

The total length is 0.3961λ, about 19 percent shorter than the unit-element design. The minimum passband gain is -0.106 dB, at a ripple just left of the corner frequency.

The particular results obtained for a given design problem depend on a number of choices external to the performance requirements. For example, if the substrate is twice as thick, the microstrip lines will be about twice as wide, but the lengths will be the same.

The designs in Figure 1 could be folded to trade length for width in the layout, but a large increase in line width would prevent this because adjacent edges couldn't be separated enough to avoid edge-coupling.

Another way to trade width for length is to use a smaller value for the minimum wave impedance. This would not help the unit-element design, but would shorten the other two, so that the relative shortening of the LC-to-short line mappings would be greater.

The frequency responses in Figure 2 and Figure 3 show properties typical of each design. The unit-element design would have an equal-ripple passband response if the length were shorter, although the ripples are not evenly spaced on a linear frequency scale. The two designs based on LC conversions are each exact at dc, and the corner-frequency mapping is exact at the corner frequency.

The quality of conversion of the low-frequency mapping is good at frequencies well below the corner frequency, and gets worse at higher frequencies. The corner frequency mapping is close around the corner frequency, and, as an approximation, gets worse at lower frequencies.

However, because this is a lowpass filter with equal resistive terminations, there is less loss than with the LC prototype. Thus, matching the response of the LC prototype at the corner frequency insures good performance for the entire passband. The stopband performances are also typical of each structure.

The unit-element design has a regular periodicity because of the equal-electrical-length lines, while the various-length-line designs have irregular unpredictable responses. If harmonic suppression is an issue, it may be necessary to test designs using different Z_hi and Z_lo values to shift the stopband peaks.

In this example, we consider a network to match 12.5Ω to 50Ω over an octave bandwidth. In 1964, G.L. Matthei¹⁰ published a frequency transformation to obtain quasi-bandpass response from an LC lowpass ladder for the purpose of matching unequal resistances. The paper, “Tables of Chebyshev Impedance-Transforming Networks of Low-Pass Filter Form,” published by the IEEE, gave 20 pages of tables of element values derived by conventional ladder-network synthesis¹⁰.

In 1966, he used the same transformation followed by Richards' transformation to synthesize unit-element-based transformers¹¹.

These designs were a great improvement over designs based on the quarter-wave line. In several papers culminating in Meschanov, Rasukova, and Tupikin's, “Stepped Transformers on TEM-Transmission Lines,” from the June 1996 IEEE Trans. On Microwave Theory and Techniques¹², Meschanov and his colleagues showed that significantly shorter designs can be obtained by using just two wave impedance values and optimizing the line lengths.

In both cases, the designs produced an even number of lines with alternating high-and low-impedance values. One can imagine them as multiple L-sections¹. Meschanov imposes some conditions on the design before the optimization is done:

1. If R₁ < R₂ are the terminating resistances, then Z_hi Z_lo = R₁ R₂.
2. The sequence of wave impedances is R₁, Z_hi, Z_lo, Z_hi … Z_lo, R₂.
3. If there are N sections of line, length L_k = L_N+1-k, 1 ≤ k ≤ N/2.

In their examples, Meschanov, Rasukova, and Tupikin, fixed the overall line length and searched on N/2-1 line lengths and one wave impedance. Conditions one through three and the total length value fixed all the other variables.

Thus, for an N = 6 design, there are three variables to search. The penalty function they minimized was the maximum reflection coefficient magnitude over the set of matching frequencies. I verified the results in the Meschanov, Rasukova, and Tupikin, article by choosing the wave impedance values they found, using conditions one through three, and leaving the total length free.

I also used a different optimizer, NelderMead simplex, as implemented in a Matlab¹³ environment. For N = 2,4, I found the same lengths and reflection coefficient regardless of initial condition on the line lengths. For N = 6, I also duplicated their results, but it was necessary to enclose the optimization in a loop that stepped through sets of initial values.

The initial line lengths are set equal, with L₀ = 0.01*k, 0 ≤ k ≤ 10, where k is an integer. I also found that the sensitivity of the mini-max reflection coefficient to condition 1 is quite sharp. A 1 percent deviation can raise the penalty an order of magnitude. Also, it was found that if condition three is not imposed and optimization is done on all the line lengths, the best solution is never found.

An alternative to use of an optimizer is to use Matthei's approach to an LC design, and then apply the high-frequency mapping. For the design goals given above, Matthei's procedure with N = 6 yields a minimum gain of -0.013 dB.

However, conversion fails with the Z_hi and Z_lo values chosen. His procedure allows one to specify an additional gain reduction. By specifying the minimum gain at -0.5 dB, the element values become convertible. The result is a design with minimum gain of -0.678 dB, and overall length of 0.3586 λ, where λ is the wavelength at band center. A single-section quarter-wave transformer for this problem has a minimum gain of -0.57 dB, a two-line design has a minimum gain of -0.151 dB.

In pursuit of the design objective stated above, I ran the optimization for a sequence of Z_hi values, with Z^lo= 15Ω, from the one that satisfies condition 1, 625/15, to 100Ω. For each value of Z_hi, the optimizer was stepped through the initial condition loop, and we selected the result with the least loss for the table below. As in the verification case, the runs with N = 2,4 gave results independent of the initial condition. There were as many as six different results for a given run with N = 6. The best results tended to be for L₀ near 0.05λ.

In Table 1, it can be seen that a trade can be made between loss and length. One can write the penalty function as:

e = (g_d-g)² + W_LL² (2)

where g_d and g are the desired and actual minimum gains in dB, W_L is a weighting factor, and L is the total length of the network.

In this form, one can experiment with the various parameters to obtain target performance and complexity goals. For example, if one wants the performance of the half-wave, two-line solution, -0.151 dB, one can either search on Z_hi between 50Ω and 60Ω, with N = 4, or one can set g_d = -0.151, W_L = 0 for Z_hi = 50Ω and run the optimizer once.

Doing the latter produces a design with an overall length of 0.3λ. Similarly, if one sets g_d = -0.57, the value for one quarter-wave transformer section, and Z_hi = 60, N = 4, one obtains L = 0.2λ. The responses for these two designs are shown in Figure 4, and the line lengths in Table 2.

While these reductions in length over designs based on quarter-wave lines are significant, they are not as dramatic as Meschanov's results. This is due to the high value of Z_lo compared to the low terminating resistance. For example, if one could set Z_lo = 10Ω, Z_hi = 62.5Ω, then L = 0.148λ for g_d =-0.57 dB, and L = 0.231λ for g_d = -0.151 dB.

I conclude from the results in Table 2 that a shorter transformer with reasonable loss can be found by violating Meschanov's condition 1, but not necessarily with the limiting values of wave impedance.

The input impedance for a simulated small antenna on a vertical handheld radio box is shown in Figure 5. It is intended for multiple uses in the 800 MHz to 1000 MHz range and at 1575 MHz.

The antenna is a volume-loaded dipole^14,15, 12.5 mm high by 25 mm diameter. The radio box is the actual metal container for the radio, not the plastic shell, and is assumed to be 9 cm high by 4 cm wide by 1.4 cm thick. The matching network is to be a printed circuit added to one of the broad sides of the box.

The impedance was calculated at 10 MHz steps. The network transducer gain when driven by a 50Ω source rises from -1.8 dB at 800 MHz to -0.18 dB at 1000 MHz, but falls to -4 dB at 1580 MHz.

Several possible methods could be used to obtain a matching network using short line, including:

1. Developing a series RLC model for the antenna, use the equations of Wai-Kai Chen¹⁶ to obtain a ladder network, then use the high-frequency mapping to convert to short lines.
2. Using a version of the real frequency method^17,18 to obtain a piecewise-linear approximation to the optimum network input resistance function, followed by a synthesis procedure, which again approximates this function by a network¹⁹. That is, a process with two approximation-optimization stages.
3. Optimize directly on the elements of an LC ladder, then convert to short lines using the high-frequency mapping.
4. Optimize directly on the short-line characteristics. (a) Use unit elements and optimize on their wave impedances; and (b) Use fixed high and low wave impedances and optimize on the line lengths.

I present results based on method 4 in this section, to continue the exploration of the use of a general-purpose optimizer. I chose the frequencies from 800 MHz to 1000 MHz, and 1550 MHz to 1600 MHz, 27 points, as the set over which transducer gain will be maximized.

In working with the unit-element structure, one has to choose a line length and then let the optimizer find the wave impedances for the minimum-loss design. If the wave impedances fall outside the allowed range, one chooses a longer length and tries again.

Thus, for each choice on a number of elements, there is a trial-and-error process for the line length, or one could write a loop to step through a range of candidate lengths. In the beginning, I obtained rather lossy results when choosing the same initial value for all the wave impedances in a given initial-condition sweep. I found that using an alternating sequence gave reasonably good results, that is, setting the initial Z vector to:

Z_init = 20*{1 H 1 H …}, H > 1 (22)

For the three- and four-element cascades, the value of H is not critical. For longer networks, it was found necessary to use a loop and step H from 3 to 6 in steps of 0.1. Some of the results are given in Table 3.

For the first four entries in Table 3, one can see that the minimum gain increases with network length. The minimum wave impedance for a given number of lines also increases with line length.

As with the filter example, this effect determines the minimum length of the network, which is required to have a minimum wave impedance of at least 15Ω. As the number of elements is increased, the minimum gain results for a sweep of H become less organized, indicating the existence of too many local minima.

I believe that the longer result for the five-line case compared to the seven-line case shows that a finer search would be needed to align these results in the pattern for networks with fewer lines.

Fixing the high and low wave impedances and optimizing on the lengths produces similar trends to the unit-element approach. That is, increasing the network length increases the minimum gain.

Increasing the number of lines also again increases the number of local search minima, and the improvement in performance decreases. For these reasons, only three results are given in Table 4. The penalty function did not include length.

It appears that the three-element design fits in with the length versus gain pattern of the three-line unit-element designs. The five-element result is much shorter and has higher gain than the five-line unit-element result. The four-element design is probably the best choice for length at the cost of loss. The response is shown in Figure 6.

I believe that the examples presented in this article support the assertion that cascade line designs in which the line lengths are the design variables are significantly shorter than designs in which the wave impedances are the design variables.

There are many possible ways to find such design solutions. I gave a new mapping from LC ladder prototype to a short-line cascade at a specified frequency. This mapping works well for lowpass filters, but, like any single-frequency solution, its bandwidth of application is limited in impedance-matching problems.

I have concentrated on the use of a general-purpose optimizer to find solutions. I have shown that, in spite of the existence of many local minima, stepping the optimizer through an initial-condition loop yields practical solutions.

It is unlikely that an optimizer will find the global minimum in a general problem involving more than three variables, even with the initial-condition looping method. This is nicely demonstrated by the fact that Meschanov's constraints are needed to help the optimizer in the resistance transformer problem. If an analytical solution exists for a given problem, it should be used. The discovery of a synthesis method in which the line lengths are the variables remains a challenge to the engineering community.

1. Herbert Krauss, Charles Bostian, and Frederick Raab, Solid-State Radio Engineering (John Wiley & Sons Ltd., 1980).

2. David Pozar, Microwave Engineering (Addison-Wesley, 1990).

3. Peter Rizzi, Microwave Engineering: Passive Circuits (Pearson Education POD, 1987).

4. D. B. Miron, “Short-Line Impedance Matching; Some Exact Results,” RF Design, March 1992.

5. D. B. Miron, “The LC Immittance Inverter,” RF Design, January 2000.

6. D. B. Miron, “The Short-Line Transformer,” Applied Microwave and Wireless, March 2001.

7. G. Matthei, L. Young, and E.M.T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures (McGraw-Hill, 1964).

8. John David Rhodes, Theory of Electrical Filters (John Wiley & Sons Ltd., 1976).

9. Ian Hunter, Theory and Design of Microwave Filters (London: IEE Publishing, 2001).

10. G.L. Matthei, “Tables of Chebyshev Impedance-Transforming Networks of Low-Pass Filter Form”, Proceedings of the IEEE, pp. 939-963, August 1964.

11. G.L. Matthei, “Short-Step Chebyshev Impedance Transformers”, IEEE Transactions On Microwave Theory and Techniques, pp. 372-383, August 1966.

12. V.P. Meschanov, I.A. Rasukova, and V.D. Tupikin, “Stepped Transformers on TEM-Transmission Lines”, IEEE Transaction On Microwave Theory and Techniques, pp. 793-798, June 1996.

13. The MathWorks Inc. (www.mathworks.com).

14. D.B. Miron, “Volume Loading-A New Principle for Small Antennas,” Journal of the Applied Computational Electromagnetics Society, July 1999.

15. U.S. Patent 5,986,610, November 1999.

16. Wai-Kai Chen, Passive and Active Filters: Theory and Implementation (John Wiley & Sons Ltd., 1986).

17. Herbert Carlin and Pier Paolo Civalleri, Wideband Circuit Design (CRC Press, 1997).

18. D.B. Miron, “Simplifications for Numerical Impedance Matching,” Proceedings of the North Dakota Academy of Science, pg. 28, April 1994.

19. D.B. Miron, “Synthesis of Long Lowpass Ladders,” Proceedings of the 38th IEEE Midwest Symposium on Circuits and Systems, 1993.

Note: In reference 9, page 146, eq. 5.52 is incorrect. The angle step in the numerator factors should be π/N, as in the denominator. The original [reference 8, p146], eq. 5.4.12, is correct.

Douglas B. Miron received his B.E. and M.E. degrees in electrical engineering from Yale in 1962 and 1963, respectively. He received his Ph.D. in EE/control and communications from the University of Connecticut in 1977.

He worked in various positions in the industry since 1963 and taught at South Dakota State University from 1979 through 1996. He has been a consultant since 1998. His current interests are in small antennas and RF circuits. He can be reached at dbmiron@paulbunyan.net.

The two-port parameters for one- and two-element networks are too dissimilar for a conversion between LC and short line circuits. Adding a third element to both the LC and short-line circuits gives enough flexibility to allow the matrices to match, and enough complexity to make the solution challenging. For an LC T with elements jX₁ Ω, jB₂ S, and jX₃ Ω, the chain (ABCD) matrix is:

For a pi with elements jB₁ S, jX₂ Ω, and jB₃ S, the chain matrix is:

A short-line analog to these circuits is a three-line cascade whose wave impedances and electrical lengths are (Z₁, θ₁), (Z₂, θ₂), and (Z₁, θ₃). For a T analog, Z₁ > Z₂ and for a pi analog, Z₁ < Z₂. Define the chain matrix for this cascade as:

These matrices are not used directly in the solution process. The elements of M_3SL are given as separate equations. By virtue of the labeling, they are the same for both short-line analogs.

The parameters and equations for the short lines use many trigonometric functions. The following shorthand is used.

C_x = cos(θ_x), S_x = sin(θ_x), T_x = tan(θ_x) (6)

F_xpy = function(θ_x + θ_y), F_xmy = function(θ_x - θ_y) (7)

Also define r = Z₁/Z₂ (8)

A = C₁C₂C₃ - rS₁S₂C₃ - S₁C₂S₃ - C₁S₂S₃/r (9)

B = Z₁S₁C₂C₃ + Z₂C₁S₂C₃ + Z₁C₁C₂S₃ - rZ₁S₁S₂S₃ (10)

C = Y₁S₁C₂C₃ + Y₂C₁S₂C₃ + Y₁C₁C₂S₃ - Y₁S₁S₂S₃/r (11)

D = C₁C₂C₃ - S₁S₂C₃/r - S₁C₂S₃ - rC₁S₂S₃ (12)

For a given LC pi or T, we can compute the desired values of A, B, C, and D. Therefore, given these values, we wish to solve for the electrical lengths. The strategy is to form differences that eliminate some terms, and use two-angle trigonometric identities to combine those that remain.

CZ₁ - BY₁ = (r-1/r)(C₁S₂C₃ + S₁S₂S₃) = (r-1/r)S₂C_1m3 (13)

D - A = (r-1/r)(S₁S₂C₃ - C₁S₂S₃) = (r-1/r)S₂S_1m3 (14)

By squaring and adding (13) and (14) one obtains a solution for S₂.

Dividing (14) by (13) gives an expression for tan(θ₁ - θ₃).

A + D = 2C₁C₂C₃ - 2S₁C₂S₃ - (r+1/r)(S₁S₂C₃ + C₁S₂S₃) = 2C₂C_1p3 - (r+1/r)S₂S_1p3

CZ₁ + BY₁ = (r+1/r)S₂C_1p3 + 2C₂S_1p3 (18)

θ₂ is known from (15), so (17) and (18) are two equations in two unknowns, cos(θ₁ + θ₃) and sin(θ₁ + θ₃). These are solved and the ratio taken to give an expression for tan(θ₁ + θ₃). This is a result that has no quadrant ambiguity.

Equations (15), (16) and (19) are sufficient to write a computer program for the three electrical lengths. The practical cases are those for which θ₂ is first quadrant and θ₁ - θ₃ is either first or fourth quadrant.

A set of functions has been written to take the LC X_j and B_k values (one for the T and another for the pi), generates the chain parameters, and then calls another function, which finds the line lengths for the desired structure. The latter function does test the expression for S₂ for realizability, and for each of the four cases of θ₂, finds θ₁ and θ₃ for each, uses (9) to (12) to flag whether the solution is valid, and then returns the minimum-length solution.

A ladder of five or more elements can be represented as a cascade of three-element sections. For example, a five-element ladder which begins and ends with shunt capacitors, C₁, L₂, C₃, L₄, C₅, can be split into two pi sections at C₃, first C₁, L₂, C₃/2, and then C₃/2, L₄, C₅. After each pi is converted to the short-line analog, the adjacent lines will both have wave impedance Z₁, so they can be combined into one line, giving a five-line result.

The chain matrix for any passive loss-less two-port has a determinant of one. That is, AD + BC = 1. This means that there are only three potentially independent parameters, and knowing the overall chain matrix for a ladder of any length would only allow one to solve for three values.

In principle, one could take the chain parameters of a seven-element filter at the corner frequency, and find a three-line cascade that gives the same values. However, this three-line cascade would not give the sharp cutoff behavior of the seven-element original.

This is why the seven-element ladder should be split into three, three-element sections for conversion. The one ladder size that can't be so subdivided is four elements. The following technique seems to work well for this case.

The extra element in a four-element ladder is, of course, either a shunt capacitor or a series inductor. Since we want to start with the chain matrix for the entire ladder and obtain a chain matrix for a three-element conversion, we need a good chain-matrix approximation for the extra element.

That is, we need a chain matrix for a single short line that is a reasonable approximation for the extra element. A numerical examination shows that the chain matrix for a single high-Z short line is a better approximation for a series inductor than is that of a low-Z short line for a shunt capacitor.

Suppose then, that we wish to convert a four-element ladder in which the extra element is L₄. If, as above, we use A, jB, jC, D to represent the chain-matrix elements for the overall ladder and A', jB', jC', D' to represent the chain-matrix elements for a three-line cascade, and (Z₂, θ₄) as the description of the short line approximating L₄, then set

sin(θ₄) = ω_cL₄ / Z₂ (20)

The chain matrix for the desired three-element cascade is found by right-multiplying the known one for the four-element ladder by the inverse of the chain matrix for the fourth short line.

Even though ω_cL₄ is only approximated by the fourth line, the overall match is exact.