Techniques Yield Tiny Hairpin-Line Resonator Filters

By Rodrigo Neves Martins
Graduate Student

Humberto Abdalla Jr.
Associate Professor

Universidade de Brasília, Departamento de Engenharia Elétrica, P.O. Box 04386, 70.710-900-Brasília-D.F., Brazil, e-mail: [email protected]

HAIRPIN-LINE resonator filters are relatively simple to design and build, although they are generally too large for mobile radio applications. Fortunately, design methodology and supporting software have been developed that support the creation of narrowband filters with miniature hairpin resonators that are compact and low in cost.

Traditionally, microstrip bandpass filters are widely used at microwave frequencies with special attention at X-band. In this band, with an appropriate dielectric substrate, it is possible to perform an analysis based on quasi-transverse-electromagnetic (TEM) modes of propagation. Among the innumerous structures that can be used, edge-coupled microstrip filters are very popular because they do not require short circuits and are easy to design and build. The main disadvantage is their large size. In order to solve this problem, hairpin-line filters are used. Conceptually, hairpin-line filters can be obtained by folding the resonators of edge-coupled filters.¹ Figure 1 shows the construction of a hairpin-line filter, where each half-wave resonator is folded into a hairpin. While the configuration is suitable for compact microwave bandpass filters, the size of the hairpin is still too large to design a filter for mobile radio communications.

Fig 1. This four-step folding procedure can be used to obtain a hairpin-line filter. Each half-wave resonator is folded into a hairpin. The configuration is suitable for compact microwave bandpass filters.

To reduce the size of hairpin resonators and create an attractive topology for applications in the A-band of mobile radio communications, the arms of the resonator are reactively loaded with parallel coupled lines.² Figure 2 shows the difference between the conventional hairpin resonator and the miniaturized hairpin resonator loaded with coupled lines. Filters using these miniaturized resonators are 50 percent more compact than filters that use conventional hairpin resonators.

Fig 2. By applying some straightforward design procedures, it is possible to transform (a) a conventional hairpin resonator and (b) a miniaturized hairpin resonator hairpin-line filter into a (c) miniature hairpin-line filter.

When it is necessary to design a bandpass filter with narrow bandwidth and high selectivity, a conventional lowpass-to-bandpass transformation, although theoretically correct, does not achieve the desired performance. Filters with these characteristics are achieved by coupling similar-type resonators together. In this case, the resonators consist of reactive parallel circuits which are tuned to the desired center frequency and coupled by a simple lumped capacitance. The design of multiresonator filters is based on the lowpass prototype aided by the concept of coupling coefficients, k_ij, and quality factors, q_I.³

Fig 3. The schematic diagrams for these three structures show (a) a lowpass filter used for defining k values, (b) a bandpass filter derived from the lowpass filter, and (c) a miniature hairpin resonator filter with 50- input and output ports. The coupling acheived is associated with coupling coefficients, k_ij, and the quality factor, Q_ij.

The coupling coefficient k_ij is defined for a lowpass prototype circuit as the ratio of the resonant frequency of two adjacent elements to the 3-dB cutoff frequency [Fig. 3(a)]:

where:

w_ij = 1/(g_Ig_j)^0.5, and

w_{3 dB} = the normalized cutoff frequency (= 1).

Therefore:

The quality factor, q_I, is defined as the quality of the reactive element influenced by the source and load resistance, if present, and it is in series or parallel. For symmetrical and lossless networks, the quality factor of each element evaluated at w = w_{3 dB} = 1 is infinity except for the end terminations:

where:

R₁ = the source termination, and

R_n = the load termination.

From the bandwidth, f, and the center frequency, f_m, the unnormalized coupling coefficients, k_ij, and the unnormalized quality factor, Q_i, for the end resonators can be obtained³:

These quantities are directly related to the bandpass circuit elements (Fig. 3b). The coupling capacitances between the ith and jth nodes are provided for:

where:

C_node = the total shunt capacitance of each node when all other nodes are shorted to ground.

Therefore, the shunt capacitors of the parallel tuned circuits are equal to the total nodal capacitance without the values of the coupling capacitors connected to that node⁴:

When all of the nodes are shorted except for the ith node, the nodal inductor, L_I, and the nodal capacitor, C_I, resonate at f_m.

If the circuit is built in distributed form, the equivalent resonance properties of the resonators, regardless of their form, are obtained through the resonant frequency and the slope parameter.⁵ For shunt LC resonators, the susceptance slope parameter is provided by:

Therefore, the quality factor can be defined by:

Considering that all of the resonators are without loss, it is possible to establish identical resonance conditions between a miniature hairpin resonator and a shunt LC resonator. Figure 3c shows a schematic of a multiresonator filter using miniaturized hairpin resonators. In this drawing, the coupling achieved between interior resonators and the first and last resonators is associated with the coupling coefficients, k_ij , and the quality factor, Q_i, respectively.

A hairpin resonator can be considered as two quadripoles represented by a transmission line and a parallel coupled line, respectively (Fig. 4), with the following parameters: Z_s is the characteristic impedance of the transmission line, _s is the electrical length of the transmission line, Z_pe and Z_po are the even- and odd-mode impedances, respectively, of the parallel coupled lines, and _pe and _po are the even- and odd-mode electrical lengths of the parallel coupled lines.

Fig 4. Two quadripoles in a parallel connection can be used to form an equivalent circuit of a hairpin resonator (d), by modifying the basic hairpin resonator (a) through input and output electrical connectors (b) while maintaining the even- and odd-mode electrical lengths and impedances (c).

These two quadripoles are connected in parallel at their input and output terminals to form a composite quadripole where the short-circuit admittance matrix is simply provided by the sum of the short-circuit admittance matrices of each quadripole component. The resonance condition can be calculated from the input admittance using the total short-circuit admittance matrix. At resonance frequency, the input admittance must be zero, and since the even- and odd-mode electrical lengths are almost equal, with _pe = _po = _p, it is possible to obtain the relationship between the resonance frequency and the electrical parameters of the transmission line and coupled line.² The result of these calculations is the resonance equation shown here:

An analysis of the resonance equation (eq. 10) shows that some parameters can be adjusted by the designer while other parameters are intrinsically linked to each other. The line impedances are design parameters, and it is evident that the resonance frequency is directly proportional to the electrical length of the resonator. Hence, if the impedance of the single transmission line and the coupling factor or one of the impedances of the parallel coupled line (the even- or odd-mode impedance) are known, the resonance condition is determinated by two parameters, _s and _p. Therefore, if the value of _s, which is associated with the resonant frequency, is known, there is only one value for _p that will cause resonance at the desired frequency. Figure 5 shows the behavior of the resonant frequency when the electrical length of the parallel coupled line is changed--for practical values of single-line impedance and coupled-line impedance used in the practical design of the miniaturized hairpin resonator filter.

Fig 5. Fine resonance frequency tuning of the hairpin-line filter can be achieved by adjusting _p.

Consequently, the variation of the electrical length of the parallel-coupled line causes a fine frequency tuning, which supports greater design flexibility.

Fig 6. The equivalent circuit (a) shows the circuit-element representation of the coupling effects. The interstage coupling between resonators (b) is achieved by parallel coupling of lines.

The non-normalized coupling coefficients represent the interstage coupling between resonators, that is achieved by parallel lines coupling with electrical length of _c (Fig. 6):

where:

k = (Z_oe - Z_oo)/(Z_oe + Z_oo),
and

(Z_oo - Z_oe) = ZS²

These equations enable the even and odd impedances to be obtained for the coupled line as a function of the coupling coefficients between the resonators, coupled electrical length, and characteristic impedance of the single transmission line of the resonator:

If the even and odd impedances are known by using the classic synthesis process, the line width and the spacing between the resonators are easily obtained.⁶ The quality factor of the first and last resonators will be used to calculate the coupling of the filter to the source and to the load.

Bandpass filters are designed for maximum power transfer, accomplished by using low-loss networks and a good matching network between the input and output impedances in the desired frequency band. In the topology analyzed here, the source-power transfer to the load is obtained by coupling to the resonator-transmission line. This kind of coupling has problems in production due to a significant reduction in the coupled section length, and for this reason, the spacing of the external coupling sections is very critical. To avoid this, the end resonators may be externally coupled by tapping instead of using a coupled section (Fig. 7).

Fig 7. This tapped bandpass filter can be simplified by using taps (b) instead of coupled sections (b) for the end resonators.

The schematic of the tapped miniaturized hairpin resonator is shown in Fig. 8, where the effective electrical length of resonance is defined.⁷. Using this effective electrical length, the equivalent circuit of the tapped miniaturized hairpin resonator is established.

Fig 8. This equivalent circuit can be used to represent a tapped miniature hairpin resonator.

The tapping location, f_l, is provided by:

In the vicinity of the resonant frequency, the input admitance at the tap point is:

where :

G_o = the input conductance of the filter,

b = the slope parameter of the equivalent circuit of the tapped miniaturized hairpin resonator, and

_o = the angular resonant frequency.

The susceptance slope parameter b of the tapped miniaturized hairpin resonator is:⁸

The condition of resonance and maximum power transfer is established when the input admittance of the structure is purely real and equals the source conductance:

Considering that the structure is symmetrical, these equations enable the calculation of the tap point of the input and output resonators:

On the basis of the derived formula, software was developed for the design of tapped miniaturized hairpin-line filters. The software is composed of three independent modules that execute the following functions:

� Construction of the filter-transfer function with Chebyschev or Butterworth frequency response. This module also contains a dedicated graphic interface that permits the visualization of the frequency response before the filter synthesis.

� Graphics visualization and analysis module of the poles and zeros of filters in the complex variable plane (the S plane).

� Synthesis of the filter in lumped and distributed parameters. The synthesis software module provides the layout of the filter, including all of the calculated dimensions, taking the chosen fabrication technology into account.

A step-by-step design procedure that uses the developed software is outlined:

1. The type of frequency response and the transmission structure is first selected. The filter specifications and the basic parameters of the transmission structure based on Fig. 9 are then completed.

Fig 9. The software program was used to predict and plot the frequency response of the hairpin-line resonator filter.

2. With the specifications selected, it is possible to access the analysis and synthesis modules of the computer program.

These software modules support the realization of the hairpin circuit using all necessary data. Figure 9 shows a typical output of the frequency response module, where it is possible to see the insertion and return losses. In the synthesis module, it is possible to obtain the design in lumped and distributed parameters. In the program, a design using distributed parameters initiates calculation of the miniature hairpin resonator and the calculation of coupling the resonators and the end-tapped resonators. A screen with the layout of the tapped miniaturized hairpin-line resonator filter is shown in Fig. 10.

Fig 10. The program shows the miniature hairpin-line resonator filter design.

Using this software, a miniature hairpin-line filter was designed and tested for the following specifications: a center frequency of 836.5 MHz, a fractional bandwidth of 3 percent, ripple of 0.1 dB, and three resonators. A microstrip transmission structure was chosen for the filter, using a substrate with a dielectric constant _r of 10.5 and thickness of 1.27 mm. To tune the filter and obtain the adjustment of the tap point, a computer-aided optimization process was used.

Fig 11. This photograph shows the miniature hairpin-line resonator filter fabricated with the help of the custom computer program.

A photograph of the filter is shown in Fig. 11 and its measured frequency response is shown in Fig. 12.

Fig 12. The frequency response of the miniature hairpin-line resonator filter agrees well with simulated data from the computer program.

The passband insertion loss was found to be approximately 1.5 dB. Measurements indicate a displacement of 1 percent at the center frequency in addition to a bandwidth that is close to the specified value.

8. Shimon Caspi and J. Adelman, "Design of Combline and Intergital Filters With Tapped-Line Input," IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-36, No. 4, pp. 759-763, April 1988.