# 3.1: Introduction

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Central to microwave filter design is the identification of a particular distributed structure that inherently has the desired frequency-shaping attributes. Then this structure is adjusted to match an ideal mathematical, or perhaps lumped-element, representation of a filter. Filter design is both science and art, matching synthesis with instinctive knowledge of appropriate physical and circuit structures.

A large number of microwave filters are realized using coupled transmission lines and the most important of these are parallel coupled line (PCL) filters. These are derived from prototypes, with the development following a strategy that enables the filter to be realized using one of several coupled-line configurations. So the strategy is to first examine coupled-line configurations and determine the types of circuit structures that can be realized. Then the steps in synthesis are designed to go from an LC filter prototype to a prototype that has the structures that can be realized by a coupled-line configuration.

Since the beginnings of microwave circuit synthesis, it has become common to present circuit concepts using examples, very often because the steps in realizing a filter may be too difficult to specify algorithmically and there are many steps that require intuition. This procedure is followed here.

This introduction concludes with an example that illustrates the art and science of microwave engineering. It is seen that a simple pair of coupled lines with appropriate matching has a bandpass filter response. The intrinsic bandpass response is supported by coupled lines, so the synthesis procedure is adapting the intrinsic response to specifications.

Example \(\PageIndex{1}\): Coupled Lines as a Basic Element of a Bandpass Filter

Design Environment Project File: RFDesign Coupled Shorted Microstrip Lines C.emp

A coupled pair of microstrip lines terminated in short circuits is shown in Figure 3.2.1. This circuit has a bandpass response centered at \(7.5\text{ GHz}\), as shown in Figure 3.2.2. So while the insertion loss reduces to about \(3.5\text{ dB}\) at \(7.5\text{ GHz}\), the return loss is not very high. This is a good indication that matching could be used to reduce the insertion loss. So the problem is then one of determining the appropriate matching network and to do this an understanding of the circuit response needs to be gained. Figure 3.2.3 plots the \(S_{11}\) and \(S_{21}\) responses on a Smith chart. By looking at \(S_{11}\) it is seen that between \(7\) and \(8\text{ GHz}\) the input impedance is primarily inductive. This indicates that a series capacitance could be used in matching. Another view is shown in Figure 3.2.4, which plots the magnitude and phase of the input impedance at Port \(1\) of the coupled microstrip lines (this will be the same as the input impedance at Port \(2\)). So around \(7\text{ GHz}\), the input impedance is inductive since the phase of the impedance is close to \(90^{\circ}\) and the magnitude of the impedance is increasing with frequency. At \(7\text{ GHz}\) the impedance is approximately \(\jmath 50\:\Omega\), which would be resonated out by a series capacitor of \(0.45\text{ pF}\). However it will not be as simple as this, as the tuning capacitor will need to be placed at both Ports \(1\) and \(2\). However, this does serve to provide an initial design point.

The schematic required to implement the design above is shown in Figure 3.2.5. In this circuit schematic, the coupled line shown in Figure 3.2.1(b) is captured as a subcircuit called ”CoupledLine.” The series capacitors at Ports \(1\) and \(2\) are ”\(C_{1}\)” and ”\(C_{2}\),” that is, \(C_{1}\) and \(C_{2}\), respectively. \(C_{1}\) and \(C_{2}\) are established as tunable elements with the governing variable being ”CC.” This capacitor is tuned to obtain the optimum bandpass response. With \(C_{1} = C_{2} = 44.8\text{ fF}\), the bandpass response shown in Figure 3.2.6 is obtained. This is an almost ideal maximally flat bandpass filter response. The main passband is at \(7\text{ GHz}\) and there is a parasitic bandpass response at the third harmonic. This spurious passband at an odd harmonic occurs often in transmission line designs. A small alteration of the tuning capacitor can change the response to have ripples in the passband and sharper filter skirts (see Figure 3.2.7), where the tuning capacitors are each \(37.6\text{ fF}\). (To convince yourself of the sharper skirt consider the insertion loss at \(100\text{ MHz}\) away from the passband. The insertion loss of the filter with the maximally flat response is \(23\text{ dB}\) there, and that of the filter with the ripple response is \(27\text{ dB}\).) This filter has two passband poles and is simple enough to design as done here. Higher-order filters (with more than two passband resonators) require a more sophisticated design approach. Still this example demonstrates that the coupled-line structure has a good passband response on its own provided that appropriate matching is used.

The bandpass characteristic is a result of the phase velocities of the even and odd modes being different. If they were the same, the transmission coefficient would be zero at \(7\text{ GHz}\).