Broadband Matching

The following sections are included:

• Dedication
• Contents
• Preface to the 3rd Edition
• Preface to the 2nd Edition
• Preface to the 1st Edition

AN electrical network is a structure composed of a finite number of interconnected elements with a set of ports or accessible terminal pairs at which voltages and currents may be measured and the transfer of electromagnetic energy into or out of the structure can be made. The elements are idealizations of actual physical devices such as resistors, capacitors, inductors, transformers and generators; and obey the established laws of physics relating various physical quantities such as current, voltage and so forth. Fundamental to the concept of a port is the assumption that the instantaneous current entering one terminal of the port is always equal to the instantaneous current leaving the other terminal of the port. A network with n such accessible ports is called an n-port network or simply an n-port, as depicted in Fig. 1.1. In this chapter, we introduce many fundamental concepts related to linear, time-invariant n-port networks. We first define passivity in terms of the universally encountered physical quantities time and energy and review the general characterizations of an n-port network. We then translate the time-domain passivity conditions into the equivalent frequency-domain passivity criteria, which are to be employed to obtain the fundamental limitations on its behavior and utility…

IN THE preceding chapter, we have indicated that the terminal behavior of a linear n-port network can be characterized by any one of the various sets of parameters such as the short-circuit admittance parameters, the open-circuit impedance parameters, the hybrid parameters, the transmission parameters, etc. However, not all of these parameters will exist. For example, a two-port network consisting only of two wires with a finite impedance connecting across them has no short-circuit admittance matrix but has the open-circuit impedance matrix. If the finite impedance is removed from the two-port network, the resulting two-port possesses neither the impedance matrix, the admittance matrix, the hybrid matrix, nor the transmission matrix. An ideal transformer possesses the hybrid matrix but neither the impedance nor the admittance matrix. The reason for this is that these parameters are defined in terms of the quantities that are obtained when one of the ports is short-circuited or open-circuited. In other words, they are defined with respect to the zero or infinite loading at the ports. The scattering parameters, on the other hand, are defined in terms of some finite stable loadings at the ports. Thus, they always exist for all nonpathological linear passive time-invariant networks…

IN CHAPTERS 4 and 5, we shall apply the concept of the scattering parameters defined in the foregoing to the design of a coupling network that matches a given load impedance to a resistive generator and that achieves a preassigned transducer power-gain characteristic over a frequency band of interest. Ideally, we hope that we can design such a coupling network having any desired gain characteristic, such as the ideal brick-wall type of response shown in Fig. 3.1, which is constant from ω = 0 to ω = ω_c and zero for all ω greater than w_c. However, such niceties cannot be achieved with a finite number of network elements. What then can be done in order to obtain a desired gain characteristic? Instead of seeking an overly idealistic performance criteria, we specify the maximum permissible loss or maximum permissible reflection coefficient over a given frequency band of interest called the passband, the minimum allowable loss or reflection coefficient over another frequency band called the stopband, and a statement about the selectivity or the tolerable interval between these two bands. We then seek a rational function that meets all the specifications and at the same time it must be realizable for the class of networks desired. This is known as the approximation problem…

IN CHAPTER 2, we have studied the properties of the scattering matrix associated with an n-port network, and indicated how it can be extended to complex normalization. In Chapter 3, we presented three popular rational function schemes for approximating the ideal low-pass brick-wall type of gain response. In the present chapter, we shall apply these results to the design of matching networks. As is well known, in the design of communication systems, a basic problem is to design a coupling network between a given source and a given load so that the transfer of power from the source to the load is maximized over a given frequency band of interest. A problem of this type invariably involves the design of a coupling network to transform a given load impedance into another specified one. We refer to this operation as impedance matching or equalization, and the resulting coupling network as matching network or equalizer. We recognize that the choice of a lossy equalizer would not only lessen the transducer power gain but also severely hamper our ability to manipulate since the scattering matrix of a lossy equalizer is not necessarily unitary. Hence, we shall deal exclusively with the design of lossless equalizers…

IN THE preceding chapter, we were concerned with the problem of matching a given strictly passive load impedance to a resistive generator to achieve a preassigned transducer power-gain characteristic. The central problem was to ascertain the restrictions imposed upon the transducer power-gain characteristic by the passive load impedance. The restrictions were stated in terms of the coefficients of the Laurent series expansions of various quantities defined by the load impedance, which were then shown to be both necessary and sufficient for the existence of a lossless two-port coupling network called an equalizer. Since we admit only passive networks in the study, it is clear that at any sinusoidal frequency the maximal attainable gain cannot exceed unity. Thus, no amplification can be achieved. On the other hand, if we admit a load impedance z₂(s) which is active

IN CHAPTER 4 we considered the problem of matching a given strictly passive load impedance to a resistive generator to achieve a preassigned transducer power-gain characteristic. In Chapter 5 we extended this result by considering a special class of active impedances and showed how it can be applied to the design of three basic amplifier configurations. In the present chapter, we shall derive explicit formulas for the synthesis of optimum broadband impedance-matching networks for a class of very useful and practical load impedances composed of the parallel combination of a resistor and a capacitor and then in series with an inductor, as shown in Fig. 6.1, which may include the parasitic effects of a physical device. The problem of matching out this type of loads over a given frequency band to within a given tolerance recurs constantly in broadband amplifier design…

SO FAR we have only considered the match between a resistive generator and a frequency-dependent load to achieve a preassigned transducer powergain characteristic over the entire sinusoidal frequency spectrum. However, in many practical situations, the internal impedances of the available electronic sources are not purely resistive, especially at high frequencies. In the design of interstage coupling networks, for example, the output and the input impedances of the stages involved may not be approximated by pure resistances. In these cases, the design of a lossless equalizer to match out two arbitrary passive impedances is necessary. Earlier work along this line was done by Fielder (1961), who considered a class of lossless ladder networks terminating in a resistor. The general solution to the problem was given by Chien (1974) and Chen and Satyanarayana (1982). Chien's approach is, by using a generalized scattering parameter representation of a lossless two-port network one is able to construct the class of all paraunitary matrices from the given impedances and the prescribed transducer power-gain characteristic. Then by imposing the physical realizability requirements for the class of scattering matrices so constructed, one can determine the necessary and sufficient conditions on the transducer powergain characteristic so that at least one of the matrices is physically realizable. The desired matching network is obtained by realizing this scattering matrix using any of the known techniques. This by itself is not a simple matter. On the other hand, Chen and Satyanarayana's method involves the realization of a positive-real impedance as the input impedance of a lossless two-port network terminating in a one-ohm resistor. The removal of this one-ohm resistor yields a desired lossless coupling network. In the present chapter, we give a unified summary of these results and show how they are interrelated.

IN CHAPTER 4 we described the broadband matching between a resistive generator and an arbitrary load to achieve a preassigned transducer power-gain characteristic over the entire real-frequency spectrum. In Chapter 7 we extended the single match to double match by considering the situation where both source and load impedances are frequency dependent. In applying these techniques, a prerequisite is that the load or source impedance be first represented by a finite circuit model, from which a driving-point function that analytically characterizes the model over the entire complex frequency plane is obtained. This function is then processed to find the theoretic gain-bandwidth limits for the approximated model of the source or load. To realize the equalizer, an analytic form of the transducer power-gain characteristic must be assumed. This together with the zeros of an all-pass function is adjusted to satisfy the gain-bandwidth restrictions. Even if the circuit model is known, the procedure presents great numerical difficulties that in principle are resolvable but in practice become almost intractable, especially if the model contains more than two reactive elements…

In Chapter 3, we have studied three common types of approximation: The Butterworth response, the Chebyshev response, and the elliptic response. These responses approximate the magnitude of a transfer function. In this chapter, we first introduce an approximating function that will give maximally-flat time delay. Delay filters are frequently encountered in the design of communication systems such as in the transmission of a signal through a coaxial cable or an optical fiber, especially in digital transmission, where delay, being insensitive to human ear, plays a vital role in performance. We then show how to design a coupling network between a given source and a given load to achieve a maximally-flat group delay characteristic. A similar introductory material on this can also be found in Chen (1986).

A diplexer is a circuit that separates a frequency spectrum into two channels of signals. The most popular configuration is composed of a low-pass two-port network and a high-pass one connected either in series or in parallel, as shown in Figs. 10.1 and 10.2. In general, a multiplexer is designed to separate a frequency spectrum into many channels. In this chapter, we first show the design techniques of a diplexer and then a multiplexer of various configurations and channel characteristics.

The following sections are included:

• APPENDICES
  • APPENDIX A: The Butterworth Response
  • APPENDIX B: The Chebyshev Response
  • APPENDIX C: The Elliptic Response
• Symbol Index
• Subject Index