# Trends In Adaptive Impedance Computer Science Essay

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In the design of any communication system, impedance matching between/at all stages is very critical aspect. When a transmission line is not terminated by its characteristic impedance, maximum power is not transferred to load. Part of incident energy gets reflected due to impedance mismatch at load stage. we say here as power getting wasted due to reflection from the load. It has a direct impact on efficiency of transmission.

Impedance matching in this paper refers to dynamic impedance matching between stages involved in high frequency communications for maximing power transfer. Impedance matching is crucial part of all high frequency communication systems from battery conservation point. Usually π and L networks made from L and C components are used to achieve impedance matching. Several methods have been proposed for fixed frequency communication systems but they all fail to offer acceptable results when ported to high frequency multiband, multimode systems where carrier frequency keeps changing based on predefined algorithm. Here arises a need for adaptive impedance matching systems. Adaptive solutions have been implemented in many domains so far and have offered very efficient results. All adaptive algorithms tune L and C components involved in π and L networks. This paper discusses methods available and been accepted for implementation practically for adaptive impedance matching.

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1. Introduction

Wireless communications is currently having an exponentially growth in communication industry. Technological enhancements are leading to nanoscale based high speed systems enabling high speed transmissions of data, voice and video possible. Reduction in operational circuit size leads to more radiation losses and high speed operations often leads to signal integrity issues like reflection and crosstalk. Such issues need to be addressed failing which efficiency of system would be tremendously compromised. Signal reflections are due to impedance mismatch between every two successive stages. Formulation of a complete mathematical model for impedance mismatch is very complex process since parameters involved are function of many factors like process variations, length variations of interconnection lines and atmospheric paramaters. To address such complex issue, knowledge based algorithms present an unmatched alternative for impedance matching.

2. Literature

The period between 1880 and 1920 proved very significant for experimental electrical engineering. the wireless revolution sweeping across the globe today has its technological underpinnings also situated during this period. The invention of the vacuum tube in the early 1900s and its subsequent maturity by the 1920s resulted in a lot of interest in RF power amplifiers. Expectedly, the impedance matching problem had also to be dealt with. T and Π circuits were being employed at the output stage of an RF power amplifier at initial stages. This work suggests their use and also discusses the utility of such circuits to couple the power from the output of the power amplifier to an antenna. As time went on, several authors proposed their other works which have dominated today with their further enhancements.

2.1. Darlingtons Theorem of Impedance Matching - 1939

Darlingtons Theorem says that an impedance function of an arbitrary assemblage of reactive and resistive elements can be represented by a reactive (lossless L and C) network terminated in a 1-ohm resistance (Darlington, 1939).[15]

Figure 2.1. Darlington Prediction of Network [15]

2.2. Bode-Fano Theory and the Analytic Approach - 1948

The first steps towards a theoretical basis for broadband matching were taken by Bode. While this work considered a simple load (parallel combination of resistor and capacitor) under the single matching category, it was significant because for the first time engineers had a bound, cast in a gain-bandwidth formulation, for a given load. Central to Bode's work is the analytical approach which requires a circuit approximation of the load impedance as it uses the data in the complex frequency plane; it has been developed for

simple load circuits only. This approach, unlike, did not provide the techniques to build a good impedance matching network. It did however give the designer a mathematical tool to compare the performance of a designed impedance matching network with a specified bandwidth in terms of the gain or the reflection coefficient. This approach was further investigated by Fano and Youla [15] .

Of particular importance is the utilization of Darlington's powerful theorem by Fano together with the work done earlier by Bode to convert the broadband impedance matching problem into a filter design problem. This theorem states that any physically realizable impedance function can be decomposed into a purely reactive lossless network

terminated into a 1 resistor. Since the source resistance can also be changed to 1 by including a transformer, the impedance matching problem gets converted to a filter design problem.

Figure 2.2: Broadband Impedance Matching

Problem a) As tackled by Fano, by applying Darlington's Theorem to the load shown b) Thus Converting it into a Filter Design Problem seen in c) [15]

2.3 The Real Frequency Technique

The progress in analytic approaches to broadband impedance matching was tempered by the fact that designers needed to approximate the load with a circuit model. The analytic techniques work well for simple load types. However, for more complicated loads the analytic approach is too difficult. The Carlins Real Frequency Technique (RFT) based gain-bandwidth optimization approach is a numerical technique that does not require any circuit approximation of the load impedance. Instead, it relies on the actual measurement data of the load. A comprehensive review has been recently provided by Newman.

Although being quite versatile, this approach, along with the pure optimization step, still requires non-unique operations with rational polynomial approximations, and further extraction of equalizer parameters using the Darlington procedure. Moreover, a transformer is required to match the obtained equalizer to the fixed generator resistance of 50 [15].

2.4 Carlin - 1977

Carlin presented a new method for designing lossless matching networks. This method utilizes directly experimental load impedance data sampled at arbitrary frequencies. The arithmetic is well organized so that required optimization step works well. Also, this technique is especially well suited for producing arbitrarily shaped power transfer functions versus frequency. This has special application in microwave amplifier network design,

where it can compensate for approx 6 dB/octave roll off of transistor gain above critical frequency [15]. Carlins method is based on a very practical application of hilbert transform which relates reactive frequency behaviour to resistive behaviour [15].

2.5 Smith Chart - a Bilinear Transformation of Complex Impedance Plane - 1939

The impedance's at several stages of communciation's system is a complicated function

of frequency and as such cannot be put into an analytic form for all frequencies. However it can be approximately represented over a frequency band by a lumped element lossless network terminated by a resistance. The complexity of this two terminal pair lossless network is dependent on the degree of approximation of the impedance curve of the antenna in the desired frequency range [15].

Prior to the advent of digital computers and calculators, engineers developed all sorts of aids (tables, charts, graphs, etc.) to facilitate their calculations for design and analysis. To reduce the tedious manipulations involved in calculating the characteristics of transmission lines, graphical means have been developed. The Smith chart3 is the most commonly used of the graphical techniques. It is basically a graphical indication of the impedance of a transmission line as one moves along the line. It becomes easy to use after a small amount of experience [15].

3 Impedance Matching in General

In general ,basic voltage and current sources can be represented as in below figure.

Figure 3.1. Practical Sources with their Internal Resistances [6]

Considering only voltage source for discussion,. As per thevenin's and max power transfer theorem, this incident energy will be completely absorbed only if output impedance of source is equal to input impedance of load. If impedances are complex, then real parts must match and reactive parts must get cancelled. If source and load impedances do not match, part of incident energy is absorbed by load and rest is reflected back on transmission line. Amplitude of reflected wave is governed by degree of mismatch. Incident and reflected waves add in-phase/out-phase giving rise to standing waves. This gives clear indication that impedance matching between every two successive stage becomes a very important aspect. Here two parameters are used to make judge of impedance matching in running state.

reflection co-efficient Ð“ =

vswr s = =

3.1 Impedance Matching Categories

In general, impedance matching between an arbitrary pair of source and load can be classified into three categories. These possible cases are shown in figure and outlined below:

1] The resistive matching problem → source and load are both resistive and unequal 2] The single matching problem → source is resistive and the load are complex 3] The double matching problem → source and load are both complex

The first case requires a network that will simply adjust the load resistance such that it is equal to the source resistance. The single matching and double matching cases however are a little more complicated. The single matching case is typically the most common impedance matching scenario encountered in antenna and RF circuit design. Since the load is complex and the source is resistive, there is reactance cancellation required. Additionally if the source and load resistances are not equal, there is the need for a step up or step down network to maximize the power transfer (assuming narrowband single frequency).

Figure 3.2. Impedance Matching Categories a] Source and Load are Resistive b] Source is Resistive and Load is Complex c] Source is Complex and Load is Complex [15]

3.2 Narrowband and Broadband

Impedance matching is achieved with the design of a network or a transducer so that a terminating impedance is transformed exactly to a desired impedance at a frequency or over a band of frequencies. These matching networks can be active or passive and can be designed using graphic and numerical methods. Matching networks when inserted in system should not bring any insertion losses(attenuation) and hence they must be compiled of L and C components having high quality factors. But practically when we increase quality factor , it has a direct impact on bandwidth of system making them operate in narrowband. Narrowband issue can be solved by cascading several networks for obtaining high Bandwidth [9] . In order to achieve impedance matching for maximum power transfer, lumped networks or transformers or distributed matching can be used. Hence one can see that if a designer wishes to design an equalizer with zero insertion loss, he will endup with narrowband equalizer. Often, multiple such equalizers can be cascaded to obtain high bandwidth. L type matching networks in cascaded configuration are known as ladder networks. Ladder networks have advantage over simple networks as higher impedance transformation ratios are possible. Q of two-stage network is significantly lower [15] [9] [13].

Figure 3.3. Typical Configurations of L,T,_, Networks and their Ladders [31]

Comparision of various methods available..

Figure 3.4.: Typical evaluations shows comparision of various methods available..[31]

Impedance matching can also be achieved by single or multiple quarter wave transformers discussed in next chapter. The biggest challenge is the mismatch of impedance caused between antenna and surrounding atmosphere which is very unpredictive and nonstatistical. The traditional methods will not be sufficient to counter this mismatch.

3.3 VSWR Measurement

Several methods have been proposed for measurement of VSWR. This task is often a critical part of entire system as error signal controlling element values of equalizers is generated based on this measurement. It is important to know that for accurate VSWR measurements of devices, the VSWR should be measured at the input of the device in question (antenna, CDN, etc). Any cable loss, or attenuation, will make the VSWR at the input of the cable appear much better than at the load or termination. The reason is that the cable loss or attenuation increases the return loss.

Figure 3.5. VSWR Measurement [13]

VSWR meter calibrates SWR into proportional analog signal which is processed with adaptive sigma Analog to digital converter to get digital signal. this signal thus obtained is used by adaptive algorithms to tune element values of matching networks.

4 Methods in Practice for Impedance Matching

Todays communication demand's can be satisfied only through multiband, multimode models especially mobile communications, radio navigation and many. Handsets (mobile terminals) operate at very high switching rates maintaining high data rates. There are broadly two approaches for impedance matching : broadband and narrowband matching. L,_ and T networks of reactive lumped elements, Single and double tuning of transmission lines and/or a hybrid approach involving a combination of both. these may also include transformers. Transformers in some cases offer good bandwidth but still not sufficient for modern wideband communications hence sometimes considered as narrowband matching. Wideband matching methods can be classified into two groups : analytical approach and real frequency technique [RFT] approach. Analytical approach requires a circuit approximation of the antenna's input impedance as it uses the data in complex frequency plane and can be used for simple load circuits only. Carlin's RFT gain bandwidth optimization approach is numerical technique does not require a load model, but involves non-unique operations with rational polynomial approximations, extraction of equalizers parameters using darlington procedure and a transformer to match obtained equalizer to the fixed generator resistance of 50 ohms [15] [13] [9]. In narrowband matching, first is distributed impedance matching which leads to modifying the antenna geometry itself by identifying appropriate degrees of freedom within the structure. Distributed model matching offers very flexibility for broadband of frequencies and hence good bandwidth [15] . The second option is the lumped element approach to impedance matching. In this approach instead of modifying the antenna geometry, a passive network attempts to equalize the impedance mismatch between the source and the antenna load. Lumped models offer less bandwidth but several lumped networks can be cascaded to improve bandwidth [15] [13] [9].

When characteristic impedance of transmission line matches the output impedance of transmitter and the impedance of the antenna itself, the SWR will be 1:1 and maximum power transfer will take place. The best way to prevent mismatch between antenna and transmission line is through correct design. In practice, when mismatch do occur, some corrections are possible. One solution is to tune antenna usually by adjusting its length to minimize the VSWR. Another is to insert an impedance matching network or antenna tuner between transmitter and transmission line such as balun or LC, L, T or Pi networks. These circuits can make impedances equal so that no mismatch occurs or atleast the mismatch is minimized. Ideally VSWR should be 1 but practically VSWR value below 2 is usually acceptable. Today, most transmitters and receivers are designed to have an antenna impedance of 50 ohms. Receivers must see an antenna system including transmission line that looks like a generator with a 50 ohm resistive impedance. Transmitters must see an antenna including transmission line as 50 ohm resistive impedance over the desired operating frequency range if the VSWR is to be close to 1 and maximum power is to be transferred to antenna. Further, an antenna that is not of correct length for the desired frequency will have an large reactive component that can severely affect the VSWR and power output. In situations in which a perfect match between antenna, transmission line, and transmitter is not possible, special techniques collectively referred to as antenna tuning or antenna matching are used to maximize power input and output. Most of these techniques are aimed at impedance matching i.e. making one impedance look like another through the use of tuned circuits or other devices .

Figure 4.1. L-Section Matching Networks [15]

4.1 Q Sections or Matching Stub

It is one quarter wavelength of coaxial or balanced transmission line of a specific impedance that is connected between a load and a source for the purpose of impedance matching. A quarter wave length transmission line can be used to make one impedance look like another according to relation

Figure 4.2. Typical T Matching network [15]

Figure 4.3. Typical Π Matching Network [15]

Two or more Q sections can be used in series to achieve the desired match with each section performing an impedance match between its input and output impedances. iplementing a transformer or coupled line-based matching network suffers from trade-off between impedance tuning range and coupling factor of coils. as a result, high Q conditions are required to obtain intented tuning range, yielding higher losses.

4.2 Baluns

Another commonly used impedance matching technique makes use of balun, a type of transformer used to match impedances. Most baluns are made of ferrite core, either a toroid or rod and windings of copper wire. Baluns have a very wide bandwidth and therefore are essentially independent of frequency. Baluns can be create d for producing impedance matching ratios of 4:1, 9:1 or 16:1. Some baluns have 1:1 impedance ratio as they are intented to convert balanced to unbalanced condition with no phase reversal. One such example is shown in following figure

4.3 Antenna Tuners

When baluns and matching sections cannot do the job, antenna tuners are used. An antenna tuner is a variable inductor, one or more variable capacitors or a combination of these components connected in various configurations. L T π networks are all widely used. The inductor and capacitor values are adjusted until the SWR indicates that the impedances match.

Figure 4.4. Toroid [13]

It is important to mention that using an antenna tuner at transmitter only tricks the transmitter into seeing a low SWR. In reality, the SR on line between tuner and antenna is still high. A typical configuration consists of a coil and and three capacitors to tune the antenna for optimal SWR as shown in fig 4.5. Capacitor C2 and C3 are ganged together and tuned simultaneously. This kind of scheme often provides huge bandwidth. A major development in antenna tuners is automatic tuners. This may be L, π or T networks self adjusting themselves to minimize SWR. Here multiple taps are along of inductor to change to desired value and banks of parallel connected capacitors that could be switched in or out. All inductor taps and capacitors are switched in or out by relay contacts. The relays in turn are operated by optimizing algorithms.

Figure 4.5. Antenna Tuner [13]

Figure 4.6. Typical antenna tuner for receiver [14]

Power meters measuring both incident and reflected power are available that provide DC voltage proportional to those voltages. These DC values are converted by ADC and are used by processors or controllers running above optimizing algorithms.

5. Methods proposed for adaptive impedance matching

Research into automatic impedance matching and antenna tuning has gone through some significant stages from traditional electro-mechanical tuning and discrete compenents to automated tuning with support of state of the art algorithms.

Figure 5.1. Typical Flow for Adaptive Impedance Matching [7]

Figure 5.2. Typical Adaptive Impedance Matching [24]

Convergence of tuning values is achieved by optimizing algorithms categorised as follows...

1] Real frequency technique 2] Mems based adaptive impedance match 3] Fuzzy controller based adaptive impedance match 4] Genetic algorithm based adaptive impedance match 5] Neural network based adaptive impedance match 6] LMS algorithm based adaptive impedance match

5.1 Real Frequency Technique

This is a practical real frequency technique for broadband impedance matching and is based on bilinear reflection behavior versus lumped/distributed element variables. An efficient grid search locates a likely-global solution so that a precise constrained gradient optimization can eliminate unnecesaary elements in networks. Broadband matching requires lossless two-port matching network (equalizer) to minimize loss in passband. This n/w is usually LC network without R part. RFT is a numerical optimization method that is applicable to a wide range of problems and utilizes frequency sampled load/source impedance data. Numerical optimization adjusts free parameters (variables) to minimize an objective function, e.g. least squares or random search. Variables are values of equalizer elements which may be mixed lumped and distributed. A well known RFT employs one or more approximation and optimization stages and ends with polynomial synthesis. The variables are polynomial co-efficients or s-plane root co-ordinates. A new algorithm also has been proposed that avoids linear /nonlinear/rational approximations and polynomial synthesis. An equalizer topology is not known in advance in this method unlike a topology is selected and optimized for equalizer values. Here a useful equalizer topology is selected and the grid approach to broadband impedance matching (GRABIM) automatically eliminates all unnecessary elements.

5.2 Mems based Adaptive Impedance Match

Due to increasing demand of self adjusting / cognitive systems in several domains, MEMS technology has started a kind of revolution by offering miniaturised electromechanic based solutions. These modules are so small that they can be integrated into integrated circuits. It has taken a start in offering solutions for adaptive impedance matching for modern communications. First article was in 2003 and next article was in 2009. As per the paper published in 2009 by authors of Swedish defence research agency, RF MEMS matching networks have been proved immensely helpful in realization of tunable LNAs , power amplifiers and filters at microwave and millimeter-wave frequencies.

Figure 5.3. Typical Mems Based Switching of Capacitor [3]

Proposed networks for impedance matching consists of traditional Π network or T network which contain capacitor banks (most often)or varactors. tuning these capacitor banks is possible by using electromechanic switches developed with help of MEMS Technology. MEMS based capacitors are fabricated in shunt with MIM capacitors on fused silica (quartz) thus forming pie or T networks acting as transformation networks. Capacitors designed thrugh MEMS are a low voltage control, wide tuning range and high quality factor Impedance matching and LNA centre frequency tuning range is controlled by MEMS capacitance ratio. LNA was based on GaAs mHEMT technology consisting of 2 active transistor stages . Proposed above methods proved excellent upto 10 Ghz and proved good upto 20 ghz. MEMs technology offers a lot of promises to communication in future.

5.3 Fuzzy controller based Adaptive Impedance Match

Briefing fuzzy logics : Fuzzy logic had offered solutions for many problems by its unique way of working. Fuzzy logic models the uncertainty of human thought and it offers a mathematical formalism, which attempts to emulate the scheme of human deduction. Fuzzy logic formalizes the treatment of vague knowledge and it approximates reasoning through inference rules, establishing the bases to generate practical solutions to problems where traditional methods, which may require precise mathematical models, could not be suitable. In this sense, fuzzy control represents a good alternative to be explored as a solution for impedance mismatch problem through adaptive mechanisms According to ieee article of 2009 : 978-7-4244-4480-9, formulation of complete mathematical model for impedance mismatch is very complex process since many parameters like process variations, length variations, temperature etc influence impedances. In article, authors proposed a two port passive network controlled by zero order takagi-sugeno-fuzzy controller to optimize reactance part of network. A typical adaptive matching using fuzzy controller an be shown as follows A typical fuzzy controller that process tuning values can be shown as follows

Figure 5.5. Typical Fuzzy Logic Based System [4]

Figure 5.6. Typical Fuzzy Controller [4]

5.4. Genetic Algorithm based Adaptive Impedance Match

Genetic algorithms are a type of optimization algorithm that are applicable to a much wider range of problems than optimizing algorithms. Considering the fact that an impedance presented to a device has a major effect on the gain, noise and output power of the device. Genetic algorithms are a type of optimization algorithm that are applicable to wide range of problems. Matching networks involved in bringing impedance transformation are typical pie/L/T networks. Here, genetic algorithm is used to optimize values of varactors based on value of reflection coefficient measured. Almost all authors who have worked on genetic algorithms have been say that methods available so far like real frequency techniques and various methods/algorithms including LMS prove good but fails to specify minimum and maximum component values. This is major issue at high frequencies where component values available are limited.

There have been many works published so far on genetic algorithm for impedance matching. Many of them have considered tapered lines [32-34]. Sun and lau[35-36] have considered optimizing component values of π network using genetic algothm. Plessis from university of Pretoria, south Africa[11] had used floating point algorithm rather than binary algorithm as it gives much precise values and better results.

5.5 LMS Algorithm Based Adaptive Matching

Least Mean Square (LMS) algorithm introduced by Widrow and Hoff in 1959 is an adaptive algorithm which uses a gradient-based method of steepest decent. (LMS) algorithm has been extensively used in many applications due to its

simplicity and robustness. LMS algorithm uses the estimates of the gradient vector from the available data. LMS incorporates an iterative procedure that makes successive corrections to the weight vector in the direction of the negative of the gradient vector which eventually leads to the minimum mean square error. Compared to other algorithms LMS algorithm is relatively simple; it does not require correlation function calculation nor does it require matrix inversions. In practical application of the LMS algorithm, a key parameter is the step size. As the step size becomes large /small, the convergence rate of the LMS algorithm will be rapid and the steady-state mean square error (MSE) will increase/decrease. Thus, the step size provides a trade off between the convergence rate and the steady-state MSE of the LMS algorithm. An intuitive way to improve the performance of the LMS algorithm is to make the step size variable rather than fixed, that is, choose large step size values during the initial convergence of the LMS algorithm, and use small step size values when the system is close to its steady state, which results invariable step size Least Mean square (VSSLMS) algorithms. By utilizing such an approach, both a fast convergence rate and a small steady-state MSE can be obtained.

6. Conclusion

Tunability, Reconfigurability, and Adaptability for RF and microwave circuits are highly desirable because they not only enhance the functionality and performance but also reduce the circuit size and cost. Adaptive Impedance Matching Techniques through various logics and algorithms are offering solutions to compensate for impedance mismatches. Enhanced and Optimised Algorithms are still being proposed for impedance matching proving unmatched solutions to practical impedance matching problems.