ÖZET ANAHTAR KELİMELER : Anahtarlı Kapasite Devreleri, Anahtarlı Kapasite Filtre Devreleri, Anahtarlı Kapasite Filtre Devrelerinin Gerçeklenmesi, Anahtarlı Kapasite Filtre Devrelerinin SPICE Programı ile Analizi. Bu çalışmada önce Anahtarlı Kapasite (AK) devrelerinde sıkça kullanılmakta olan matematiksel dönüşümler ele alınmıştır. Daha sonra ise AK devreleri tanıtılarak AK devrelerinde kullanılan anahtarların çalışması, anahtarlama frekansı, anahtarlamada kullanılan darbelerin özellikleri, düğüm denklemlerinin elde edilmesi, veri örneklemek" gerilim dalga şekilleri ve AK devrelerinde kullanılan transfer fonksiyonu bağıntıları konulan geniş bir şekilde ele alınmıştır. Bu bölümde ayrıca AK yapılarından olan AK entegral alıcı, AK türev alıcı, AK tersleyici ve AK birim gecikme elemanı devreleri incelenmiştir. Üçüncü bölümde ise UR ve FIR AK filtre devrelerinin gerçeklenmesinde kullanılan Aktif-RC devresi kullanılması ve polinom dönüşümleri yöntemleri geniş bir şekilde incelenmiştir. AK filtrelerin aktif-RC devresi kullanılarak gerçeklenmesinde s-domeni transfer fonksiyonu kullanılmaktadır. Bu metod ile istenilen derecede filtre gerçeklenebilmekte ve bu gerçeklemede sadace AK entegral alıcı devreler kullanılmaktadır. Polinom dönüşümleri yöntemiyle AK filtrelerin gerçeklenmesinde ise z-domeni transfer fonksiyonu kullanılmaktadır. Bu metodla yapılan AK filtrelerde entegral alıcı ve türev alıcı AK devrelerinin her ikisi de kullanılabilmekte ve istenilen derecede Anahtarlı Kapasite filtre devresi gerçeklenebilmektedir.Bu bölümde filtre dizaynı için gereken teorik bilgi yer almaktadır. Dördüncü bölümde ise sayısal örnekler verilerek aktif-RC devresi kullanılması ve polinom dönüşümleri yönterrderinin her ikisi için AK filtre devreleri gerçeklenmiştir. Bu gerçeklemelerde gerekli hesaplamalar yapılmış ve devre elemanlarının değerleri hesaplanmıştır. Bu bölümde her iki metod kullanılarak HR ve FIR AK filtre devreleri gerçeklenmiştir. Beşinci bölümde ise AK filtre devrelerinin SPICE programı ile analizi için gerekli altyapı verilmiştir. Birkaç örnek filtre için gerekli hesaplamalar yapılarak SPICE programı giriş dosyası yazılmış ve program çalıştırılarak filtrenin frekans cevabı elde edilmiştir. AK devrelerinin özelliği nedeniyle bu devrelerde dönüşüm yapılmadan SPICE programında ac analizleri elde edilemez. Bu tip bir program ile yalnızca belirli bir frekansta verilen giriş sinyaline karşılık çıkış elde edilebilmekte, ve oldukça uzun bir analiz süresi de karşımıza çıkmaktadır. Bu bölümde bu türden bir analiz yapılmış ve çıkış değerleri elde edilmiştir. AK filtre devreleri uygun bir teknikle analog devreye dönüştürülmek suretiyle elde edilen devre için SPICE programı yazılırsa devrenin ac analizi yapılabilmekte ve frekans cevabı elde edilebilmektedir. Bu bölümde bu dönüşümün temelleri verilmiş ve dönüştürülen devre için SPICE programı giriş dosyası yazılarak frekans cevabı eğrileri elde edilmiştir. vııı
REALIZATION OF SWITCHED CAPACITOR FILTERS AND ANALYSING BY SPICE PROGRAM KEYWORDS: Switched Capacitor Circuits, Switched Capacitor Filters, Realization of Switched Capacitor Filters, Analysis of Switched Capacitor Filters by SPICE Program. SC technique has been widely applied to the design of analog sampled-data filters. In Ihe conventional design method of SC filters, the number of operational amplifiers is proportional to the order of the realized transfer functions. Thus the realization of high-order filters requires large chip area and power dissipation. SC filters are commonly used in voice and audio frequency communication circuits. The applicable frequency range of an SC design is particularly determined by the relation between the number of clock phases, the number of sampling phases, and the maximum available settling time for the charge transfer that is provided by the design technique. SC circuits can operate on minimally two clock phases. At least one phase is required for the charge transfer. Design techniques using more than two clock phases are not suitable for the realization of high-frequency SC filters, since the available settling time for the charge transfer is inversely proportional to the number of clock phases for a fixed clock frequency. For this reason and because of the need for stray insensitivity, the realization of high frequency SC filters should be based on the use of stray-insensitive bi-phase SC integrators. Depending on the switch operation, an inverting and a non inverting integrator can be realized. Since only one switch samples the input voltage, all signals in SC circuits built with these bi-phase integrators will have a uniform periodicity. Monolithic switched-capacitor circuits have been widely applied to analog signal processing with a good accuracy and less chip area. Most SC circuits are based on the SC integrator, although it has certain limits in realizing analog functions. Recently, SC differentiators have been developed and various applications have been explored. With both SC differentiators and integrators, therefore, the application field of SC circuits can be extended and their design versatility can be enhanced. Using the stray-insensitive integrators, it is possible to construct simple cascadable filter sections. Using the approximation z = esT, H(z) (biquad) becomes a biquadratic transfer function H(s) in s domain. In constructing the SC biquads, we IXwill make use of active-RC biquads developed to realize H(s). Firstly, the block diagram of a continuous-time system containing two cascaded integrators and coupling branches is constructed from H(s). Than an equivalent active-RC circuit is found. Each resistor in the active-RC section is replaced by an equivalent SC branch containing a capacitor and four switches. The value of the switched capacitor Cj replacing a resistor Rj is given by Cj = T /Rj. Using an op-amp with a unit-valued feedback capacitor to realize each integrator with a voltage/current transfer function (-1/ s), and RC admittances to realize the coupling branches, the active-RC filter section results. Next, each resistor is replaced by an equivalent circuit similar to the input branch of the integrator. The second method proposed by Davis and Smith relies on the synthetic division. By using this method, SC circuits can be constructed efficiently and directly from z- domain specifications. This method is a good design skill for SC circuits because the design procedure is clear and easy. This method is also applicable for SC differentiators. Two canonical structures are proposed for the realization of UR transfer functions using SC differentiators and the synthetic division technique. In the proposed realization method, the differentiator type of the element (r1 -1) is used instead of the (z*1) element. The resultant SC DR filters have shown the superiority over the conventional structures using (zr1) or integrator type in component sensitivity. They can also retain the advantages of SC differentiators, such as simple structure, stray-insensitive, low sensitivity to offset voltage and power supply variations, and good noise performance at low frequency. The technique is applied to synthesize a fourth-order elliptic filter using only forward difference integrators (FDI), or, alternatively, only backward difference integrators (BDI). A specific state- variable topology, the so-called observer canonical form, was chosen; the technique, however, is more generally applicable. The second section, includes a brief instruction on transformation methods and active-SC circuits as building-blocks. Switched-Capacitor filters are sampled-data circuits, with analog signal representation. Hence, their analysis requires, in general, the mathematical tools of both analog signals (Laplace and Fourier transformations) and those of sampled signals (z-transformation). Furthermore, the relations between these two groups of transformations must be correctly formulated and used. For these reasons, this section gives a summary of the basic definitions of analog, digital, and sampled-analog systems. Data sampling waveforms and transfer function equations are examined. Then, active-SC integrator and active-SC differentiator circuits are depicted and the transfer functions of these circuits are given. And also two types (BDI and FDI) of active-SC integrator are described. In addition, a basic delay element is also given. The third section deals with design principles, transfer functions, block diagrams and actual circuits of SC integrator and SC differentiator filters. Two different HR/FIR filter design techniques are discussed. These are cascaded SC sections and polynomial transformation (synthetic division) methods. Also synthetic division, which is used for transforming the transfer function H(z) or H(7rl) to H(z:1-1) is. shown with an example. In the fourth and fifth section we describe and demonstrate techniques for the analysis and design of active switched capacitor filters. Because of their sampled-datacharacter, switched capacitor circuits are most conveniently analyzed and designed in the z-transform domain, like digital circuits. However, switched capacitor circuits are analog circuits. Like active-RC circuits, there are many SC topologies that can be used to realize a given z-domain transfer function. There are several sampled-data wave forms which can be modeled as special cases of the wave form. One can immediately invoke the z-transform to mathematically describe these wave forms. Since switched capacitor circuits can be characterized in terms of charge-transfer operation, discrete-time voltages and discrete-time charge variations or transfer are used as port variables. For single capacitor switched capacitor blocks z-transformed nodal charge equations lead directly to simple z-domain equivalent circuits. The z- transformed voltage transfer functions plays an equivalents important role in specifying and designing active-SC filters. The two port can be represented by an equivalent four-port. In general, a 2x2 transfer matrix is required to fully characterize the input-output relations for this four-port network. The initial step in the synthesis of an SC network is to obtain an appropriate z- domain transfer function. Since filters are typically specified by frequency-domain requirements, it is convenient to have a mathematical expression that allows us to transfer functions. In lieu of computing a z-domain transfer function and synthesizing the SC network in the z-domain, analog active RC circuits can be transformed into z-domain circuits using the frequency transformation. This type of synthesis enables us to transform low-sensitivity active-RC filter design into active- SC realizations of compatible quality. In switched capacitor circuits, switches are controlled by a two-phase, non overlapping clock of frequency fc = (2T)-1. Note that VS2 is used to denote the even clock phase, which instantaneously closes the S2(e) switch on the even 2nT times. Similarly, VS1 denotes the odd clock phase, which instantaneously closes the Sl(o) switch on the odd (2n+l)T times. The switches are assumed to have a 50% duty cycle with equal on and off time periods. In practice, the clock rate is typically chosen no higher than is required to achieve the desired degree of anti-aliasing protection with a second order continuous filter of sufficiently high cutoff frequency to render its main pass band variation acceptably small. The purpose of this thesis is to introduce switched capacitor filter realization in detail and to find realization possibilities for computer aided simulation of such circuits. Today as electronic circuits advance, predesigning of the circuits and defining the working conditions of the circuits are as important as the physical implementation. That is why the computer aided design became popular. In computer aided design, defining the circuit elements and circuit conditions properly is very important. The final chapter gives examples of the simulation of switched capacitor filters using the SPICE program. Graphical results are presented using appropriate SPICE models. XI
