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Appendix A Useful MATLABrfunctions The following are useful MATLABrfunctions for circuit analysis and synthesis. The reader is encouraged to consult the reference guide and the “Help” directory in any MATLABrwindow for a detailed description of each function.1 Ax=b: Linear equation Ax=b is solved by invoking x = Anb. See also linsolve. Bode plot: [mag, phs, w]=bode(n, d, w) places the magnitude in vector mag and phase in vector phs at each frequency in vector w of a transfer function whose numerator polynomial is n and denominator polynomial is d. Convolution: y=conv(p,q) obtains the product of two polynomials, represented by their vectors of coefficients p and q. Eigenvalues: [V,D]=eig(A) places the eigenvalues of A in a diagonal matrix D and their corresponding eigenvectors in V. Elliptic functions: [sn cn dn]=ellipj(u,m) yields sn(u;k), cn(u;k), and dn(u;k); and [K, E]=ellipke(m) yields K(k) and E(k), where m = k2. Factors: y=factor(f) obtains the factors of a symbolic polynomial f . Imaginary part: y=imag(z) returns the imaginary part of z. Linear equations: y=linsolve(A,b) obtains the symbolic or numerical solution of linear equation Ax = b, A and b being numeric or symbolic. Matrix exponential: H=expm(At), where A is a matrix of simple rational numbers and t declared symbolic, will obtain the impulse response matrix H(t) in analytic form. Numerator and denominator: [n,d]=numden(f) places the numerator of symbolic rational function f in n and the denominator in d. Ordinary differential equations: dsolve(...) obtains the symbolic solution of an or- dinary differential equation. Polynomial from its roots: p=poly(r) obtain the coefficients of polynomial p in descending order in a row vector whose roots are placed in a column vector r. 1 The Academia area of The MathWorks Web site provides many resources for professors and students. Check there for contributed course materials, recommended products by cur- riculum, tutorials on MATLABrsoftware, and information on MATLABrStudent Version: www.mathworks.com/products/academia/ 285 286 A Useful MATLABrfunctions Polynomial to symbolic: p=poly2sym(q,’s’) converts a vector of coefficients of a polynomial q to its symbolic representation in s. Random numbers: y = m + s ¤ randn generates a random number taken from a universe of numbers which are normally distributed with mean m and variance s 2 . Random numbers: y=a + (b-a)*rand generates a random number taken from a universe of numbers which are uniformly distributed over [a;b]. Real part: y=real(z) returns the real part of z. Residues: [r,p,k]=residue(b,a) obtains the partial fraction expansion of rational function b=a where a and b are expressed as vectors of coefficients of polynomi- als a and b. The residues are placed in vector r and the poles in vector p. Vector k contains the coefficients of any excess polynomial. Roots of a polynomial: r=roots(p) obtains a column vector of roots of polyno- mial p expressed as a row vector of coefficients in descending order. Step response: y=step(num,den,t) produces the step response of a transfer func- tion whose numerator polynomial is num and denominator polynomial is den, both expressed as vectors of coefficients, for a vector of time points in t. Simplify: y=simplify(f) obtains a simplified version of a symbolic function f . Substitution: y=subs(f, ’s’, w) replaces each occurrence of s by each member of vector w in the symbolic function f . This function is useful in computing the frequency response of a rational function f if w is a vector of frequency points. Symbolic to polynomial: q=sym2poly(p) converts a symbolic representation of a polynomial p to its representation as a vector q of coefficients. Transfer function: [num,den]=ss2tf(A,B,C,D,k), where A;B;C;D are the matri- ces of a state equation in normal form, with an input component Uk = 1 and all others being zero, will return a vector of transfer functions whose numerator polynomials are in the vector num and whose denominator polynomial, being the same for all, is in den. Transfer function to zeros and poles: [z,p,k]=tf2zp(num,den) places the zeros of a transfer function with numerator num and denominator den in vector z, the poles in vector p and the gain in k. The vector num contains the coefficients of the numerator polynomial and den those of the denominator polynomial. Zeros and poles to transfer function: [num,den]=zp2tf(z,p,k) forms the transfer function with numerator polynomial num and denominator polynomial den from a vector of zeros z, a vector of poles p and a multiplicative constant k. References 1. Abramowitz, M., (ed.), I.A.S.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, D.C. (1967) 2. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998) 3. Bashkow, T.R.: The A matrix, new network description. IRE Trans. on Circuit Theory CT-4, 117–120 (1957) 4. Belevitch, V.: Th´eorie des circuits de t´ele ´communication. Librairie Universitaire Louvain (1957) 5. Belevitch, V.: An alternative derivation of Brune’s cycle. IRE Trans. Circuit Theory CT-6, 389–390 (1959) 6. Belevitch, V.: On the Bott-Duffin synthesis of driving-point impedances. IRE Trans. Circuit Theory CT-6, 389–390 (1959) 7. Blostein, M.L.: Sensitivity analysis of parasitic effects in resistance-terminated LC two-ports. IEEE Trans. on Circuit Theory CT-14, 21–26 (1967) 8. Bode, H.W.: Network Analysis and Amplifier Design. D. Van Nostrand, Princeton (1945) 9. Bott, R., Duffin, R.J.: Impedance synthesis without use of transformers. J. Appl. Phys. 20, 816 (1949) 10. Branin Jr., F.H.: The inverse of the incidence matrix of a tree and the formulation of the algebraic- first-order differential equations of an RLC network. IEEE Trans. on Circuit Theory CT-10, 543–544 (1963) 11. Brown, D.P.: Derivative-explicit differential equations for RLC graphs. J. Franklin Institute 275, 503–514 (1963) 12. Brune, O.: Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency. J. Math. Phys. 10, 191–236 (1931) 13. Bryant, P.R.: The explicit form of Bashkow A matrix. IRE Trans. on Circuit Theory CT-9, 303–306 (1962) 14. Carlin, H.J.: A new approach to gain-bandwidth problems. IEEE Trans. on Circuits and Sys- tems CAS-23(4), 170–175 (1977) 15. Carlin, H.J., Civalleri, P.P.: Wideband Circuit Design. CRC Press, Boca Raton (1998) 16. Cauer, W.: Die Verwirklichung von Wechselstromwidersta ¨nden vorgeschriebener Frequenzabha ¨ngigkeit. Arch. Elektrotech. 17(4), 355–388 (1926) 17. Chen, W.K.: Theory and Design of Broadband Matching Networks. Pergamom Press, Oxford (1976) 18. Daniels, R.W.: Approximation Methods for Electronic Filter Design. McGraw-Hill, New York (1974) 19. Darlington, S.: Synthesis of reactance 4-poles which produce prescribed insertion loss char- acteristics. J. Math. Phys. 18, 257–353 (1939) 287 288 References 20. Darlington, S.: Realization of a constant phase-difference. Bell System Technical Journal 29, 94–104 (1950) 21. Director, S.W.: LU factorization in network sensitivity computations. IEEE Trans. on Circuit Theory CT-18, 184–185 (1971) 22. Ellinger, F.: Radio Frequency Integrated Circuits and Technologies. Springer, Berlin; New York (2007) 23. Fano, R.M.: Theoretical limitations on the broad-band matching of arbitrary impedances, Part I. J. Franklin Institute 249(1), 57–83 (1950) 24. Fano, R.M.: Theoretical limitations on the broad-band matching of arbitrary impedances, Part II. J. Franklin Institute 249(2), 139–154 (1950) 25. Filanovsky, I.M.: Sensitivity and selectivity. In: W.K. Chen (ed.) The Circuits and Filters Handbook, chap. 68, pp. 2205–2236. CRC Press, Boca Raton (1995) 26. Foster, R.M.: A reactance theorem. Bell System Journal 3, 259–267 (1924) 27. Fujisawa, T.: Realizability theorem for mid-series or mid-shunt low-pass ladders without mu- tual induction. IRE Trans. PGCT CT-2(4), 320–325 (1955) 28. Guillemin, E.A.: The Mathematics of Circuit Analysis. John Wiley and Sons, New York (1953) 29. Guillemin, E.A.: Synthesis of Passive Networks. John Wiley and Sons, New York (1957) 30. Hazony, D.: An alternate approach to the Bott-Duffin cycle. IRE Trans. Circuit Theory CT-8, 363 (1961) 31. Hazony, D.: Two extensions of the Darlington synthesis procedure. IRE Trans. Circuit Theory CT-8, 284–288 (1961) 32. Hazony, D., Schott, F.W.: A cascade representation of the Bott-Duffin synthesis. IRE Trans. Circuit Theory CT-5, 144 (1958) 33. Huang, Q., Sansen, W.: Design techniques for switched capacitor broadband phase splitting networks. IEEE Transactions on Circuit Theory CT-34, 1096–1102 (1987) 34. Hurwitz, A.: Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit nega- tiven reelen Theilen besitzt. Math. Ann. 46, 273–284 (1895) 35. II, L.W.C.: Digital and Analog Communication Systems. Prentice Hall, Upper Saddle River, NJ (1997) 36. Jeffrey, A.: Handbook of Mathematical Formulas and Integrals. Academic Press (1995) 37. Kim, W.H.: A new method of driving-point function synthesis. Technical report 1, University of Illinois Engineering Experimental Station (1956) 38. Kishi, G., Kida, T.: Energy theory of sensitivity in LCR networks. IEEE Transactions on Circuit Theory CT-14(4), 380–387 (1967) 39. Lathi, B.P.: Modern Digital and Analog Communication Systems, 3rd edn. Oxford University Press, New York (1998) 40. Luck, D.G.C.: Properties of some wide-band phase splitting networks.
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