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. 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