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System Poles and Zeros: The transfer function provides a basis for determining important system response characteristics without solving the complete differential equation. As defined, the transfer function is a rational function in the complex variable s = σ + jω, that is It is often convenient to factor the polynomials in the numerator and denominator, and to write the transfer function in terms of those factors Where the numerator and denominator polynomials, N(s) and D(s), have real coeffi
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  System Poles and Zeros: The transfer function provides a basis for determining importantsystem response characteristics without solving the complete differential equation. As defined,the transfer function is a rational function in the complex variable  s = σ +  jω , that isIt is often convenient to factor the polynomials in the numerator and denominator, and to writethe transfer function in terms of those factorshere the numerator and denominator polynomials,  N  !  s and  D !  s , have real coefficients defined by the system#s differential equation and  K = bm/an . As written in $q. !% the  zi #s are the roots of the equation  &!s =' and are defined to be the system (eros, and the pi#s are the roots of the equation )!s = ',and are defined to be the system  poles . In $q. !% the factors in the numerator and denominator are written so that when  s =  zi the numerator  N  !  s = ' and the transfer function vanishes, that isand similarly when  s =  pi the denominator polynomial  D !  s = ' and the value of the transfer function becomes unbounded,  All of the coefficients of polynomials  N  !  s and  D !  s are real, therefore the poles and (eros must be either purely real, or appear in complex con*ugate pairs. In general for the poles, either  pi = σi ,or else  pi, pi + = σi+jωi . The existence of a single complex pole without a correspondingcon*ugate pole would generate complex coefficients in the polynomial  D !  s . imilarly, thesystem (eros are either real or appear in complex con*ugate pairs. The Pole-Zero Plot A system is characteri(ed by its poles and (eros in the sense that they allow reconstruction of theinput-output differential equation. In general, the poles and (eros of a transfer function may becomplex, and the system dynamics may be represented graphically by plotting their locations onthe complex  s plane, whose axes represent the real and imaginary parts of the complex variable  s . uch plots are /nown as  pole-zero plots . It is usual to mar/ a (ero location by a circle ! ◦ and a pole location a cross ! × . The location of the poles and (eros provide qualitative insights into theresponse characteristics of a system. System Stability: The stability of a linear system may be determined directly from its transfer function. An n th order linear system is asymptotically stable only if all of the components in thehomogeneous response from a finite set of initial conditions decay to (ero as time increases, or   here the  pi are the system poles. In a stable system all components of the homogeneousresponse must decay to (ero as time increases. If any pole has a positive real part there is acomponent in the output that increases without bound, causing the system to be unstable. Inorder for a linear system to be stable, all of its poles must have negative real parts that are theymust all lie within the lefthalf of the  s plane. An 0unstable pole, lying in the right half of the ‖  s  plane, generates a component in the system homogeneous response that increases without boundfrom any finite initial conditions. A system having one or more poles lying on the imaginary axisof the  s plane has nondecaying oscillatory components in its homogeneous response, and isdefined to be marginally stable.  TWO PORT NETWORK: A pair of terminals through which a current may enter or leave a networ/ is /nown as a  port  .Twoterminal devices or elements !such as resistors, capacitors, and inductors result in oneportnetwor/s. 1ost of the circuits we have dealt with so far are twoterminal or oneport circuits,represented in 2igure %!a . e have considered the voltage across or current through a single pair of terminals3such as the two terminals of a resistor, a capacitor, or an inductor. e have alsostudied fourterminal or twoport circuits involving op amps, transistors, and transformers, asshown in 2igure %!b . In general, a networ/ may have n  ports. A port is an access to the networ/ and consists of a pair of terminals4 the current entering one terminal leaves through the other terminal so that the net current entering the port equals (ero.Thus, a twoport networ/ has two terminal pairs acting as access points. As shown in 2igure%!b , the current entering one terminal of a pair leaves the other terminal in the pair. Threeterminal devices such as transistors can be configured into twoport networ/s. 5ur study of two port networ/s is for at least two reasons. 2irst, such networ/s are useful in communications,control systems, power systems, and electronics. 2or example, they are used in electronics tomodel transistors and to facilitate cascaded design. econd, /nowing the parameters of a two port networ/ enables us to treat it as a 0blac/ box when embedded within a larger networ/. To ‖ characteri(e a twoport networ/ requires that we relate the terminal quantities V , V %, I , and I %in 2igure % !b , out of which two are independent. The various terms that relates these voltagesand currents are called  parameters . 5ur goal in this chapter is to derive six sets of these parameters. e will show the relationship between these parameters and how twoport networ/s
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