EE 529 Circuits and Systems Analysis Mustafa Kemal Uyguroğlu Lecture 7

Eastern Mediterranean University Analysis of Circuit Containing OP-AMP Show that

Eastern Mediterranean University Analysis of Circuit Containing OP-AMP Equivalent circuit

Eastern Mediterranean University Analysis of Circuit Containing OP-AMP Circuit Graph

Eastern Mediterranean University Analysis of Circuit Containing OP-AMP Fundamental Cut-set Equations

Eastern Mediterranean University Analysis of Circuit Containing OP-AMP By using the terminal equations, the fundamental cut-set equations can be written as:

Eastern Mediterranean University Analysis of Circuit Containing OP-AMP Fundamental Circuit Equations

Eastern Mediterranean University Analysis of Circuit Containing OP-AMP Subst. of fundamental circuit equations into fundamental cut-set equations yields:

Eastern Mediterranean University Mathematical Models of Energy Storage Elements  Capacitor

Eastern Mediterranean University Mathematical Models of Energy Storage Elements  Inductor

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  RLC Circuit ●Branch Voltages Method a b o (L) (C) (R) (S)

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  RLC Circuit ●Branch Voltages Method ♦ Fundamental Cut-Set Equations a b o (L) (C) (R) (S) Terminal Equations

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  Subst. of terminal equations into fundamental cut-set equation yields:

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  RLC Circuit ●Branch Voltages Method ♦ Fundamental Circuit Equations a b o (L) (C) (R) (S)

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  Subst. of fundamental circuit equations into Eq.(1) yields: Integro-Differential Equation

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  Taking the derivate wrt t and dividing by C results in

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  RLC Circuit ●Chord Currents method a b o (L) (C) (R) (S)

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  RLC Circuit ●Chord Currents Method ♦ Fundamental Circuit Equations a b o (L) (C) (R) (S) Terminal Equations

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  Subst. of terminal equations into fundamental circuit equations yield:

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements ♦ Fundamental Cut-Set Equations a b o (L) (C) (R) (S)

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  Subst. of fundamental cut-set equation into Eqns. (1) and (2) yields:

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  Taking the derivate wrt t results in

Eastern Mediterranean University Analysis of Circuits Containing Energy Storage Elements  Subst. Eq. (5) into (6) gives: Taking the integral wrt t and dividing by RLC results in

Eastern Mediterranean University Solutions of n-th order DE  Consider an n-th order DE Let

Eastern Mediterranean University Solutions of n-th order DE  Therefore Eq.(1) can be written as or in matrix form or in more compact form Companion matrix

Eastern Mediterranean University Solutions of n-th order DE The right hand side of the equation may contain known u i (t) functions and their different order of derivatives.

Eastern Mediterranean University State Variables Method  So far, we explain, how a circuit containing energy storage elements are analysed using Branch Voltages method and Chord Currents Method. The circuit equations are Integro-differential equations. It means circuit equations contains integral and derivative terms together.  Now, another method called state variables method will be introduced. In this method, the circuit equations called circuit state equations do not contain integral terms. The unknown quantities in state equations are called state variables and as it will be seen the state variables are capacitors voltages and inductors currents

Eastern Mediterranean University State Variables Method  Selection of tree is an important step in order to obtain the state equations of a RLC circuit.  The folllowing steps are used in selection of the tree. ●The edges in the system graph corresponding to the voltage sources must be placed in the tree. ●The edges corresponding to as many capacitors as possible must next be placed in the tree. ●If the tree is not complete, the edges corresponding to the resistors must be chosen and as many resistors as possible must be included. ●If still the tree is not completed, then, the edges corresponding to the inductors will be chosen until the tree is completed ●All the edges corresponding to the current sources must be placed in the co-tree.

Eastern Mediterranean University State Variables Method  After the selection of proper tree, the state variables are branch capacitor voltages and chord inductor currents. The terminal equations of branch capacitors and chord inductors are: where C bC and L cL are non-singular and diagonal matrices.

Eastern Mediterranean University State Variables Method  In order to obtain state equations i bC and v cL must be written in terms of state variables and source functions or/and derivatives of the source functions.

Eastern Mediterranean University State Variables Method EXAMPLE  Consider the following RLC circuit. a b d(L) (C) (R3)(R3)(S) (R2)(R2) f In this circuit, there are one branch capacitor and one chord inductor.

Eastern Mediterranean University State Variables Method EXAMPLE  State variables are v c (t) and i L (t). The terminal equations for states variables are a b d(L) (C) (R3)(R3)(S) (R2)(R2) f

Eastern Mediterranean University State Variables Method EXAMPLE  Now, let’s assume that the branch (C) is a voltage source and chord (L) is a current source in the system graph. a b d(L) (C) (R3)(R3)(S) (R2)(R2) f Therefore, we have only one unknown branch voltage v 3, and only one chord current i 2.. In order to find v 3, branch voltages method is used.

Eastern Mediterranean University State Variables Method EXAMPLE In order to find i 2, chord currents method is used. a b d(L) (C) (R3)(R3)(S) (R2)(R2) f

Eastern Mediterranean University State Variables Method EXAMPLE Now, let’s look at the terminal equations of the states variables. a b d(L) (C) (R3)(R3)(S) (R2)(R2) f From the system graph, we can write that and

Eastern Mediterranean University State Variables Method EXAMPLE Therefore, the circuit state equations are