CIRCUITS and SYSTEMS – part I Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 4 Methods of analysis of complex circuits

The basic method - Kirchhoff’s equations In this method we describe the circuit by the set of equations following from Kirchhoff’s laws. At b branches there is b unknown currents and the same number of equations. We formulate (n-1) independent KCL equations (n – the number of nodes) and the rest (b-n+1) equations follows from KVL. The KVL equations should be written for arbitrary chosen (independent) meshes of the circuit (avoid choosing the mesh containing ideal current source).

Thevenin theorem Any linear circuit (from the point of view of the terminals AB) may be replaced by an equivalent circuit formed by series connection of the ideal voltage source U AB and the impedance Z AB. The voltage source is equal to the voltage across the terminals AB of the original circuit. The Thevenin impedance Z AB is the equivalent resistance from the terminals AB after elimination of all independent sources in the original circuit (voltage sources – short circuit, current sources – open circuit).

Thevenin theorem (cont.) The graphical interpretation of Thevenin equivalence

Analysis of circuit using Thevenin theorem 1.Eliminate the branch of the current under interest 2.Replace the original circuit by the Thevenin equivalent circuit (the circuit below) 3.Calculate the needed current from the equation

Example Using Thevenin theorem calculate the current I of the circuit below. Assume e(t)=14.1sin(ωt), R 0 =7.5Ω, R 1 =5Ω, R 2 =5Ω, X L =ωL=5Ω, X C =1/(ωC)=10 Ω, X C0 =1/(ωC 0 )=10 Ω.

Solution The parameters of the Thevenin equivalent circuit are calculated for the circuits presented below (after elimination of the branch R 0, C 0 ). The subcircuits for calculation a) impedance Z AB, b) voltage U AB

Thevenin equivalent parameters Equivalent impedance Z AB Equivalent voltage U AB

Equivalent Thevenin circuit Current I

Norton theorem Any linear circuit (from the point of view of the terminals AB) is equivalent to the parallel connection of the ideal current source I Z and the impedance Z AB. The current source is equal to the short-circuit current of the terminals AB. The Thevenin impedance Z AB is the equivalent resistance from the terminals AB after elimination of all independent sources in the circuit (voltage sources – short circuit, current sources – open circuit). It is the same impedance as for Thevenin theorem.

Norton theorem (cont.) The graphical interpretation of Norton equivalence

Equivalence of Thevenin and Norton circuits Both equivalent circuits should results in the same external current in the branch attached to AB. Hence the following equivalnce rules hold.

Nodal method Nodal method (known also as nodal potential method) allows to simplify the calculations of all currents in the circuit. One node is treated as the reference one (the ground node). The voltages of other independent nodes measured with respect to the ground one are treated as the nodal potentials. All branch currents are expressed through these potentials. Application of KCL to all (N) independent nodes leads to the set of N equations with respect to nodal potentials. Its solution enables to calculate all branch currents..

Nodal method (cont.) Any RLC circuit may be described by th enodal equation in the matrix form

Automatic creation of nodal description Elements of the main diagonal Y ii of the admittance matrix Y are equal to the sum of all admittances connected to ith node. The admittance Y ij is equal to the common admittance joining node ith with jth, taken with minus sign (all valid strictly for passive RLC circuit only). The elements of the current excitation vector I zr are formed by the sum of current sources attached to the proper node. The current entering the node is taken with plus sign and that leaving the node with minus sign. Only current sources are allowed. The voltage sources should be transformed to current ones by applying Thevenin- Norton eqiuvalence..

Solution of circuit 1) Form the nodal equation in the form 2) Solve for nodal potentials V 3) Calculate branch currents using KVL for each branch at tknown nodal potentials

Example of nodal analysis Circuit diagram under analysis

Nodal description Nodal matrix equation

Solution Vector of node potentials Branch currents

Remarks of nodal analysis Nodal potential method requires solution of N equations, where N is the number of independent nodes (always smaller than number of branches). For passive RLC networks nodal desription is formed automatically. When circuit contains controlled sources we apply 2 steps. 1.In the first step treat controlled sources like independent ones and form automatically the nodal type description 2.In the second step express all controlled sources through node voltages and move these terms to the left side of equations, forming the final nodal matrix equation.

Mesh method In mesh (loop) analysis we assume the independent meshes covering the whole circuit. Each mesh is associated with a mesh current I oi circulating in the mesh (usually of the same direction for all meshes ). Example of choosing the meshes and mesh currents in the circuit

Mesh matrix description Matrix mesh equation Mesh description results from application of KVL to all meshes of the circuit. The real current of the branch is the superposition of mesh currents adjacent to this particular branch.

Automatic creation of mesh description Elements Z ii on the main diagonal of Z are the sum of impedances existing in the ith loop. The off-diagonal element Z ij is equal to the impedance common to loop ith and jth, taken with minus sign (at assumption that all mesh currents have the same directions). The kth element of the excitation vector E is equal to the sum of voltage sources existing in kth mesh. If the direction of source is identical with mesh current it is taken with plus sign, in opposite case with minus sign.

Example of mesh analysis Circuit diagram

Mesh description Since the circuit contains 3 independent loops the mesh desription is composed of 3 equations and is of the form ZI o =E

Solution Mesh currents Branch currents

Remarks of mesh analysis Mesh method requires solution of N equations, where N is the number of independent loops (always smaller than number of branches). For passive RLC networks mesh desription is formed automatically. When circuit contains controlled sources we apply 2 steps. 1.In the first step treat controlled sources like independent and form automatically the mesh type desription 2.In the second step express all controlled sources through mesh currents and move these terms to the left side of equations, forming the final mesh matrix equation.

Superposition principle The time response of the linear circuit at many excitations is the sum of time responses to each source acting independently in turn, while the others are replaced in the circuit by their internal resistances (zero for voltage source and infinite for current source) Illustration of superposition theorem