ECE 552 Numerical Circuit Analysis Chapter Two EQUATION FORMULATION Copyright © I. Hajj 2015 All rights reserved

Equation Formulation There are many ways of writing circuit equations. We will try to answer the following questions. How are different forms of circuit equations related? Why is nodal formulation popular? Why does it work? Why, sometimes, it does not work? What is the 'modified' nodal formulation? Why do we choose a circuit node as a reference when writing the nodal equations?

Circuit equations consist of: I- Element characteristics (explained in Chap. 1) II- Topological equations II. Topological Equations Kirchoff's Current Law: KCL Kirchoff's Voltage Law: KVL KCL and KVL depend only on circuit interconnection, and are independent of element characteristics.

Example 2.1 Undirected graph representation Circuit with 2-terminal elements 1

Directed graph representation (Associated Reference Directions) Netlist Representation El. Namen1n2Value I131* R213* R312* R414* R534* R624*

Netlist Representation El. Namen1n2n3Value I113* R212* R323* T4123 **** Circuit with 3-terminal elementDirected graph representation Example 2.2 R2

Netlisl El. Namenln2n3n4Value V112* C213* R334* P41234**** Example 2.3 Circuit with 2-port elementDirected graph representation C2

A a => Incidence Matrix KCL equations of Example 3.1 node 1: -i 1 + i 2 + i 3 + i 4 = 0 node 2: -i 3 + i 6 = 0 node 3: i 1 - i 2 + i 5 = 0 node 4: - i 4 – i 5 - i 6 = 0 Kirchoff's Current Law (KCL) For any lumped network, and at any time, the algebraic sum of all branch currents leaving (or entering) a node is zero.

Some properties of A a : 1.Consists of n rows and b columns; n = # of nodes, b = # of branches 2.Each column contains exactly a "+1" and a "-1“ 3.Can be constructed by inspection from the netlist 4.Has rank (n - 1); the n KCL equations are dependent => they add up to zero 5.Can delete any one row (i.e., one KCL equation) to obtain (n - 1) independent KCL equations KCL Equations: Ai = 0 A is the "reduced" incidence matrix of rank (n - 1)

Loop: A subgraph L of a graph G is called a loop if: (1)The subgraph L is connected. (2) Precisely two branches of L are incident with each nod e. Kirchoff's Voltage Law (KVL) For any lumped network, for any of its loops, and at any time, the algebraic sum of the branch voltages around the loop is zero.

Problem: How to construct the maximum number of independent loop equations for a given circuit. For Planar graphs (i.e., graphs that can be drawn in a plane such that no two branches cross at points, other than at nodes) the number of "windows" give the maximum number of independent loop equations.

However, not all graphs are planar. In addition, windows cannot be easily recognized from a netlist. We will thus use the concept of a tree of a graph.

Properties of a tree of a connected graph Is a subset of the graph. Contains all the nodes. Contains no loops. Contains (n - 1) branches called tree branches. The remaining (b - n + 1) branches are called links or chords. For a given graph, there are many different trees (countable in number). Each link forms a unique loop with tree branches only (why?). The (b - n + 1) loops generated by the links (by taking one link at a time and forming a loop with tree branches only) form a maximum set of independent loop equations. (Why?)

Example KVL wrt tree

KCL Revisited (Cutset Equations) Cutsets A set of branches of a connected graph is called a cutset if: The removal of all the branches of the set causes the remaining graph to have two separate parts. The removal of all but any one of the branches of the set leaves the remaining graph connected.

Example of cutsets

General Form of KCL For any lumped network, for any of its cutsets, and at any time, the algebraic sum of all the branch currents traversing the cutset branches is zero Each tree branch forms a unique cutset with links only (Why?) The (n - 1) KCL equations generated by taking one tree branch at a time and forming a cutset with links only produce a maximum set of independent KCL equations (Why?)

Remarks The characteristic equation of a short-circuit is v = 0; i.e. a voltage source with value 0 V The characteristic equation of an open-circuit is i = 0; i.e. a current source with value 0 A Short-circuits and open circuits can be considered as circuit elements

Example

Theorem : QB T = 0 and BQ T = 0 When both Q and B are constructed with the same tree and same branch orientations and ordering. Proof: Qi b = 0, i b = B T i l (KCL)  Q B T i l = 0 for arbitrary i l Therefore, QB T =0 Bv b = 0, v b = Q T v t (KVL)  BQ T v t = 0 for arbitrary v t Therefore, BQ T =0

Also: AB T = 0 and BA T = 0 using the same branch orientations and ordering. Proof: Bv b =0, v b =A T v n (KVL) BA T v n = 0 for arbitrary v n Therefore BA T = 0

Tellegen’s Theorem Provided v b and i b follow the associated reference directions, v b satisfies KVL and i b satisfies KCL, independent of element charactersitics. Proof: v b = Q T v t, i b = B T i l  v b T i b = v t T QB T i l = 0 since QB T =0.  In a lumped circuit, energy is conserved at all times.

An Algorithm for Constructing a Tree 1.Construct the (reduced) incidence matrix A from the netlist. 2.Apply elementary operations on A: –Interchange any rows or columns –Multiply any row’ by ‘-’ –Add or subtract any rows These operations will keep the transformed matrix entries +1, -1, or 0.  Until the A matrix is transformed into the form: The first (n — 1) columns correspond to a tree. The B matrix can then be generated from the Q matrix: Note: The transformation of A into Q is not unique.

Example

Another way:

Circuit Equations I.Element characteristics (Linear): Total number of equations: b-equations in 2b unknowns For simplicity, consider sinusoidal steady-state analysis:

II.Topological Constraints (n — 1) equations in b-unknowns Total number of equations: b-equations in 2b unknowns In Matrix Form (Tableau Equations): Using Q and B wrt a tree:

Better Ordering Block Elimination: OR: link currents and tree voltages uniquely determine all other variables. (

Tableau Equations Using the Incidence Matrix (2b + n - 1) equations in (2b + n - 1) unknowns. There are more equations and more variables and more equations, but there is no need to find a tree

Nodal Analysis Elements allowed are: conductors (admittances), independent current sources, and voltage- controlled current sources. Elements not allowed are: independent voltage sources, short-circuits, current- controlled voltage sources, current-controlled current sources, and voltage-controlled voltage- sources. Put (2) in (3) Put (3’) in (1)

Nodal Equations from the Tableau Equations Using block elimination, one gets [AGA T ]v n = -As or [Y]v n = -As

Tableau Equations Example Q is generated from the A matrix by elementary transformation. The A matrix is constructed from the netlist.

Nodal Equations Some Properties of Y: If G is strictly diagonal (no controlled sources) and positive, then: 1- is nonsingular 2- diagonally dominant 3- All diagonal elements are positive, and all off-diagonal elements are negative or zero.

Formulation of the Y matrix “by Inspection” from the Netlist Order the nodes so that the reference node is ordered last. For each element whose characteristics fit the nodal equation formulation (i = G v + s) use a “stamp” as follows. (1) Resistor

(2) Capacitor i C = jωCV c (3) Inductor

(4) Independent Current Source (5) Voltage-Controlled Current Source

(6) Three-Terminal Element (7) Two-Port Element

Example Node 4 is chosen as a reference node.

The “Modified” Nodal Equations When not all element characteristics fit nodal analysis (e.g. voltage sources) Assume i 1 = G 1 v 1 + H 1 i 2 +s 1 (1) v 2 = H 2 v 1 + Z 2 i 2 + s 2 (2) Partition the Incidence Matrix A accordingly KCL[A 1 A 2 ] = 0 ORA 1 i 1 + A 2 i 2 = 0 (3)

Substitute (4) into (1) and (2): Put (l') in KCL eqn. (3) Final equations are (3') and (2')

Derivation of Modified Nodal Equations from the Tableau Equations:

Formulation of Modified Nodal Equations using the “Stamp” Approach Partition circuit elements into two sets: El. 1 and El. 2 according to their characteristic equations, as shown above. El. 2 includes voltage sources, short-circuits, inductors, and any element whose current is a declared as a controlling variable Declare v n and i 2 as the circuit variables

(1) Resistor * If current in R is not declared a variable Stamps

(2) Capacitor (3) Inductor

(4) Independent Current Source (5) Independent Voltage Source E

(6) Short-Circuit (7) Voltage-Controlled Current Source

(8) Current-Controlled Current Source i α = α i x (9) Voltage-Controlled Voltage Source (vn 1 – vn 2 - α (vn 3 - vn 4 ) = 0

(10) Current – Controlled Voltage Source v nl -v n2 - αi x = 0 (11) Two-Port (Current-Controlled) Example V 1 = jωL 1 I 1 +jωMI 2 V 2 =jωMI 1 +jωLI 2

(12) Three-Terminal Element (Voltage Controlled) i 1 =a 11 v ] + a 12 v 2 + s 1 i 2 = a 21 v 1 + a 22 v 2 + s 2 Another way (12') i l = a 11 v 1 + ai 2 v 2 + s 1 i 2 = a 21 v 1 + a 22 v 2 + s 2 Increases circuit equation variables unnecessarily.

(13) Two-Port (Hybrid Representation) i l = a ll v 1 + a 12 i 2 + s 1 v 2 = a 2l v 1 + a 22 i 2 + s 2 vn 3 -vn 4 -a 21 (vn l -vn 2 )-a 22 i 2 = s 2 Another way (13') i l = a ll v 1 + a 12 i 2 + s 1 v 2 = a 2l v 1 + a 22 i 2 + s 2

Example

MNA Equations in Sinusoidal Steady- State (G + jωC)x = y jω  d/dt

Extended Nodal Analysis (ENA ) i 1 = G 1 v 1 + H 1 i 2 + N 1 z + s 1 M 2 v 2 = H 2 v 1 + Z 2 i 2 + N 2 z + s 2 z includes variables other than currents and voltages, such as charge, flux, temperature or other parameters, which on turn are functions of voltages and currents.

Extended Nodal Analysis (ENA) I : -A 1 T v 1 0 I : -A 2 T v 2 0 -G 1 I : -H 1 -N 1 i 1 s : = A 1 : 0 A 2 0 v n 0 -H 2 M 2 : 0 -Z 2 -N 2 i 2 s 2 z

Extended Nodal Analysis (ENA) I : -A 1 T v 1 0 I : -A 2 T v 2 0 I : -G 1 A 1 T -H 1 -N 1 i 1 s : = : A 1 G 1 A 1 T (A 1 H 1 +A 2 ) A 1 N 1 v n - A 1 s 1 : (-H 2 A 1 T +M 2 A 2 T ) -Z 2 -N 2 i 2 s 2 z

Extended Nodal Analysis (ENA) A 1 G 1 A 1 T (A 1 H 1 +A 2 ) A 1 N 1 v n - A 1 s 1 (-H 2 A 1 T +M 2 A 2 T ) -Z 2 -N 2 i 2 s 2 z