Systems in eqilibrim Graph models and Kirchoff s laws nna-karin ornberg Mathematical Models, nalysis and Simlation Fall semester, 2011 system in eqilibrim [Material from Strang, sections 21 and 22] Consider a system of springs and masses in eqilibrim [NOES] o obtain or system of eqations, we have applied three eqations: 1) he forces shold be in eqilibrim 2) Hooke s law for springs 3) Relation between spring length and mass positions x i C 1 F b he system is on the form:,where 0 x f the colmns of are linearly independent, and C is a diagonal matrix with positive entries on the diagonal his yields Cx f + Cb Stiffness matrix K C is symmetric positive definite Strang works with displacements i instead of positions x i nthiscase, replacing x by in the system, we will have b 0 [NOES]
Minimization We have K f, with K C From earlier: he soltion to K f is the that minimizes the qadratic form P() 1/2 K f - Stretching increases the internal energy, 1/2e Ce 1/2 K (e is the elongation of the spring) - he masses loose potential energy by f (force times displacement, work done by external forces) his is NO a conserved system: external forces (in or example gravitational forces) are active Hence, the configration of this system at eqilibrim is the configration that minimizes the potential energy of the system ncidence matrix is the so called incidence matrix [NOES - EXMPLES] n or example, the strctre was simply a line Can have a graph instead Will develop these graph models frther and apply it to electric circits (Strang, Section 23) Nmbers i can represent the heights of the nodes, the pressre at the nodes, or the voltages at the nodes, dependent on the applications Common langage: potentials : the potential difference, ie the difference across an edge he nllspace of contains the vector that solve 0 f we do not set any potential i, the nllspace is a line he colmns of is a line, it has dimension 1 he colmns of are linearly dependent and is singlar n or example with masses, we fixed one location For electrical circits, we say that we grond one node his will remove one colmn of, and the remaining colmns will be linearly independent hen is invertible
he graph Laplacian matrix he matrix trns ot to have a very specific form: - On the diagonal: ( ) jj degree nmber of edges meeting at node j - Off the diagonal: ( 1 if nodes j andk share an edge ) jk 0 if no edge goes between nodes j and k nd for C: - On the diagonal changes to ( C) jj sm of all spring constants/condctivities/ in edges meeting at node j - Off the diagonal: the 1 changes to the negative of the spring constant/condctivity/ in the branch between node j and k Kirchoff s laws Consider one node with brach crrents w 1,,w l entering this node - Kirchoff s crrent law (KCL): w 1 + + w l 0 Sm of all branch crrent entering a node eqals zero Consider one loop with brach voltages (or voltage drops) e 1,,e n - Kirchoff s voltage law (KVL): e 1 + + e n 0 Sm of all branch voltages in a loop eqals zero n a practical network, many nodes and loops Need a systematic way to derive all these eqations for a given network
Eqations Kirchoff s crrent law (KCL) yields w 0 Kirchoff s voltage law is atomatically satisified with Ohm s law yields e w Ce, where C is a diagonal matrix with entries c i > 0, where c i is the condctance in branch i (have c i 1/R i,wherer i is the resistance vale e i R i w i ) m edges, n nodes Grond one node is m (n 1) Node potentials, branchcrrentsw, branch voltages (or voltage drops) e, 1 n 1, w w 1 w m, e e 1 e m dding crrent and voltage sorces dding voltage sorces and crrent sorces to the system, yields the modifications: - w f (crrent sorce in f), - e b, (voltage sorce in b) - Ohm s law, w Ce remains the same s a large system, this reads: C 1 0 w b f where C is a diagonal matrix with positive entries on the diagonal (the condctances) is the incidence matrix his yields C Cb f Stiffness matrix K C is symmetric positive definite if the colmns of are linearly independent - grond a node
Modified Nodal nalysis (MN) t is common in engineering context to work systematically with incidence matrices also for voltage sorce and crrent sorce branches We will denote - R, the incidence matrix of the resistive branches - V the incidence matrix for all voltage sorce branches - the incidence matrix for all crrent sorce branches - ssme N R resistive branches, N V voltage sorce branches, N crrent sorce branches With m nodes, R is N R m, V is N V m, is N m - Let i v : branch crrents trogh voltage sorces (N V 1) : vector of vales of all crrent sorces (N 1) E: vector of vales of all voltage sorces (N V 1) he notation we will se differs slightly from common MN notation (mainly st MN R R etc) Choosing or notation to stay closer to the notation of Strang Modified Nodal nalysis (MN), contd We obtain the system R C R V V 0 i V E - R, V and incidence matrices for resistive, voltage sorce and crrent sorce branches, respectively - C diagonal matrix with condctances (inverse of resistances) - : vector of vales of all crrent sorces, E: vector of vales of all voltage sorces - : vector of node potentials, i v : branch crrents trogh voltage sorces NOES: - How to get to this system - Example earlier done with Strang s notation in this notation
RLC circits and C analysis R-resistor,L-indctor,C-capacitor Voltage drops V, crrent (Strang s notation) Characteristic eqations: V R, C dv dt, V L d dt (if capacitance vale C and self-indctance vale L are constant) ssme sinsoidal forcing term of fixed freqency ω ypical voltage V (t) ˆV cos(ωt) ( ˆVe jωt ) Crrent (t) (Îe jωt ) Since d dt ejωt jωe jωt, we get the simple algebraic law ˆV ZÎ, with Z 1/R for resistors, Z jωl for indctors and Z 1/(jωC) for capacitors jω analysis - Strang s approach Same system as before: C 1 0 w b f Bt now, the diagonal matrix C contains the complex impedances Z Consider an example where branch 1 has a resistor with resistance R 1, branch 2 a indctor with self-indctance L 2,branch3acapacitorwith capacitance C 3 and finally, branch 4 a resistor with resistance R 4 hen: C 1/R 1 1/(jωL 2 ) jωc 3 1/R 4
jω analysis by MN Before, we had incidence matrices R, V and,forresistivebranches, voltage sorce branches and crrent sorce branches, respectively Now, introdce also the incidence matrix C for capacitive branches, and L for indctive branches he admittances Y are the inverses of the impedances Z ntrodce the diagonal matrices Y R, Y C and Y L that contain the admittances of the resistive, capacitive and indctive branches, respectively (Before, Y R C) he system can now be written as R Y R R + C Y C C + L Y L L V V 0 i V E (Derivation in NOES on board)