1. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Electrical CircuitsI This lecture discusses the mathematical modeling of simple electrical linear circuits. When modeling a circuit, one ends up with a set of implicitly formulated algebraic and differential equations (DAEs), which in the process of horizontal and vertical sorting are converted to a set of explicitly formulated algebraic and differential equations. By eliminating the algebraic variables, it is possible to convert these DAEs to a state-space representation. September 20, 2012

2. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Table of Contents Components and their models The circuit topology and its equations An example Horizontal sorting Vertical sorting State-space representation Transformation to state-space form September 20, 2012

3. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Linear Circuit Components Resistors Capacitors Inductors R i v a v b u C i v a v b u L i v a v b u u = v a – v b u = R·i u = v a – v b i = C· du dt u = v a – v b u = L· di dt September 20, 2012

4. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Linear Circuit Components II Voltage sources Current sources Ground U 0 = v b – v a U 0 = f(t) I 0 I v a v b u 0 u = v b – v a I 0 = f(t) V 0 V 0 V 0 - V 0 = 0 U 0 i v a v b U 0 |  September 20, 2012

5. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Circuit Topology Nodes Meshes v a = v b = v c i a + i b + i c = 0 v a v b i a i b i c v c v a v b v c u ab u bc u ca u ab + u bc + u ca = 0 September 20, 2012

6. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier An Example I September 20, 2012

7. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Systems of Equations I The component and topology equations contain a certain degree of redundancy. For example, it is possible to eliminate all potential variables (v i ) without problems. The current node equation for the ground node is redundant and is not used. The mesh equations are only used if the potential variables are being eliminated. If this is not the case, they are redundant. September 20, 2012

8. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Systems of Equations II If the potential variables are eliminated, every circuit component defines two variables: the current (i) through the element and die Voltage (u) across the element. Consequently, we need two equations to compute values for these two variables. One of the equations is the constituent equation of the element itself, the other comes from the topology. September 20, 2012

9. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier An Example II Component equations: U 0 = f(t)i C = C· du C /dt u 1 = R 1 · i 1 u L = L· di L /dt u 2 = R 2 · i 2 Node equations: i 0 = i 1 + i L i 1 = i 2 + i C Mesh equations: U 0 = u 1 + u C u L = u 1 + u 2 u C = u 2 The circuit contains 5 components  We require 10 equations in 10 unknowns September 20, 2012

10. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Horizontal Sorting I The time t may be assumed as known. The state variables (variables that appear in differentiated form) may be assumed as known. U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2 U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2  September 20, 2012

11. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Horizontal Sorting II Equations that contain only one unknown must be solved for it. The solved variables are now known. U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2  U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2 September 20, 2012

12. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Horizontal Sorting III Variables that show up in only one equation must be solved for using that equation. U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2  U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2 September 20, 2012

13. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Horizontal Sorting IV All rules may be used recursively. U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2  U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2 September 20, 2012

14. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2  U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2 U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2  The algorithm is applied, until every equation defines exactly one variable that is being solved for. September 20, 2012

15. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Horizontal Sorting V The horizontal sorting can now be performed using symbolic formula manipulation techniques. U 0 = f(t) u 1 = R 1 · i 1 u 2 = R 2 · i 2 i C = C· du C /dt u L = L· di L /dt i 0 = i 1 + i L i 1 = i 2 + i C U 0 = u 1 + u C u C = u 2 u L = u 1 + u 2  U 0 = f(t) i 1 = u 1 /R 1 i 2 = u 2 /R 2 du C /dt = i C /C di L /dt = u L /L i 0 = i 1 + i L i C = i 1 - i 2 u 1 = U 0 - u C u 2 = u C u L = u 1 + u 2 September 20, 2012

16. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Vertical Sorting By now, the equations have become assignment statements. They can be sorted vertically, such that no variable is being used before it has been defined. U 0 = f(t) i 1 = u 1 /R 1 i 2 = u 2 /R 2 du C /dt = i C /C di L /dt = u L /L i 0 = i 1 + i L i C = i 1 - i 2 u 1 = U 0 - u C u 2 = u C u L = u 1 + u 2  U 0 = f(t) u 1 = U 0 - u C i 1 = u 1 /R 1 i 0 = i 1 + i L u 2 = u C i 2 = u 2 /R 2 i C = i 1 - i 2 u L = u 1 + u 2 du C /dt = i C /C di L /dt = u L /L September 20, 2012

17. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Rules for Systems of Equations III Alternatively, it is possible to work with both potentials and voltages. In that case, additional equations for the node potentials must be found. These are the potential equations of the components and the potential equations of the topology. Those had been ignored before. The mesh equations are in this case redundant and can be ignored. September 20, 2012

18. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier An Example III The circuit contains 5 components and 3 nodes.  We require 13 equations in 13 unknowns. Component equations: U 0 = f(t)U 0 = v 1 – v 0 u 1 = R 1 · i 1 u 1 = v 1 – v 2 u 2 = R 2 · i 2 u 2 = v 2 – v 0 i C = C· du C /dt u C = v 2 – v 0 u L = L· di L /dt u L = v 1 – v 0 v 0 = 0 Node equations: i 0 = i 1 + i L i 1 = i 2 + i C v1v1 v2v2 v0v0 September 20, 2012

19. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Sorting The sorting algorithms are applied just like before. The sorting algorithm has already been reduced to a purely mathematical (informational) structure without any remaining knowledge of electrical circuit theory. Therefore, the overall modeling task can be reduced to two sub-problems: 1.Mapping of the physical topology to a system of implicitly formulated DAEs. 2.Conversion of the DAE system into an executable program structure. September 20, 2012

20. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier State-space Representation Linear systems: Non-linear systems: dxdx dt = A · x + B · u y = C · x + D · u x(t 0 ) = x 0 ; dxdx dt = f(x,u,t) y = g(x,u,t) ; x(t 0 ) = x 0 x  n u  m y  p x = State vector u = Input vector y = Output vector n = Number of state variables m = Number of inputs p = Number of outputs A  n  n B  n   m C  p  n D  p  m September 20, 2012

21. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Conversion to State-space Form I U 0 = f(t) u 1 = U 0 - u C i 1 = u 1 /R 1 i 0 = i 1 + i L u 2 = u C i 2 = u 2 /R 2 i C = i 1 - i 2 u L = u 1 + u 2 du C /dt = i C /C di L /dt = u L /L du C /dt= i C /C = (i 1 - i 2 ) /C = i 1 /C - i 2 /C = u 1 /(R 1 · C) – u 2 /(R 2 · C) = (U 0 - u C ) /(R 1 · C) – u C /(R 2 · C) di L /dt = u L /L = (u 1 + u 2 ) /L = u 1 /L + u 2 /L = (U 0 - u C ) /L + u C /L = U 0 /L  For each equation defining a state derivative, we substitute the variables on the right-hand side by the equations defining them, until the state derivatives depend only on state variables and inputs. September 20, 2012

22. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier Conversion to State-space Form II x 1 = u C x 2 = i L u = U 0 y = u C We let:  x 1 = -  R 1 · C  R 2 · C   [ ] x1x1 R 1 · C    u  x 2 =  1 L  u y = x 1 September 20, 2012

23. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier An Example IV September 20, 2012

24. Start of Presentation Mathematical Modeling of Physical Systems © Prof. Dr. François E. Cellier References Cellier, F.E. (1991), Continuous System Modeling, Springer-Verlag, New York, Chapter 3.Continuous System ModelingChapter 3 Cellier, F.E. (2001), Matlab code of circuit example.Matlab code of circuit example September 20, 2012