1. CSE245: Computer-Aided Circuit Simulation and Verification
Lecture Note 2: State Equations
Spring 2008
Prof. Chung-Kuan Cheng

2. State Equations Motivation Formulation Analytical Solution
Frequency Domain Analysis
Concept of Moments

3. Motivation Why What Where Whole Circuit Analysis
Interconnect Dominance
Wires smaller  R increase
Separation smaller  C increase
What
Power Net, Clock, Interconnect Coupling, Parallel Processing
Where
Matrix Solvers, Integration For Dynamic System
RLC Reduction, Transmission Lines, S Parameters
Whole Chip Analysis
Thermal, Mechanical, Biological Analysis

4. Formulation General Equation (a.k.a. state equations)
Equation Formulation
Conservation Laws
KCL (Kirchhoff’s Current Law)
n-1 equations, n is number of nodes in the circuit
KVL (Kirchhoff’s Voltage Law)
m-(n-1) equations, m is number of branches in the circuit.
Branch Constitutive Equations
m equations

5. Formulation State Equations (Modified Nodal Analysis):
Desired variables
Capacitors: voltage variables
Inductors: current variables
Current controlled sources: control currents
Controlled voltage sources: currents of controlled voltage sources.
Freedom of the choices
Tree trunks: voltage variables
Branches: current variables

6. Conservation Laws KCL KVL n-1 independent cutsets
m-(n-1) independent loops

7. Branch Constitutive Laws
Each branch has a circuit element
Resistor
Capacitor
Forward Euler (FE) Approximation
Backward Euler (BE) Approximation
Trapezoidal (TR) Approximation
Inductor
Similar approximation (FE, BE or TR) can be used for inductor.
v=R(i)i
i=dq/dt=C(v)dv/dt

8. Branch Constitutive Laws
Inductors
v=L(i)di/dt
Mutual inductance
V12=M12,34di34/dt

9. Formulation - Cutset and Loop Analysis
Select tree trunks and links
find a cutset for each trunk
write a KCL for each cutset
find a loop for each link
write a KVL for each loop
cutset matrix
loop matrix

10. Formulation - Cutset and Loop Analysis
Or we can re-write the equations as:
In general, the cutset and loop matrices can be written as

11. Formulation – State Equations
From the cutset and loop matrices, we have
Combine above two equations, we have the state equation
In general, one should
Select capacitive branches as tree trunks
no capacitive loops
for each node, there is at least one capacitor (every node actually should have a shunt capacitor)
Select inductive branches as tree links
no inductive cutsets

12. Formulation – An Example
Output Equation
(suppose v3 is desired output)
State Equation

13. Responses in Time Domain
State Equation
The solution to the above differential equation is the time domain response
Where

14. Exponential of a Matrix
Calculation of eA is hard if A is large
Properties of eA
k! can be approximated by Stirling Approximation
That is, higher order terms of eA will approach 0 because k! is much larger than Ak for large k’s.

15. Responses in Frequency Domain: Laplace Transform
Definition:
Simple Transform Pairs
Laplace Transform Property - Derivatives

16. Responses in Frequency Domain
Time Domain State Equation
Laplace Transform to Frequency Domain
Re-write the first equation
Solve for X, we have the frequency domain solution

17. Serial Expansion of Matrix Inversion
For the case s0, assuming initial condition x0=0, we can express the state response function as
For the case s, assuming initial condition x0=0, we can express the state response function as

18. Concept of Moments
The moments are the coefficients of the Taylor’s expansion about s=0, or Maclaurin Expansion
Recall the definition of Laplace Transform
Re-Write as
Moments

19. Concept of Moments
Re-write Maclaurin Expansion of the state response function
Moments are

20. Moments Calculation: An Example

21. Moments Calculation: An Example
A voltage or current can be approximated by
For the state response function, we have

22. Moments Calculation: An Example (Cont’d)
(1) Set Vs(0)=1 (suppose voltage source is an impulse function)
(2) Short all inductors, open all capacitors, derive Vc(0), IL(0)
(3) Use Vc(i), IL(i) as sources, i.e. Ic(i+1)=CVc(i) and VL(i+1)=LIL(i), derive Vc(i+1), IL(i+1)
(4) i++, repeat (3)