LECTURE 4: Circuit Simulation: KCL
Fundamentals
Ohm's Law states the voltage across a resistor, R (or impedance, Z) is directly
proportional to the current passing through it (the resistance/impedance is the
proportionality constant)
Kirchhoff's Voltage Law (KVL): the algebraic sum of the voltages around any loop of
N elements is zero (like pressure drops through a closed pipe loop)
Kirchhoff's Current Law (KCL): the algebraic sum of the currents entering any node is
zero, i.e., sum of currents entering equals sum of currents leaving (like mass flow at a
junction in a pipe)
Nodal Analysis
Nodal analysis is generally best in the case of several voltage sources. In nodal analysis,
the variables (unknowns) are the "node voltages."
Nodal Analysis Procedure:
1. Label the N node voltages. The node voltages are defined positive with respect to
a common point (i.e., the reference node) in the circuit generally designated as the
ground (V = 0).
2. Apply KCL at each node in terms of node voltages.
a. Use KCL to write a current balance at N-1 of the N nodes of the circuit
using assumed current directions, as necessary. This will create N-1
linearly independent equations.
b. Take advantage of supernodes, which create constraint equations. For
circuits containing independent voltage sources, a supernode is generally
used when two nodes of interest are separated by a voltage source instead
of a resistor or current source. Since the current (i) is unknown through the
voltage source, this extra constraint equation is needed.
c. Compute the currents based on voltage differences between nodes. Each
resistive element in the circuit is connected between two nodes; the
current in this branch is obtained via Ohm's Law where Vm is the positive
side and current flows from node m to n (that is, I is m --> n).
3. Determine the unknown node voltages; that is, solve the N-1 simultaneous
equations for the unknowns, for example using Gaussian elimination or matrix
solution methods.
Nodal Analysis Example
1. Label the nodal voltages.
2. Apply KCL.
a. KCL at top node gives IS = IL + IC
b. Supernode constraint eq. of VL = VS
c.
Solving Equation: Numerical Method
For a given equation d/dt (X) = AX + BU[t]
Where X and U are vector quantities.
The numerical solution in discrete term can be given as
{ X(k+1) – X(k) } /Δt = AX(k) + BU[k].
If the initial condition X[0] is give each of the next are determined recursively.
Such systems are called Causal and the method used is called Forward Euler Method.
X
--[t] 
‘
The forgoing equation can be solved in frequency domain using Laplace Transform
on the time domain function.
The condition number of a matrix
For example, the condition number associated with the linear equation
Ax = b
gives a bound on how inaccurate the solution x will be after
approximate solution.
Matrix Condition Number
Note that this is before the effects of round-off error are taken into account;
conditioning is a property of the matrix, not the algorithm or floating point accuracy
of the computer used to solve the corresponding system.
It may be thought as the rate at which the solution, x, will change with respect to a
change in b.
Thus, if the condition number is large, even a small error in b may cause a large error
in x. On the other hand, if the condition number is small then the error in x will not
be much bigger than the error in b.
What do you mean by ill-conditioned and well-conditioned system of equations?
A system of equations is considered to be well-conditioned if a small change in the
coefficient matrix or a small change in the right hand side results in a small change in
the solution vector.
A system of equations is considered to be ill-conditioned if a small change in the
coefficient matrix or a small change in the right hand side results in a large change in
the solution vector.
The Δt value taken in the numerical calculation makes different impact depending on
the type of function it is used.
Relatively bigger step size prone to higher error in X2(t) than X1(t).