Example 14.9: Circuit Estimation

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The MODEL Procedure

Example 14.9: Circuit Estimation

Consider the nonlinear circuit shown in Figure 14.81.

Figure 14.81: Nonlinear Resistor Capacitor Circuit

The theory of electric circuits is governed by Kirchhoff's laws: the sum of the currents flowing to a node is zero, and the net voltage drop around a closed loop is zero. In addition to Kirchhoff's laws, there are relationships between the current I through each element and the voltage drop V across the elements. For the circuit in Figure 14.81, the relationships are

C[(dV)/dt] = I

for the capacitor and

V = (R₁ + R₂ (1-exp(-V))) I

for the nonlinear resistor. The following differential equation describes the current at node 2 as a function of time and voltage for this circuit:

label dvdt

C[(dV₂)/dt] - [(V₁ -V₂)/(R₁+R₂(1-exp(-V)))] = 0

This equation can be written in the form

[(dV₂)/dt] = [(V₁ -V₂)/((R₁+R₂(1-exp(-V))) C)]

Consider the following data.

      data circ;
         input v2 v1 time@@;
         datalines;
      -0.00007 0.0 0.0000000001 0.00912 0.5 0.0000000002
       0.03091 1.0 0.0000000003 0.06419 1.5 0.0000000004
       0.11019 2.0 0.0000000005 0.16398 2.5 0.0000000006
       0.23048 3.0 0.0000000007 0.30529 3.5 0.0000000008
       0.39394 4.0 0.0000000009 0.49121 4.5 0.0000000010
       0.59476 5.0 0.0000000011 0.70285 5.0 0.0000000012
       0.81315 5.0 0.0000000013 0.90929 5.0 0.0000000014
       1.01412 5.0 0.0000000015 1.11386 5.0 0.0000000016
       1.21106 5.0 0.0000000017 1.30237 5.0 0.0000000018
       1.40461 5.0 0.0000000019 1.48624 5.0 0.0000000020
       1.57894 5.0 0.0000000021 1.66471 5.0 0.0000000022
      ;

You can estimate the parameters in the previous equation by using the following SAS statements:

      proc model data=circ mintimestep=1.0e-23;
         parm R2 2000  R1 4000 C 5.0e-13;
         dert.v2 = (v1-v2)/((r1 + r2*(1-exp( -(v1-v2)))) * C);
         fit v2;
      run;

The results of the estimation are shown in Output 14.9.1.

Output 14.9.1: Circuit Estimation

The MODEL Procedure

Nonlinear OLS Parameter Estimates
Parameter	Estimate	Approx Std Err	t Value	Approx Pr > \|t\|
R2	3002.465	1556.5	1.93	0.0688
R1	4984.848	1504.9	3.31	0.0037
C	5E-13	1.01E-22	4.941E9	<.0001

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