Approximating Perturbed Systems
- The Circuit and Test Frequency
- Nominal Values of Branch Currents and Node Voltages
- Perturbed Values of Branch Currents and Node Voltages
- Exact Value of deltaIV
- Approximating deltaIV From a Recursion Relationship
- Performing an Iteration
- Mapping the Component Space to the Node Voltage Space
- Plotting the Node Voltage Space
- Plotting the Component Value Space
- Summary and Conclusions
A ladder circuit is used in this example showing how the changes in branch currents and node voltages can be approximated.
The circuit and tableau objects are created and the test frequency defined below.
ckt=circuit('test circuits\r2rLadder.cir'); T=tableau(ckt); f=1000;
Nominal branch currents and node voltages (IVn) can be calculated from the tableau matrix. The nominal values will be used to compare to the perturbed values, first by exact calculations and second by an approximate method.
Tmat_n=tableau_matrix(T,f); IVn=Tmat_n\T.S; disp(full(IVn))
0.0020 0.0010 0.0010 0.0010 10.0000 5.0000
The exact solution for the perturbed values is obtained by changing the randomly selected values in varyKi and varyKv. For this example, the values are changed to the maximum tolerance value of the resistors. This process is begun by obtaining a random value for varyKi and varyKv.
Having obtained a random value for varyKi and varyKv, tolerance and component values are obtained via the deltaComp_vector. The deltaComp variable is a structure containing the negative and positive fractional limits and the value of the components.
The random selection is removed and the maximum tolerance value substituted in varyKi and varyKv. These substitutions are completed below.
[row,~]=find(T.deltaKi); varyKi(logical(T.deltaKi))=1+deltaComp.pTol(row); [row,~]=find(T.deltaKv); varyKv(logical(T.deltaKv))=1+deltaComp.pTol(row);
The perturbed matrix is generated and the branch currents and node voltages computed. This operation finds the exact solution for perturbed branch currents and node voltages.
Tmat=tableau_matrix(T,f,varyKi,varyKv); IV=Tmat\T.S; disp(full(IV))
0.0020 0.0010 0.0010 0.0010 10.5000 5.2500
The exact value of deltaIV is obtained by subtracting the nonimal value from the perturbed value.
0 0 0 0.0000 0.5000 0.2500
At first glance, the value of deltaIVexact may not appear correct as the changes in branch currents are zero and only the node voltages change. The first row in the deltaIVexact vector is zero because the value of the constant current source is not changed. The values of the second, third, and fourth rows are also zero, indicating that branch currents have not changed. This appears to be anomalous. In the nominal condition R1 = R2 + R3. Since the resistor tolerances are all 5%, the relationship R1 = R2 + R3 still holds at maximum tolerance. When the current is constant, the relative current split between R1 and R2 + R3 is the same. The change in current is therefore zero in all three resistors. The result that appears incorrect is actually correct.
A circuit with its components at their nominal values can be represented in tableau format as
Tn*IVn = S
where Tn is the tableau matrix, IVn is the vector of branch currents and node voltages, and S is the vector of independent sources. The variation of branch currents and node voltages can be explored by letting the components vary from their nominal value. (Here we will assume that the independent sources remain constant since variation would simply be a linear response to the stimulus.) In this situation, the perturbed circuit represented in tableau format is
When the first equation is subtracted from the second equation, the result is
This equation can be simplified to
Since the tableau matrix has an inverse, then
The tableau system of equations has a property not shared by all methods of formulating circuit equations; a component value appears in one and only one row of the tableau format at a single location within the row. Each location in a row of deltaT has a number representing the change-in-value for that component. For each row, the change in component value can be extracted from deltaT and replaced by the appropriate value of (IVn+delatIV) because there is only one matrix element in a row. The equation becomes
The variable deltaT_IV is the matrix where the change in component value has been interchanged with (IVn+delatIV). The variable deltaComp is the vector of changes in component values.
In this form, the matrix
directly maps the change-in-component-value space to the change-in-current-and-voltage space.
Note that deltaT_IV forms a recursion relation that can be iterated to find deltaIV. To begin the iteration, the values which do not change are pre-calculated. As part of the assumptions, deltaIV is assumed to be zero initially.
invTmat_n=inv(Tmat_n); deltaIV=0; deltaComp=deltaComp_vector(T); deltaPosTol=deltaComp.pTol.*deltaComp.compVal;
To begin the iteration IV is just IVn, since deltaIV was assumed to be zero initially.
IV=IVn+deltaIV; deltaT_IV=deltaTableau_matrix(T,f,IV); deltaIV=-invTmat_n*deltaT_IV*deltaPosTol;
After one iteration, the approximate and exact values of deltaIV are:
0 0 -0.0000 0 0.0000 0 0.0000 0.0000 0.5000 0.5000 0.2500 0.2500
The difference between the approximate and the exact values are:
ans = 1.0e-015 * 0 -0.0000 0.0000 -0.0002 -0.0555 -0.9437
Typically, the recursion relation converges quite quickly as can be seen above.
To begin the mapping, a logical variable is defined whose columns are all logical combinations of three things.
x=logical([1 1 1 1 0 0 0 0;1 1 0 0 1 1 0 0 ;1 0 1 0 1 0 1 0]); disp(x)
1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0
The logical variable "x" is now used to create all possible combinations of tolerance extremes.
index=find(deltaComp.pTol); tolExtremes=x.*repmat(deltaComp.pTol(index),1,8)-~x.*repmat(deltaComp.nTol(index),1,8); disp(tolExtremes)
0.0500 0.0500 0.0500 0.0500 -0.0500 -0.0500 -0.0500 -0.0500 0.0500 0.0500 -0.0500 -0.0500 0.0500 0.0500 -0.0500 -0.0500 0.0500 -0.0500 0.0500 -0.0500 0.0500 -0.0500 0.0500 -0.0500
Having created the extremes in tolerance (fractional) values, the extremes in change-in-component values is computed.
compExtremeValues=repmat(deltaComp.compVal,1,8); compExtremeValues(index,:)=tolExtremes.*compExtremeValues(index,:); disp(compExtremeValues)
0 0 0 0 0 0 0 0 500 500 500 500 -500 -500 -500 -500 250 250 -250 -250 250 250 -250 -250 250 -250 250 -250 250 -250 250 -250 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
The values of deltaIV for each extreme in component values is created. The recursion relation is iterated twice and the difference between the current iteration and the last iteration tested to determine if the difference is less than 1e-15. When the test shows that the difference is less than 1e-15, iteration is terminated and the value deltaIV copied to the matrix of extreme points.
[row,col]=size(compExtremeValues); extremePoints=zeros(row,col); for I=1:length(x) % Initialize values deltaIV=zeros(row,1); deltaIVlist=zeros(row,4); % Make room for up to four iterations % First iteration count=1; IV=IVn+deltaIV; deltaT_IV=deltaTableau_matrix(T,f,IV); deltaIV=-invTmat_n*deltaT_IV*compExtremeValues(:,I); deltaIVlist(:,count)=deltaIV; % Second iteration count=count+1; IV=IVn+deltaIV; deltaT_IV=deltaTableau_matrix(T,f,IV); deltaIV=-invTmat_n*deltaT_IV*compExtremeValues(:,I); deltaIVlist(:,count)=deltaIV; % Subsequent iterations (must have two iterations to compare first) while all((deltaIVlist(:,count)-deltaIVlist(:,count-1)>1e-15)) count=count+1; IV=IVn+deltaIV; deltaT_IV=deltaTableau_matrix(T,f,IV); deltaIV=-invTmat_n*deltaT_IV*compExtremeValues(:,I); deltaIVlist(:,count)=deltaIV; end extremePoints(:,I)=deltaIVlist(:,count); end
Since the stimulus is a constant current, the node voltages are a more more interesting space. The space formed by the change in node voltages is:
The node voltage space, a two dimensional space, is plotted. The number of edges in the space matches the number of faces in the component space.
fill(voltageExtremePoints(pointOrder,1),voltageExtremePoints(pointOrder,2),'g') xlabel('Change in Voltage at N1'); ylabel('Change in Voltage at N2'); title('Node Voltage Space');
The component space is plotted next. It is a three dimensional space since there are three components. The component space is a rectangular solid.
xyz=compExtremeValues(index,:)'; pointOrder=convhulln(xyz); trisurf(pointOrder,xyz(:,1),xyz(:,2),xyz(:,3),'FaceColor','green') xlabel('Change in R1 (Ohms)') ylabel('Change in R2 (Ohms)') zlabel('Change in R3 (Ohms)') title('Component Value Space')
The recursion relation provides a means of mapping the component space into a branch current and node voltage space. This example demonstrates that the recursion relation converges quite quickly. (None of the extreme points took more that two iterations to converge to an acceptably small number.)
Each point within the green area of the component space represents an instance of a circuit where all of the components are within the stated tolerance. Each of the points in the green area of the node voltage space also represents an instance of a circuit where all of the components are within tolerance. In a practical application, verifying that all the components are in tolerance will usually require removing them from the circuit and measuring each one. Removing components is usually not practical and replacing them afterward opens up the possibility of a loading error or part damage. In contrast, measuring the node voltages and determining if they fall within the node voltage space is much less expensive and without the downside risk of a loading error or part damage.