Introduction:



Matrix Method for Describing Differential Equations:

A more systematic way to find the differential equations is to express them directly in terms the independent voltages and currents in the system. Consider the circuit below as an example:

[pic]

A general approach:

1. Identify the independent voltages and/or currents in the circuit. These are values that you need initial conditions for in order to solve for the unique complete solution (i.e. capacitor voltages and inductor currents).

2. Find an equation for each independent value that relates its first derivative to zero-order derivatives of the other independent values and sources.

3. Set up a system of first order differential equations in a matrix form.

4. Express the particular output value in terms of a linear combination of the independent values. Use matrix notation.

Example:

For the circuit below find the differential equations and output equation in matrix form, where the output is v0:

[pic]

Show:

[pic] [pic]

Example:

For the circuit below find the differential equations and output equation in matrix form, where the output is v0:

[pic]

Show:

[pic] [pic]

Solutions to Differential Equations in Matrix Form:

Once the differential equations for the circuit are in matrix form, a solution can be systematically determined.

Example: Find the solution for system with the following differential equations given is = u(t) , vC(0+) = 1, and iL(0+) = 0.

[pic]; [pic];show[pic]

The strategy is to find the complete solution for both iL and vC , then substitute into the output equation to get v0. However, since only iL is multiplied by a nonzero number in the output equation, all that needs to be found is iL.

1) Find all required initial conditions for iL. Substitute into differential equation at t = 0+ and solve for [pic]

2) Determine the form of the natural solution from the Eigenvalues of the differential equation matrix (natural solution has same form for both iLn and vCn ).

3) Find forced solution by substituting guess in for iLf and vCf and solve equations.

4) Combine natural and forced solution and apply initial conditions.

5) Substitute into output equation.

Notation for a general system is proposed: [pic]

[pic]

If the system has a single input, then the input matrix B reduces to a vector denoted by a bold small case letter b.

Determining Initial conditions:

Given [pic] find [pic]by substituting - [pic]

Determining Natural solution:

Natural solutions for both x1 and x2 have the same form determined from the Eigenvalues of matrix A. The roots of the characteristic equation obtained from:

[pic]

Determining forced solution:

This will vary depending on the nature of the source, but the solution for both x1 and x2 can be determined by substituting a guess into the matrix equation and solving for the undetermined coefficients in the guess. It can be shown for general

DC sources, the forced response is:

[pic]

where vs [pic] is a vector of DC sources in the circuit

For sinusoidal sources a phasor substitution can be performed to limit the number of resulting equations and the magnitude and phase of the force response can be determined with complex arithmetic.

If source has form [pic] (consider only a single source, if several sinusoids exist at different frequencies, you must use superposition), then the magnitudes and phases of the forced solution are determined by:

[pic]

Example:

Derive the previous forced response relationships by substituting a general guess into the matrix equation.

Example:

Find the complete solution for [pic] for the system described by the differential equations below, where [pic] A, and [pic]:

[pic]; [pic]

Show[pic]

Find the complete solution for [pic] for the same system when [pic] A, and

[pic].

Show[pic]

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