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EGR 272 – Solving Differential Equations using MATLAB Solving Differential Equations using MATLAB: Notes on using dsolve('eqn1','eqn2', ...) • Symbolic equations: • ‘eqn1’, ‘eqn2’ represent symbolic differential equations and initial conditions • Example: dsolve(‘Dy + 4y = 2’, ‘y(0) = 10’) • Independent variable: • The default independent variable is t, but can be changed by listing it. • Example: dsolve(‘Dy + 4y = 2’, ‘y(0) = 10’) % Dy means dy/dt • Example: dsolve(‘Dy + 4y = 2’, ‘y(0) = 10’, x) % Dy means dy/dx • Derivatives: • Dy – 1st derivative (note: avoid using D as a symbolic variable) • D2y – 2nd derivative, D3y = 3rd derivative, etc. • Initial conditions: • Examples: ‘y(0) = 2’, ‘Dy(0) = 4’, ‘D2y(0) = 6’, etc • If fewer than N initial conditions are provided for an Nth order DE, the solution will contain constants C1, C2, … 1 EGR 272 – Solving Differential Equations using MATLAB Example: Solve for y(t) in the following first-order DE if y(0) = 8 dy 10y(t) 20, t 0 dt MATLAB Solution: Answer : v(t) vn vf 6e-10t 2 V, t 0 2 EGR 272 – Solving Differential Equations using MATLAB Example: Solve for y(t) in the following second-order DE if y(0) = 2 and y’(0) = -4 d2v dv 7 12v(t) 10e-2t , t 0 2 dt dt Answer : v(t) vn vf - 6e-3t 3e-4t 5e-2t , t 0 MATLAB Solution: Note: This example uses a forcing function of 10e-2t (right-side of the equation). In EGR 270 we will only deal with DC sources, so the forcing functions will always be constants. 3 EGR 272 – Solving Differential Equations using MATLAB Graphing the Solution to a Differential Equation: Since the solution to a differential equation is a symbolic equation, it can easily be graphed using ezplot. Example: Graph the solution to the previous example: d2v dv 7 12v(t) 10e-2t , t 0 2 dt dt Answer : v(t) vn vf - 6e-2t 3e-2t 5e-2t , t 0 Discussion: Does this graph satisfy the initial and final values for v(t)? 4 EGR 272 – Solving Differential Equations using MATLAB 5 Class Example: A) Solve for v(t) in the following 1st - order DE if v(0) = 12 dv 10v(t) 30, t 0 dt B) Repeat this example using MATLAB Graph the solution using ezplot( ) from 0 to 5 Answer : v(t) 3 9e -10t , t 0 EGR 272 – Solving Differential Equations using MATLAB 6 Class Example: A) Solve for i(t) in the following 2nd -order DE if i(0) = 6, i’(0) = 38 d 2i di 6 9i(t) 18, t 0 dt 2 dt B) Repeat this example using MATLAB Graph the solution using ezplot( ) from 0 to 5 Answer : i(t) 2 50t 4e -3t , t 0 EGR 272 – Solving Differential Equations using MATLAB 7 Significance of overdamped, critically damped, and underdamped solutions A circuit with an overdamped response is called an overdamped circuit (similar for the other types of damping). What does this mean about the circuit? First, some definitions: Damping – the act of removing oscillations Example: A shock absorber might be adjusted so that it doesn’t oscillate, but smoothly returns a wheel to its initial position after an impact. Example: A switch is thrown in a circuit and an output voltage adjusts to a new level. • In an overdamped circuit, it would reach the new level smoothly and without oscillation (ringing). • In an underdamped circuit, it would oscillate as it approached the new level. • See the illustration on the following slide. EGR 272 – Solving Differential Equations using MATLAB 8 Example: Different types of damping in an RLC circuit For the circuit below we can easily see that v(0) = 0V and v() = 10V. So when the switch closes at t = 0, how does v(t) get to 10V? This is a 2nd-order circuit, so it must have a 2nd-order response. So it must be overdamped, underdamped, or critically damped. Sketch v(t) below. t=0 R 10 V + _ L + C v (t) _ v(t) t EGR 272 – Solving Differential Equations using MATLAB Example: The DE for v(t) in the circuit below if L = 100 H and C = 10 uF is: t=0 R 10 V + _ L + C v (t) _ d 2 v R dv 100v(t) 1000, t 0 dt 2 10 dt i(0) 0, i' (0) 0 A) What value of R results in a critically-damped circuit? R The characteristic equation is s 2 s 100 0 10 For critical damping it should be in the form (s 10) 2 s 2 20s 100 0, so R 20, or R 200 for critical damping 10 B) Use MATLAB so solve the DE and graph the response for: • R = 200 (critically damped) • R = 350 (overdamped) • R = 50 (underdamped) 9 EGR 272 – Solving Differential Equations using MATLAB Example: (continued) Solve and graph the DE for R = 350, 200, and 50 using MATLAB. 10