Description
1. The circuit shown in the figure below contains a nonlinear inductor and is driven
by a time-dependent current source. Suppose that the nonlinear inductor is
described by iL=I0sin(kϕL), where ϕL is the magnet flux of the inductor and I0 and
k are constants. Using ϕL and vC as state variables, find the state equations.
2. Use Matlab/Simulink to simulate the stable electronic oscillator in Example 8 in
Lecture 1. Choose two sets of initial conditions that are different from the ones on
pages 28-30 in this lecture, and produce the phase plane (or XY plane) plots and
plot output responses with the various initial conditions. In your simulation,
please choose A=1.5, V1=V2=1, L=1H, C=1F, and R=0.1Ω.
3. For the following system, find the equilibrium points and determine the type of
each isolated equilibrium point.
൜
ẋଵ = 2xଵ − xଵxଶ
ẋଶ = 2xଵ
ଶ − xଶ
4. By plotting trajectories starting at different initial conditions, draw the phase
portrait of the following LTI systems:
൜
𝑥̇ଵ = 𝑥ଶ
𝑥̇ଶ = −10𝑥ଵ − 10xଶ
Hint: Use Matlab command “initial”, and the hold function when plotting.
5. The phase portrait (or phase-plane plot) of the following system is shown below.
Mark the arrowheads and discuss the stability of each isolated equilibrium point.
൜
𝑥̇ଵ = 𝑥ଶ
𝑥̇ଶ = 𝑥ଵ − 2tanିଵ(xଵ + xଶ)
Please note the equilibria of the system above are (0, 0), (2.33, 0), and (-2.33, 0).
