the state vector for the circuit shown below at t = 0+.
voltage vg(t) changes
instantly from −vo to
vo at t = 0.
Ans:[i1(0+), v1(0+), v2(0+)] = [0, −vo, −vo]
state variables are always the inductor currents and capacitor voltages (which
are also the variables we use to calculate stored energy). Thus, our state
variables are iL1, vC1, and vC2. We denote these as i1, v1, and v2.
their values cannot change instantly, the state variables have the same values
at time t = 0− as they do at time t = 0+.
the circuit has had the same DC current source for an infinitely long time at t = 0−, (from time −∞ to 0−), the circuit will have reached equilibrium and time
derivatives of state variables will be zero. In other words, currents and
voltages are no longer changing.
we have that vL1, iC1, and iC2 are all zero based on the basic
means the inductors look like wires and the capacitors look like open circuits.
We get the equivalent circuit shown below at time t = 0−.
the inductor is in series with an open circuit, the inductor current will be
zero. Similarly, i2 = 0, and it follows that i3 = 0.
Thus, there is no current in any of the resistors, and no voltage drop across
any of the resistors. Consequently, vg(t) appears across C1 and across C2.
At time t = 0, vg(t) = −vo.
they can't change instantly, these are the same values as the initial
conditions at t = 0+.