Problem-Solving Strategy for solving the Second-Order Circuit
STEP 1. Write the differential equation that describes the circuit.
STEP 2. Derive the characteristic equation, which can be written in the form $s^2 + 2ζω_0s + ω^2_0 = 0$, where $ζ$ is the damping ratio and $ω_0$ is the undamped natural frequency.
STEP 3. The two roots of the characteristic equation will determine the type of response. If the roots are real and unequal ($ζ > 1$), the network response is overdamped. If the roots are real and equal ($ζ = 1$), the network response is critically damped. If the roots are complex ($ζ < 1$), the network response is underdamped.
STEP 4. The damping condition and corresponding response for the aforementioned three
cases outlined are as follows:
Overdamped: $x(t) = K_1e^{−(ζω_0 − ω_0\sqrt{ζ^2−1})t} + K_2 e^{−(ζω_0 + ω_0\sqrt{ζ^2−1})t}$
Critically damped: $x(t) = B_1 e^{−ζω_0 t} + B_2te^{−ζω_0 t}$
Underdamped: $x(t) = e^{−σt} (A_1 \cos ω_dt + A_2 \sin ω_d t)$, where $σ = ζω_0$, and $ω_d = ω_0 \sqrt{1 − ζ^2}$
STEP 5. Two initial conditions, either given or derived, are required to obtain the two unknown coefficients in the response equation.
PROBLEM:
Note: This post is based on Irwin's Basic Engineering Circuit Analysis 11th ed.
If there are any errors in solution, please let me know in the comments
SOLUTION:
When $t<0$, $i(0-)$ and $V_c(0-)$ are as follows:
$i(0-)=3A$, $V_c(0-)=-6V$
Using KVL, we can find the differential equation.
$-12+7.5i(t)+\frac{1}{0.1} \int i(t)+1.25\frac{di(t)}{dt}=0$
$\frac{d^2i(t)}{dt^2}+6\frac{di(t)}{dt}+8i(t)=0$
The characteristic equation is
$s^2+6s+8=0$
Hence the roots are
$s_1=-2$
$s_2=-4$
The circuit response is overdamped, and therefore the general solution is of the form
$i(t)=K_1e^{-2t}+K_2e^{-4t}$
Since $i(0+)=i(0-)=3A$,
$K_1+K_2=3$
Note that $V_c(0+)=V_c(0-)=-6V$
$-12+7.5i(0+)+V_c(0+)+1.25\frac{di(0+)}{dt}=0$
$K_1+2K_2=1.8$
Solving the two equations for $K_1$ and $K_2$ yields
$K_1=4.2$
$K_2=-1.2$
Thus, the general solution is
$i(t)=4.2e^{-2t}-1.2e^{-4t}$
Related posts: First-Order Circuit(Using the Differential Equations)
Comments
Post a Comment