Edupanda » Circuit Theory » Operator Method
Operator Method
The operator method involves replacing the system of differential equations describing a given circuit with a system of algebraic equations in the variable s
Solution procedure:
- Compute the initial conditions by solving the system in the pre-switching state
- Replace the elements with their operator versions
- Solve the newly formed system
- Compute the inverse transform
Operator Schemes for RLC Elements
Resistor
Inductor
Capacitor
Computing the Inverse Transform
For the overwhelming majority of considered circuits, the transform can be written in the form:
The roots of the numerator \(L(s)=0\) are called the function's zeros
The roots of the denominator \(M(s)=0\) are called the function's poles
To compute the original of such a transform, we can use the formulas listed below:
No. | Transform | Roots | Original |
---|---|---|---|
\(1\) | \(F(s)=\frac{L(s)}{M(s)} \) | real or complex roots of the denominator | \(f(t)=\sum_{k=1}^n \frac{L\left(s_k\right)}{M^{\prime}\left(s_k\right)} \mathrm{e}^{S_k t}\) |
\(2 \) | \(F(s)=\frac{L(s)}{s M(s)} \) | real roots of the denominator and one s=0 (zero) | \(f(t)=\frac{L(0)}{M(0)}+\sum_{k=1}^m \frac{L\left(s_k\right)}{s_k M^{\prime}\left(s_k\right)} \mathrm{e}^{S_k t}\) |
\(3\) | \(F(s)=\frac{L(s)}{M(s)}\) | double root of the denominator | \(f(t)=\operatorname{res}_{s=s_k}\left[\frac{L(s)}{M(s)}\right]=\lim _{s \rightarrow s_k} \frac{d}{d s}\left[\frac{L(s)}{M(s)}\left(s-s_k\right)^2 \mathrm{e}^{s t}\right]\) |
\(4 \) | \(F(s)=\frac{L(s)}{M(s)} \) | complex conjugate poles | \(f(t)=2 \operatorname{Re}\left[\frac{L\left(s_k\right)}{M^{\prime}\left(s_k\right)} \mathrm{e}^{S_k t}\right]\) |
\(5 \) | \( F(s)=\frac{L(s)}{s M(s)} \) | complex conjugate poles and one s=0 | \(f(t)=\frac{L(0)}{M(0)}+2 \operatorname{Re}\left[\frac{L\left(s_k\right)}{s_k M^{\prime}\left(s_k\right)} \mathrm{e}^{S_k t}\right]\) |
Example 1
Content
Using the operator method, determine the voltage \(u_c(t)\)
Data: \(E_1=300 V, E_2=100 V, R_1=15 \Omega, R_2=25 \Omega, R_3=30 \Omega, C=50 mF=50 \cdot 10^{-3} F, L=2 H\)
Solution
Solution for pre-switching state (to determine boundary conditions):
Replace the elements with their operator descriptions, calculate using the nodal analysis method:
As you can see, \(u_C(s)=V\), so we just need to calculate the inverse transform:
As you can see, this is variant 2 from the table:
We just need to calculate: