Transfer function and state space model are developed for a circuit with resistor, inductor and capacitor in series as shown below.
(input) | v(t) | = | source voltage | |
i(t) | = | current | unknowns | |
(output) | v_{R}(t) | = | voltage across resistor | |
(output) | v_{L}(t) | = | voltage across inductor | |
(output) | v_{C}(t) | = | voltage across capacitor | |
R | = | resistance | ||
L | = | inductance | ||
C | = | capacitance |
4 unknowns so 4 equations are needed.
Laplace transform (eq. 1), (eq. 2), (eq. 3) and (eq. 4):
Solve for the 4 unknowns:
Find output/input for each output:
When there are inductors in a system, current through these inductors are commonly chosen as state variables. When there are capacitors in a system, voltage across these capacitors are commonly chosen as state variables. Therefore, current through the inductor and voltage across capacitor are chosen as the state variables. Current through the inductor is the same as the current through resistor and capacitor. It can also be noted that source voltage (input), current, and voltage across capacitor are sufficient to determine this system's future output values. This confirms that the selected state variables are valid.
State vector:
Input vector:
Output vector:
Rewrite (eq. 1), (eq. 2), (eq. 3), (eq. 4) in these new notations:
Rearrange equations to express αΊ‹(t) and y(t) in terms of x(t) and u(t):
Rearrange in matrix format: