TITLE:
Resistor-Capacitor Circuit as a Dynamic System
AUTHORS:
Slavko Đurić
KEYWORDS:
RC Circuit, Voltage, Dynamic System
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.10,
October
11,
2024
ABSTRACT: The paper considers the response to the accumulated energy in the resistor (R)-capacitor (C) circuit. In the (RC) circuit, the capacitor C is initially charged with the “capacitive” voltage
U
0
. At that moment
t=0
, the P circuit switch turns on. By using Kirchhoff’s laws on the elements, a homogeneous differential equation of the first order with constant coefficients is obtained with the initial condition
U
C
(
0
)=
U
0
. The solution of the differential equation is presented in exponential form
U
C
(
t
)=
U
0
⋅
e
−t/τ
. Qualitative analysis RC of the circuit gives a phase portrait on the line. From the phase portrait on the line, it can be seen that the charge
U
C
(
t
)→
U
C
∗
=0
when
t→∞
stabilizes, regardless of the initial conditions. It is shown that from
U
C
(
t
)=
U
0
⋅
e
−t/τ
a dynamic system defined by the function
φ(
t,
U
C
)=
U
C
⋅
e
−t/τ
can be formed from. It has also been shown that, from the formed dynamic system, an autonomous system (circuit equation RC) can be found whose solution describes the formed dynamic system. It is also shown that the dynamic system
φ(
t,
U
C
)=
U
C
⋅
e
−t/τ
has one attractive fixed point
U
C
=0
.