Question 6.2.8: Series RC Circuit The resistor and capacitor in the circuit ...
Series RC Circuit
The resistor and capacitor in the circuit shown in Figure 6.2.10 are said to be in series because the same current flows through them. Obtain the model of the capacitor voltage v_{1}. Assume that the supply voltage v_{s} is known.
From Kirchhoff’s voltage law, v_{s} − R_{i} − v_{1} = 0. This gives
i = \frac{1}{R} (v_{s} − v_{1}) (1)
For the capacitor,
v_{1} = \frac{1}{C} \int^{t}_{0} {i dt + \frac{Q_{0}}{C}}
Differentiate this with respect to t to obtain
\frac{d v_{1}}{d t} = \frac{1}{C} i
Then substitute for i from (1):
\frac{d v_{1}}{d t} = \frac{1}{RC} (v_{s} − v_{1})
This the required model. It is often expressed in the following rearranged form:
RC \frac{d v_{1}}{d t} + v_{1} = v_{s} (2)