RC-Circuit
Snapshot of the front panel of the simulator:
Description of the simulated system
In this simulator an RC-circuit is simulated. It consists of a resistor R [Ohm]
and a capacitor C [Farad] connected in a circuit, see the front panel of the simulator.
In the tasks below the dynamic properties of the RC-circuit will be observed
through simulations. In the simulator the input signal is a sum of two
independent sinusoids and a bias (a constant).
Aim
The aim of this simulator is to increase the understanding of the RC-circuit
as a dynamic system.
Motivation
In applikations where a simple analog lowpass filter is needed, the
RC-circuit is commonly used, as in I/O equipment (input/output) for attenuation
of measurement noise.
Mathematical model
It can be shown that the relation between the input voltage v_{1}
and the output voltage v_{2} is given by the following
differential equation:
(1) RC*dv_{2}/dt = v_{1}
- v_{2}
By taking the Laplace transform of this differencial equation we find
the following transfer function, H(s), from v_{1} to v_{2}:
(2) H(s) = 1/(Ts+1)
where
(3) T = RC [s]
is the filter time constant.
Using
frequency response theory, it can be found that the bandwidth of the
filter is
(4) f_{b} = (1/T)/(2p) [Hz]
Unless otherwise stated you should use default values of the various
parameters (you get the the default value via right-click on the front panel element).
- The step resonse of the filter:
In this subtask, you should suppress the sinusoids (by setting the
amplitudes to zero).
- Calculate (by hand) the time constant T according to Eq. (3)
above. Is the result the same as can be seen on the front panel of
the simulator when the simulator runs? Then run a simulation where
you adjust the signal component B as a step, and read off the time
constant from the response. Is the observed time constant the same
as the calculated time constant?
- Run a simulation with some constant input signal, say V_{1}.
What is the corresponding steady-state value, v_{2s}, of the
output voltage response? From these results, what is the relation
between V_{1} and v_{2s}? Can you calculate this
relation directly from the model (1)?
- Frequency response of the filter:
Set the signal component B to zero. Use default values of R and C.
- Let the sinusoid v_{1a} have amplitude 0.5 and frequency 0.05Hz,
and let sinusoid v_{1b} have amplitude 0.5
and frequency 1Hz. Thus, signal component v_{1a} has a
smaller frequency than the bandwidth, which is 0.16Hz, while the
component v_{1b} has larger frequency than the
bandwidth. In other words, v_{1a} are in the passband
of the filter, while v_{1b} is in the
stoppband of the filter. Run the simulator! Can you observe
from the simulation that signal component that component 1 is
in the passband, while v_{1b} is in the stoppband?
- The bandwidth is defined as the frequency where the amplitude
gain of the filter is 1/sqrt(2) = 0.71 = -3dB. In other words,
if the sinusoidal input signal has frequency equal to the bandwidth,
the amplitude of the output signal is 71% of the amplitude of the
input signal. Verify this by running a simulation.
[SimView] [TechTeach]
Updated 30. May 2008.
Developed by
Finn Haugen.
E-mail: finn@techteach.no. |