RC-CircuitSnapshot of the front panel of the simulator: [HR][/HR] [HR][/HR] 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). [HR][/HR] Aim The aim of this simulator is to increase the understanding of the RC-circuit as a dynamic system. [HR][/HR] 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. [HR][/HR] Mathematical model It can be shown that the relation between the input voltage v[SUB]1[/SUB] and the output voltage v[SUB]2[/SUB] is given by the following differential equation: (1) RC*dv[SUB]2[/SUB]/dt = v[SUB]1[/SUB] - v[SUB]2[/SUB] By taking the Laplace transform of this differencial equation we find the following transfer function, H(s), from v[SUB]1[/SUB] to v[SUB]2[/SUB]: (2) H(s) = 1/(Ts+1) where (3) T = RC is the filter time constant. Using frequency response theory, it can be found that the bandwidth of the filter is (4) f[SUB]b[/SUB] = (1/T)/(2p) [Hz] [HR][/HR] Tasks 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[SUB]1[/SUB]. What is the corresponding steady-state value, v[SUB]2s[/SUB], of the output voltage response? From these results, what is the relation between V[SUB]1[/SUB] and v[SUB]2s[/SUB]? 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[SUB]1a[/SUB] have amplitude 0.5 and frequency 0.05Hz, and let sinusoid v[SUB]1b[/SUB] have amplitude 0.5 and frequency 1Hz. Thus, signal component v[SUB]1a[/SUB] has a smaller frequency than the bandwidth, which is 0.16Hz, while the component v[SUB]1b[/SUB] has larger frequency than the bandwidth. In other words, v[SUB]1a[/SUB] are in the passband of the filter, while v[SUB]1b[/SUB] 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[SUB]1b[/SUB] 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.