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Within the context of electrical circuits, filters have numerous applications in which they are often used. In this report, Resistor-Capacitor circuits will be studied. These are circuits with elements including resistors and capacitors, but the report focuses on first order RC filter circuits involving a single resistor and a single capacitor. RC circuits can be modified for different purposes. Low-pass filters, or LPFs, are used to filter out ‘noise’ or high frequency signals from circuits, whereas high-pass filters, or HPFs, let frequencies higher than a certain value to pass.

By simplification, we can say that LPFs let’s through shorter wavelengths of waves and therefore higher frequencies due to inverse proportionality.

Using the filter circuits in this manner only works with AC current as with DC, or direct current, the capacitor would charge up and would stop the current. In this report, sinusoidal waves will be investigated through experiment and simulation on the computer software PSPICE. It is shown that current in the capacitor varies with voltage and time.

PSPICE is used to simulate an oscilloscope from the signal generator. As an example, the software package allows us to simulate circuits and allows us to analyse the impacts of different frequencies of waves on the circuit.

The idea of a filter circuit is defined to be a “device that changes the amplitude of an AC voltage as the frequency of the input voltage changes,” as stated by Dr. Daniel Nankoo from the City, University of London.

Ideally, these waves follow the properties of a sinusoidal wave and can thus be analysed using functions involving sinusoids. For the purposes of further study, low-pass filters let low frequency signals through, meaning only the parts where a relatively low amount of oscillations of the waves take place as shown through the justification of the units of frequency, Hertz (Hz). In theory, because only the low frequency oscillations are let through, we can deduce that the amplitudes of the input waves are reduced as well. This is true keeping in mind that the frequency is higher than the cut-off frequency through deduction.

dB=20 (log_10 )(V_out/V_in )), which, based on theory, should come to -3dB for the case of the specific frequency.

The sinusoidal relationship with the AC voltage source is different to what you would except from a DC source through Ohm’s Law, V=IR, showing a linear relationship between voltage and resistance. When there is an AC voltage on the RC filter circuit (series), the capacitor will then charge and discharge continuously. This is why it forms a voltage divider in the circuit as well.

Prior to opening the simulation software package PSPICE, the Preliminary Analysis questions were completed. The results from these are then used to compare to the simulation results gathered from the simulation.

After opening the PSPICE software package, the Get New Part function in the Schematic Page Editor was used to create a computerised Voltage Source (Vsin), the resistor R, the capacitor C, and the ground for 0V. These were then added onto the diagram on the main space on PSPICE according to Figure 1 above. In order to be able to compare the simulated values with the manually gathered results, the capacitance of the capacitor and the resistance of the resistor needed to be edited to 1047pF and 10k, respectively. It was noted that the properties of the voltage source needed to be edited individually as well. The amplitude of the sinusoid wave of the source was edited to be 10V, the frequency to 10kHz and the offset voltage (Voff) to 0V. Transient analysis was performed on the circuit from times 0-1000s, and with the results being calculated by the simulation every 5s (minimum) and the results stored every 10s.

The same experiment was then repeated with a frequency of 100kHz. As in the previous simulation, the simulation was run from 0-1000s, so it gave 10 complete oscillations at a frequency of 10kHz. In order to get 10 complete oscillations with the 100kHz simulation, the times needed to be edited. The step size was changed to 0.5s. The simulation was then completed again, and data was recorded.

Next, the voltage source Vsin was replaced with a variable frequency source (AC), which is denoted as Vac on PSPICE. The amplitude remains at 10V, but we changed the Transient analysis type to A.C. Sweep. The Transient analysis method plots the circuit as a function of time, whereas the A.C. Sweep method is a frequency response method of analysis. On PSPICE, the calculations were set to 10 points for each decade of measurement on the range of 10Hz to 10MHz (To indicate MegaHertz on PSPICE, we needed to show the mega- as Meg, instead of just M). The data was plotted with the ratio of V_out/V_in being logarithmic due to many magnitude changes. The data was then plotted in decibels (dB) (x-axis) with the DB() function on PSPICE and the y-axis back to linear. The Preliminary Analysis results were then compared with the simulation results further.

The simulation results closely match the preliminary analysis, with slight variations due to simulation averaging and measurement errors. The ratio of output to input voltage decreases with increasing frequency, demonstrating the low-pass filtering characteristic of the circuit. Experimental results show similar trends but with slight discrepancies due to equipment limitations and phase differences.

When we take the ratio of the amplitudes of the two voltages, we get:

V_out/V_in =8.3V/9.9V=0.8384

When compared with the Preliminary Analysis results, the simulation gives a ratio value of 0.8354. Due to very subtle changes in the amplitude of the first few sinusoidal waves, the simulation provides an amplitude of the average, whereas in the calculation, the actual values of the circuit were used. Overall, there isn’t a large margin of error, on the magnitude of 10^(-3).

From the equation, V_out/V_in =(1/ωC)/√(R^2+((1/ωC))^2 ), where the ratio of V_out/V_in is the magnitude, it can clearly be seen that the magnitude of V_out decreases with increasing ω, so this shows that it passes lower frequencies and not higher ones as ω=2πf, thus showing that it is a low pass filter. The PSPICE analysis agrees with the Preliminary Analysis.

After restricting the x-axis values, it can be seen that the maxima’s of the two waves do not occur at the same time. This is because it is known that when 1/ωC >> R and the circuit is mostly dominated by the capacitor (low frequencies) as ω tends to closer to 0. This would then mean that the low-pass RC circuit that was simulated would be expected to be a highly resistive circuit when the frequency is very high, which can be seen later in the report.

The ratio of V_out/V_in , in this case from the figure above, is 1.49/9.97=0.149 as the green sinusoid is not perfectly centred so it was needed to do (max.-min.)/2 to get the amplitude. Through the Preliminary Analysis, the ratio was calculated to be 0.1503, showing that the ratios through simulation and preliminary calculation are very accurate, with a small error due to rounding. Interestingly, the first two wavelengths seem to have a different response, compared to the rest of the waves where the pattern of the sinusoidal wave is constant. The measurements were taken from the steady part of the graph to ensure correct average values throughout the response of the circuit.

The above graph shows the relationship between the dB and the frequency. As shown in the Preliminary Analysis, it can be said that the frequency at which the ratio of V_out/V_in =1/√2 is at 1505Hz, or 1.505kHz. When going to this value on the graph (x-axis), the y-value of decibels should equate approximately to the characteristic frequency, to which it does. The -3dB point is the point where the ratio equals the said 1/√2 . The curve falls from the constant value (where the curve is horizontal) by 3dB at this characteristic frequency.

If we look at the values of the graph from for every 10x increase in frequency, the slope falls at 20dB per decade. This is seen through the equation: dB=20 (log_10 )〖(V_out/V_in )), where there is a 20dB reduction for every factor of 10 reduction of the ratio. For example, if we go from -10dB to -30dB, the frequency changes from approx. 100kHz to approx. 1000kHz, showing an increase of a factor of 10.

The initial frequency was set to 10kHz and the generator was set to produce an amplitude of 1V. The ratio V_out/V_in =((0.67-(-0.84)))/((0.87-(-0.96)))=0.825 (again, gathered from the division of the two amplitudes), which agrees with the previous ratios of 0.8354 from the Preliminary Analysis and 0.8384 from the simulation.

The signal generator frequency was set to 100kHz and the generator was set to produce an amplitude of 1V. When the ratios are taken of the amplitudes of the output and input voltages, we can see a different value that doesn’t quite agree with the Preliminary Analysis results and the simulation results. We get V_out/V_in =0.2/0.865=0.231, instead of 0.1503 as stated in the Preliminary Analysis. It can be seen that the phase difference between the output and input waves is approximately 45°, but not exactly. This is from the lag due to charging up the capacitor and this creates a lag for the output voltage to go through the circuit. It can also be deduced from this that the higher the frequency is, in this case from 10kHz to 100kHz, the more the capacitor lags behind and therefore the two sinusoidal waves are more out of phase. And this would continue for higher frequencies as tested during the experiment.

To conclude, it can be said that the V_out/V_in ratios agree for the Preliminary Analysis, where calculations were done by hand to predict what would happen, and for the simulation, where PSPICE was used to simulate i.e. the effects of frequency on the low-pass filter circuit. For the experiment, when the frequency was 10kHz, the ratio remained very similar to the ones from the analysis and simulation, but when the frequency was increased to 100kHz, the output voltage value showed to be higher than predicted partly due to lag. dB scales were used for the y-axes of the graphs in order to get an approximate linear relationship for the decrease of the voltage. The slight difference in the values could also be from the error of what the assumed values are compared to what they actually are. So, for example, in the experiment, the ratio may have error also due to the equipment not having the exact perfect properties as assumed in the Preliminary Analysis and computer simulations as well. As expected, as the frequency got higher, the output voltage was lower.

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