Instructions for preparing a manuscript for submission to the Samahang Pisika ng Pilipinas Physics Congress

Observing RC Circuits in varying frequencies and in different kinds of waves inputs

Aiko Love Del Rosario and Roseanne A. Novesteras*

1National Institute of Physics, University of the Philippines Diliman, Quezon City 1100 Philippines*Corresponding author: rosean817@yahoo.comAbstract

Capacitors are devices that are usually made up of two conducting parallel plates that store energy in an electric field in units of capacitance. Circuits with resistors and capacitors and are driven by voltage or current source are called RC circuits. They may act as an integrator or differentiator. In charging and discharging the capacitor the oscilloscope displays longer wavelength as the frequency is increased. The theoretical time constant matched with the experimental time constant. In differentiators, when an input wave is a square wave, the output at low frequency forms a wave that exhibit charging and discharging and triangular wave output at high frequency. When the input is a triangular wave, the wave output at low frequency also exhibit charging and discharging and sinusoidal wave output at high frequencies. 1. Introduction

A capacitor, previously known as condenser, is an electrical component that stores energy in an electric field in units of capacitance, usually made up of two conducting parallel plates and has the property:

(1)

Where Q is charge in coulombs, C is capacitance in farads and V is voltage in volts. This says that a capacitor with capacitance C and of voltage V across the terminals, it has charge Q stored on one plate and charge Q on the other plate. To some extent, these devices may be considered as resistors that are frequency dependent. [1]

Another property of capacitors is that because the voltage and current are 90 out of phase, it cannot dissipate power even though current can flow through it. This follows the equation:

(2)

The effective capacitance when they are connected in series is calculated by:

(3)

And for parallel:

(4)

The circuits used in this experiment are RC Circuits or circuits that are mainly composed of resistors and capacitors driven by a voltage source or current source. [2]

Figure 1. Simple RC Circuit

In a simple RC Circuit (Figure 1), equation 2 becomes:

(5)

The solution of this differential equation is:

(6)

In equation 6 the product RC is called the time constant of the circuit and is in seconds. [1]There are methods of using capacitors and resistors to achieve alterations in the AC signals. They may be used as filters with sine wave signals of varying frequencies. These are also used to vary the shape of non-sinusoidal waves. One kind of filter is a High Pass Filter. It is used when a CR circuit just like in figure 3 is used with sinusoidal signals. Its purpose is to allow high frequency sine waves to pass easily from the input to the output, and at the same time reducing the amplitude of lower frequency signals. Another kind filter is a Low Pass Filter. It is a low pass filter when an RC circuit just like in figure 4 is used so that at low frequencies the high reactance offered by the capacitor allows almost the entire input signal to be developed as an output voltage. However, at higher frequencies, the capacitive reactance of the capacitor in the circuit becomes much less than R and a small amount of the input signal is developed across the capacitive reactance. With this, the circuit reduces the amplitude of higher frequencies applied to the input. [3] When the signal is in low frequency and is passed from input to output with minimal attenuation, signals that have high frequencies are attenuated to almost zero. However for high pass filters, it is the opposite. The point at which the response has fallen 3dB is called the cut-off frequency fc and is given by the equation:

(7)

The cut-off frequency is used to define the bandwidth of the filters and a loss of 3dB resembles the reduction in output voltage to 70.7% of the original output voltage. [7]Another thing about RC circuits is when they are used with sine waves; they do not alter the shape of the wave. The amplitude and phase of the wave may alter but the shape of the wave does not change. But if the input wave is not a sine wave, say, a square wave or a triangle wave, the effects of RC circuits are quite different.[4]When a high pass filter is used and sine wave is the input, we expect a sine wave as well but the amplitude is reduced and the phase is shifted if the frequency is low, as it is a high pass filter. However, if a square or triangular wave is the input, we expect a differentiated form of the wave. The CR circuit when used with a non-sinusoidal input is then called a differentiator.[4]Opposite to the differentiator is the, integrator. The RC circuit is called an integrator when the input wave is a non-sinusoidal wave and the time constant RC of the circuit is much longer than the period of the wave. [5]2. MethodologyThis experiment has five parts. In the first part, the charging and discharging behavior of a capacitor is observed. A square wave input with Vpp of 5V and frequency of 500Hz is used. The setup is shown in figure 2.

Figure 2. Charging and Discharging Capacitor Set-upThe input and output waveforms were noted, as well as the important values and points on the plot. The time constant was determined from the charging and discharging curves of the plot and was compared to the theoretical value. The frequency was also increased and the output was observed.For the second part of the experiment, the circuit in figure 3 was used.

Figure 3. Circuit used in the second part (Vpp = 5V, f=100kHz)The input and output waveforms were observed with square, triangle and sine waves as input.

For the third part of the experiment, the circuit in figure 4 was used.

Figure 4. Circuit used in the third part (Vpp=5V, f=100kHz)The input and output waveforms were observed with square, triangle and sine waves as input.

For the fourth part of the experiment, the circuit in figure 5 was used.

Figure 5. Circuit used in the fourth part (Vpp=5V)With varying frequencies the experimental f3dB was determined and was compared to a theoretical value. From the data, an experimental Bode plot was constructed.

For the fifth part of the experiment, the circuit is figure 6 was used.

Figure 6. Circuit used in the fifth part (Vpp=5V)With varying frequencies the experimental f3dB was determined and was compared to a theoretical value. From the data, an experimental Bode plot was constructed.3. Results and DiscussionsUsing the oscillator, the wave output was observed and recorded using a cell phone camera. From equation 4, the charging curve of the plot is expected to have a logarithmic increase, whereas for a discharging curve, it must have an exponentially decaying plot. The wave output taken is shown below:

Figure 7. The time constant may be calculated using the values R and C from the resistor and capacitor used in the circuit or by taking the range from the uncharged curve to 63% of Vin or from the Vmax to 37%Vin.From the plot, the time constant is 10 milliseconds, corresponding to one tick mark. This value is the same with the time constant calculated from the 10k resistor and 0.01F.When the frequency of the input is increased, the cycles of each wave output in the oscillator increased. However, the time constant remained the same. This means that a lower frequency (f < 1/RC) makes the capacitor to stay fully charged longer as well as it stay fully discharged. On the other hand, a higher frequency (f > 1/5RC, depending on the wave input period) will cause the capacitor to not have enough time to charge or discharge, and will have a voltage less than the input voltage.The second circuit created was a differentiator. Its components are arranged that of Fig. 3. The differentiator creates a wave output that is the derivative of the wave input.

Figure 8. The wave output recorded from the differentiator circuit with different wave inputs used for the voltage source: (a) sine wave, (b) square wave, (c) triangle wave.With a square wave input, it produced high frequency spikes (fig. 8b) the capacitor is quickly charged. As the square wave input changes each cycle, the wave changes from positive to negative output spike.

Triangular waves are produced using a square wave passed through an integrator. If a triangular wave is made to pass through a differentiator, it must give a square wave output. [6]An integrator was made using the circuit in Fig. 4. This circuit converts the wave input into its integral. Same with the differentiator, three different wave inputs were made to pass through this circuit: sine, square and triangular wave inputs. The outputs recorded are shown below:

Figure 9. The wave output recorded from the integrator circuit with different wave inputs used for the voltage source: (a) sine wave, (b) square wave, (c) triangle wave.The square wave input produces a charging and discharging wave with lower frequencies. However, when high frequencies are introduced, the oscilloscope produces a triangular wave, which agrees with the output from the differentiator with a triangular wave input. On the other hand, when a triangular wave is made to pass through the integrator, the output is a sine wave.It can be observed that when a sine wave is made to pass through the differentiator and integrator, its waveform remains unchanged except for its amplitude and frequency. To observe this, different frequencies were tested with a sine wave input and with a differentiator and an integrator.The cut-off frequency f3dB is essentially the same as 70.7% of the input voltage. Using the plot of the output wave of the integrator and differentiator with sine wave input, the value of f3dB is obtained.

Figure 10. Using the period of the sine wave, the f3dB may be calculated.With 1 div = 0.2ms, the calculated cut-off frequency is 1 kHz. Using equation 7, the theoretical value obtained is 1.06 kHz. Comparing the two values, the deviation is 0.06.

To further explain the frequency response of the two circuits, the graphs of their experimental Bode plot were made.

Figure 11. The Bode plot of the differentiator on the left, and integrator on the right.

It can be observed that for high frequencies, the integrator gives a higher voltage gain, whereas the differentiator gives a lower voltage gain. These plots shows that an integrator is a low pass filter and a differentiator is a high pass filter.4. Conclusions and Recommendations

The time constant of the RC circuit can be experimentally obtained using the values 0.63Vin or 0.37Vin. Increasing the frequency of the input voltage causes the output wave to have a longer wavelength.

RC Circuits may also act as either an integrator or differentiator. The square, triangle, and sine waveforms are related to each other as follows: the integral of a square wave is a triangular wave (at high frequencies), the derivative of a triangular wave is a square wave and the integral of a triangular wave is a sine wave. The square wave also produces high frequency spikes when introduced to a differentiator.Sine wave input does not change in form when made to pass through a differentiator or integrator. However, when the frequency responses of the two circuits were investigated, the integrator is found to be a low pass filter and the differentiator is a high pass filter.

It recommended to use Arduino as an oscilloscope for more accurate measurement of output voltage. More accurate measurement of frequencies and more data points for the Bode plot is suggested by the researchers.References

1. P. Horowitz, The Art of Electronics, Chapter 1, Press Syndicate of the University of Cambridge, Cambridge, 1989. 2. Wikipedia. November 27 2012. RC Circuit. Retrieved December 11 2012.

3. E. Coates. 2012. Learn About Electronics-AC Theory: How Filters Work. Retrieved December 11 2012.4. E. Coates.2012.Learn About Electronics-AC Theory: Differentiators. Retrieved December 11 2012.

5. E. Coates.2012.Learn About Electronics-AC Theory: Integrators. Retrieved December 11 2012.

pp. 565-578

6. A. P. Malvino, Electronic Principle, 3rd ed., McGraw-Hill Inc., USA, 1984.

7. Storr, W. December 2012. Electronics Tutorials.ws: RC Waveforms. Retrieved December 11 2012

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