Lab 2: Capacitors Topics: Differentiator Integrator Low-Pass Filter High-Pass Filter Band-Pass Filter Integrator and Differentiator Circuits The simple RC circuits that you built in a previous section are also used as a low pass and high pass filter. The reactance of the capacitor affecting the voltage output of the circuit so that depending how it was connected with the resistor whether the circuit passed low frequencies or high frequencies. In this lab you will look at the same combinations but explore how the circuits integrate the voltage over time or differentiate the voltage over time. Theory: From mathematics you know that to integrate you are taking a running sum of the area under a function over a range such as time. You also know that the opposite of integration is differentiation or the rate of change over time. If you think about a sinusoidal waveform as an input into an integrator you would have an output with a waveform of a sinusoidal wave 90 out of phase with the supplied sine wave, i.e. -cos. A derivative of said same sine wave would also produce a +cos function or a sinusoidal wave 90 out of phase with the supplied sin wave. Basic Integrator: The circuit to the right should look familiar, an RC circuit that can act be used as a low pass filter. It can also be used as an integrator circuit. The voltage across the capacitor V C = V out. The integrator is across the capacitor. The input voltage V in equals the total series voltage of the circuit V in = V series = IZ = ( ). When C >> 1/R V in IR So at frequencies where >> 1/(RC) V in IR V out = V C = q/c = =
With the integrator, the RC time constant MUST be long in comparison to the measured period of the supplied signal. Procedure: Integrator 1. Build the integrator circuit using at 20K resistor and a 0.05 F capacitor. 2. Set the generator to a sine wave function at 20K Hz. 3. Connect channel 1 of the oscilloscope to view the input voltage. Think about what the output voltage should look like. Connect channel 2 to the output voltage. 4.While observing the oscilloscope and remembering that time increases as the trace moves from left to right, which waveform, input or output, seems to be leading. Does this make sense? 5. Change the function to a triangular waveform. View the output. Note on what you observe. 6. Change the function to a square wave. Note on what you observe. 7. Change the range to the next lowest range. What happens to the output? 8. For a square wave the integrator should produce a triangular waveform at the design frequencies. At high frequencies the amplitude will diminish. At very low frequencies the output will resemble the input and at mid frequencies the output looks more like a sharks tooth. 9. For this circuit, explore and try to define the frequency ranges for the 3 different waveforms. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Stuff below here is for your reference only on how Basic Differentiator works. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX Basic Differentiator: The circuit to the right should look familiar, an RC circuit that can act be used as a high pass filter. It can also be used as a differentiator circuit. The voltage across the resistor V R = V out. The differentiator is across the resistor. For the differentiator, the RC time constant MUST be short in comparison to the measured period of supplied signal. The input voltage V in equals the total series voltage of the circuit V in = V series = IZ = ( ).
For R << 1/( C) V in I/( C) At frequencies << 1/(RC) V in = V C The output voltage is the voltage across the resistor. V out V R = ir = R dq/dt =C dv c /dt R = RC dv in /dt Now one thing to ponder on is how the two circuits affect different functions. If you were to put a square wave into each of the two circuits what may the output waveform look like? How about a ramping waveform or triangular wave? Two common filter types for AC signals are the low-pass and high pass filters. A top-end stereo speaker system may utilize such filters. The low-pass allowing only the low frequencies to the bass speaker, the high pass filter allowing on high frequencies to the tweeter, and a combination of the two called a band-pass filter sending midrange frequencies to the mid-range speakers. Both filter types consists of two components, a resistor and a capacitor. The order in which they are connected determines the type of filter. The circuits are designed so that at a desired frequency the output voltage reaches a breaking point or half power point. This point is when the input voltage has been reduced by-3db. -3db point occurs when. The breaking point is determined from the equation. This is true for both the high pass and low pass filters. Low-Pass: The simplest circuit for the low-pass filter is a resistor in series with a capacitor. The graph shows that at low frequencies the output voltage is near the input voltage but drops quickly as the frequency increases. For the graph shown here V in is 1volt. The output voltage can be predicted from the following equation. The -3db point occurs when V out = 0.707 volts and from the graph it can be seen that this occurs at 1500Hz. This circuit was design using a 2200 resistor and a 0.047 F capacitor.
Since it is a series circuit the current through each component is the same and the output voltage is measured across the capacitor. At low frequencies the capacitive reactance is very high and much greater than the resistance of the resistor therefore most of the voltage is dropped across the capacitor. As the frequency increases the capacitive reactance becomes smaller and the voltage drop across the capacitor decreases. High Pass Filter: As with the low pass filter the simplest circuit is a capacitor and resistor connected in series. This time the resistor connects to ground and the output voltage is measured across it. As with the low pass filter at low frequencies the capacitive reactance is very high and most of the voltage is dropped across the capacitor leaving only a small portion dropped across the resistor. As the frequency increases the reactance decreases allow for more voltage, the output voltage for the filter, to be dropped across the resistor. Again the -3db point is the breaking point for the filter. For a input voltage V in = 1V the break point will occur at the -3db point and the output voltage is 0.707 V. You can see from the graph that the break point frequency is 1500Hz. The output voltage can be predicted using the following equation. Band-Pass filter: A band pass filter is used to pass only frequencies between a lower end and higher end frequency range, for example 1000 to 12000Hz. It is a combination of the two types of filters with the high-pass filter feeding C 1 R 2 into the low pass filter. However this is a bit deceptive. The high pass filter is design to pass the R 1 C 2 frequencies above 1000 Hz and the low pass filter is designed to cutoff the frequencies above 12000 Hz. The frequency for the low and high end is still determined from. If you were to design the circuit using two 10k resistors for 1000 Hz, C 1 would need to be 16nF. For 12000Hz the capacitor C 2 would need to be 1.3 nf. Ideally the output would look similar to that shown to the right.
Procedure: Low Pass Filter: Determine the capacitance need to design a low-pass filter using a 20K resistor to have a breaking point around 1700 Hz. Build the circuit and supply a sinusoidal AC signal at 200 Hz to the filter. Measure the input and output voltages. Increase the frequency by 200 Hz. Monitor the input voltage and adjust the amplitude of the signal source as needed to maintain a constant input voltage value. Why do we need to do this? Think about the change in impedance of the circuit and of the generator. Measure the output voltage. At increments of 200 Hz obtain data for the output voltage through 4400 Hz. On the 1K range start from 0.1 and slowly adjust the dial to 10. Observe the output voltage. Does it behave as expected? IMP: Keep this setup for Part B. Make a new setup for High pass filter on another end of the bread-board. High Pass Filter: Use the same components. Configure them for the High Pass filter. Will the break frequency be the same? Repeat the process to gather data for the high pass filter. Band Pass Filter: Using 20k resistors determine the capacitors needed to pass frequencies from 1700 Hz to 17 KHz. Recall that the high pass filter is design to pass frequencies above the low end frequency of the band pass and that the low pass filter is design to cutoff the high end frequencies. Build the circuit shown in the theory. Explore the low end range of the filter and gather enough data to reassure that it is functioning as expected. Explore the midrange area of the filter. Reassure that it behaves as expected. Lastly explore the high end range of the filter.
Expected Lab Report format: 1. Introduction with objective 2. Theory (optional, as it is already given in this manual) 3. Experimental Setup (With pictures taken from your cell phone) 4. Results and Discussion a. For each topics, show the graphs with brief explanation b. The graphs should show the experimental and the theoretical data for each topics. 5. Conclusions 6. Appendix with data table The report must be typed. MS Excel or any other plotting software can be used to analyze the data.