Lab 5 - Lab experiment report PDF

Preview

Summary

Lab experiment report...

Description

Experiment 5: RC Circuit Frequency and Time Response Figure 2. High Pass Filter; Measuring Voltage Across R1

Abstract— The purpose of this lab is to introduce students to the concepts of circuit response in the frequency and time domains.

INTRODUCTION

In this lab, simple RC series circuits and their behaviors are observed while they are excited by an AC signal. However, this time, it is the gain of the circuit throughout various elements which is studied and compared, accordingly. For the simple RC circuit, the effects of alterations in different elements of the circuit such as frequency, input amplitude, resistor value and capacitor value are looked at. Additionally, gains across the capacitor or the resistor are observed and given the voltage division across different elements, differing results are expected. The simple RC circuit has been previously analyzed in terms of gains and the result was that there are very few parameters that the gain depends on. To commence, two circuits were provided to work with (they are really the same circuit but the reading is taken across different elements). They are shown in Fig. 1 and 2 below.

Clearly, it can be seen that the circuits really are the same and it would be reasonable to only measure the output from one element or the other.

I.

FREQUENCY DOMAIN RESPONSE:

Before being allowed to set up the circuits and jot down measurements from the bread board, the lab assignment asks for the analysis of the two circuits using equations to see exactly how the gain behaves and what it is dependent on. The calculations below show the gain equation for Figures 1 and 2. For Figure 1:

V ( out ) =

ZC V (¿) R+Z C

where

ZC=

1 j2 πfC

so

1 j 2 πfC V (¿) V ( out ) = 1 R+ j2 πfC simplifying, it gives

1 j2 πfC V ( out ) =Gain= V (¿ ) Rj 2 πfC +1 j2 πfC finally,

Gain= Figure 1. Low Pass Filter; Measuring Voltage Across C1

1 Rj 2 πfC +1

This equation clearly has, in the phasor form, a magnitude and angle associated, so let’s change both parts into the phasor domain:

Gain=

1 ∠0 °

√ ( 2 πRfC ) +1∠ arctan ( 2 πRfC ) ° 2

From this equation, it is seen that the magnitude is:

|Gain|=

1

√ ( 2 πRfC )

2

+1

And the phase difference is given by:

Phase=∠−arctan(2 πRfC )° For Figure 2:

V ( out ) =

From here, it is noted why these circuits are called high and low pass filters. For the low pass, it is seen that for low

R V (¿) R+Z C

Using the same thing as done before, it is obtained:

V ( out ) =Gain= V (¿ )

R R+

1 j 2 πfC

R Rj 2 πfC +1 j 2 πfC Rj 2 πfC Gain= Rj 2 πfC +1 Gain=

II. TIME DOMAIN RESPONSE

Again, it can be seen this has a magnitude and phase, so in phasor form, it is obtained:

Gain=

2 πRfC ∠ 90°

√ ( 2 πRfC ) +1∠ arctan ( 2 πRfC ) ° 2

The magnitude is then:

|Gain|=

It was now required to plot the V(out) of both circuits in the time domain to see how they behave with a step impulse of 1 V. The equations given are given below. For Figure 1: −t

V ( out ) =V (¿ )(1−e RC ) For Figure 2:

2 πRfC

√ ( 2 πRfC )

frequencies, the same output as the input yields so get a gain of 1 is obtained. At higher frequencies however, the gain drops off linearly. So, this circuit allows the low frequencies to “pass” and restricts the gain for higher frequencies. This is the same case for the high pass filter; it allows high frequencies to “pass” while restricting low frequencies.

2

−t

+1

V ( out ) =V (¿ )(e RC ) it was asked to graph these equations within the range of t = 0, 6RC on the same plot. This is the result:

And the phase difference is given by:

Phase=∠ 90−arctan(2 πRfC )° Clearly, as previously noted, the gain of an RC circuit is dependent only on the frequency of the input and not on its magnitude. So, no matter what amplitude of input is put in, it will always yield the same magnitude of gain. At this point it was asked to plot only the magnitude of the gain for these specified circuits over the range of 10 Hz to 10 MHz with gain on the y-axis. This is our result: Figure 4. Time Domain Response of Low Pass and High Pass Filters

From the graph, it is apparent that when measured across a resistor, over time, the capacitor reaches an optimal value of 1V (our input) while the resistor reaches a minimum value of 0 V (as if the circuit were opened). This makes sense because that is the behavior or capacitors: once fully charge, they allow no current to pass and so elements in series with it see no voltage drop.

III. MULTISIM SIMULATIONS OF LOW, HIGH, AND BANDPASS FILTERS

Figure 3. Gain plotted against Input Frequency

The next objective was to combine both the high pass and low pass filters to create a band pass filter, which allows only a certain region of frequencies to pass. By picking the right choice of resistors and capacitors, a band pass filter was

created that allowed only certain frequencies to pass through. High Pass Frequencies Low Pass Band Pass Gain Gain (V/V) (Hz) Gain (V/V) (V/V) 100 0.98148 0.12222 0.22476 300 0.94444 0.30189 0.51429 1,000 0.66981 0.66981 0.85577 10,000 0.11698 0.94340 0.92308 100,000 0.05769 0.96154 0.37255 300,000 0.05769 0.96154 0.14706 1,000,000 0.05660 0.96154 0.06796 The circuit diagram used to make the calculations is given below:

With the band pass circuit analyzed, we go back to our low and high pass filter circuits and run simulations in MultiSim. Next, the time domain response is observed by running a square wave into the input with the lowest value at 0 V and the highest value at 1 V. basically successive impulses are run in repeated equal intervals and the output across the same elements is measured same as the previous section. The parameters of the pulse period are then changed to observe effects on the output. For the low pass filter:

Pulse Period=12 RC End Time=5 mS

Pulse Period=4 RC End Time =5 mS For the high pass filter:

Pulse Period=12 RC End Time =5 mS Figure 5. Band Pass Circuit Diagram for MultiSim

After running an AC analysis of the circuit across C2 all the way on the right and ploting the gain against the frequencies, which was ran from 10 Hz to 10 MHz again, this is what resulted:

2 RC 3 End Time=1 mS Pulse Period=

these simulations will be plotted later on along with the simulated, calculated, and measured values.

IV. LAB MEASUREMENTS FOR LOW, HIGH, AND BANDPASS FILTERS ON PROTO-BOARD After all the calculations and simulations were taken care of, the actual measurements of the lab were jotted down. It was begun with the low pass filter where it was built on the breadboard and the corresponding measurements were taken down as shown in Fig.1. The input used was a sinusoidal signal of 1VPP with a variable frequency. This way, the computer can take down the gain based on these two values. Before that however, the instruction were to take down some measurements by hand. Figure 6. Band Pass Gain Plotted against Frequencies

From the graph, a clear picture of what a band pass filter does is portrayed. It combines the features of a high and low pass filter to create a “band” pass filter that allows the gain of the circuit to be near 1 for a certain “band” of frequencies. For our circuit, those frequencies lie between 103 and 104 Hz.

To make sure the circuit was working properly, it was asked to vary the frequency across a multitude of ranges similar to the calculations and simulations. Table 1 below shows the values for the hand measurements for gain by varying the frequencies by hand. Table 1. Gains for Low, High, and Bandpass Filters

These values make sense as explained earlier; that the low pass filter will give a gain of near 1 for low frequencies and much lower gains for higher frequencies with the opposite being true for the high pass filter. When compared to the values in the simulation and calculated, similar results were observed when the gain was high enough but saw some discrepancies at lower gains. This will be explained later on in the lab. Once the accuracy of the obtained readings were ascertained, a Frequency Response measurement was run to get the gain of the low pass filter circuit. 40 points ranging from 10 Hz to 10 MHz were taken, in the same fashion as the calculations and simulations. Similarly, the data for the high pass and band pass filters were saved for analysis later during the lab.

Figure 7. Measured and Calculated Values for Gain in Low Pass Filter

The same was done for the high pass filter.

After these analyses, the data for the time functions for only the low and high pass filters can be obtained. The same approach utilized in the simulation was employed by setting a square wave pulse train from the function generator with the same parameters as the simulation (with 2 simulations for each circuit). With that, all the data needed to make observations of the graph data is acquired.

V. ANALYSIS A. Frequency Domain Response At this stage it was asked to create multiple graphs using the data collected during the calculation (pre-lab), simulation and laboratory phases. The first graph includes the measured and calculated values of the gain for the low pass filter. Figure 7 below shows this graph.

Figure 8. Measured and Calculated Values for Gain in High Pass Filter

For the band pass filter, it was required to compare the simulated and measured gain on the same frequency/gain scale as the other graphs.

Now the series circuit from Fig.1 is set up on the Proto-Board, the voltage source is set to 5 V and the voltages as well as the errors across the voltages are checked. Table 2 gives these values.

Figure 9. Measured and Simulated Values for Gain in Band Pass Filter

It was noticed that, for the most part, the graphs almost overlap giving the impression that the measured values agree

Figure 10. Measured Input, Measured Output and Simulated Output for Low Pass Filter (Settings: Pulse Period = 12RC, End Time = 5 mS)

with both the calculated and simulated values. However, something odd happens around the 10 -1 V/V gain line. It is seen that the oscilloscope yields very odd readings for the low pass filter, and does not give a reading for the high pass filter, and gives odd readings for the band pass filters, too. We are lead to believe that our oscilloscope is unable to correctly read very low values of output. The same problem was observed when hand measurements were being performed. The oscilloscope was having a hard time getting measurements for very small values of the output when the gain was very tiny. It would jump from 240 mV to 180 mV, which threw off our measurements a lot. As LabVIEW was taking down data for the gain, it was noticed that the graph for the output for very low gains looked very odd. The values would jump erratically

Figure 11. Measured Input, Measured Output and Simulated Output for Low Pass Filter (Settings: Pulse Period = 4RC, End Time = 5 mS)

and the gain wouldn’t be the same as the calculated or simulated value. In terms of the frequency response, as explained before, low pass filter outputs a gain of near 1 at low frequencies, high pass filter outputs a gain of near 1 at high frequencies, and band pass filters output a gain of near 1 at a “band” of frequencies governed by the resistors and capacitors that are chosen to be used in the circuit. The cut-off frequencies for our filters are as follows: For low pass filter: around 103 Hz For high pass filter: around 103 Hz For band pass filter: from around 103 and 104 Hz B. Time Domain Response It was then asked to compare all of the measured, calculated and simulated data for the low and high pass filters. For convenience, all of them have been graphed on the same plot in different subplots.

Figure 12. Measured Input, Measured Output and Simulated Output for High Pass Filter (Settings: Pulse Period = 12RC, End Time = 5 mS)

Figure 13. Measured Input, Measured Output and Simulated Output for High Pass Filter (Settings: Pulse Period = 2RC/3, End Time = 1 mS)

For every single one of these graphs, it is seen that they are nearly identical for both measured and calculated. The only real difference seen is in the last graph where the Pulse Period was set to 2RC/3. Here, the simulated output starts at 1V but rests at just above 0.5V later on. However, for the measured values, we they rest at just above 0.5V all throughout. This is because the simulated output is measuring from time = 0 where the voltage is read as the total voltage input. Since the circuit doesn’t have enough time to totally charge or discharge, there is a cutoff voltage that the circuit cannot surpass in either the positive or negative direction. That is the voltage that the circuit reaches later on as seen in the measured values. This effect has been observed before when a pulse that doesn’t have enough time to fully charge the capacitor was input and the signal abruptly drops just as the input drops. In terms of the rise and fall times of the circuits, from our measured values it is obvious that from a max of 1V to a min of 0V, a time of just about 2.4*10 -7 s is given, an extremely small amount of time for the input parameter to make such a jump. This does, however, give a pretty good reading for the output when graphed. In terms of the RC time constant, and the equation below:

V ( out ) =V (¿ )(e

−t RC

)

we can use our measured values to find our RC time constant by rearranging the equation above. We now get:

ln

( VV( out¿ ) )= −tRC

RC=

( )

−t V ( out ) ln V ( ¿)

(

)

All the variables on the right side can be found from the measurements at certain values of t, V(out) and V(in). Only a portion of a charging or discharging case need to be taken, then setting the initial time to zero, and making readings from that reference for all t, V(out) and V(in), the actual RC time constant would be: −5

RC=

−7.48∗10 s =1.4643∗10−4 s 0.6V ln 1.0 V

(

)

Comparing this value to the ideal value with R = 1500 ohms and C = 100 nF, −4 RC= ( 1500 Ω) ∗ (100 nF )=1.5∗10 s

which is an extremely close approximation of the actual value of RC. It

must be realized, however, that sample

measurements were taken from a data sheet of over 100,000 data points so the time value will be skewed a lot. the measurements for voltage are also pretty skewed since the same value for a long time was possessed (the oscilloscope can’t exactly measure that miniscule of a reading in that time frame) so that would throw off the RC constant as well. Either way, very good readings were attained and the error stands at:

Error=

1.4643∗10−4 s−1.5∗10−4 s ∗100 %=2.438 % 1.4643∗10−4 s

which is pretty good considering that these measurements taken by hand weren’t as great as expected.

VI. CONCLUSION Either way, this lab provided the information needed to differenciate between low and high pass filters, what they do in terms of gains, compare data results for the time and frequency domains for multiple circuits and learn what comes out of combining low and high pass filters into band pass filters.

VII. REFERENCES

[1] Electrical Engineering Laboratory I, CCNY Departent of Electrical Engineering, New York, NY Spring 2016...