MEC 220 Practical Electronics Mechanical Engineers. Stony Brook University. Lab #8 Final. Time Depen

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Lab #8 Time Dependent Circuit Analysis

04/25/2018

Table of Contents Abstract ------------------------------------------------------------------------------------------- 3 Introduction -------------------------------------------------------------------------------------- 3 Procedure ----------------------------------------------------------------------------------------- 6 Results -------------------------------------------------------------------------------------------- 7 Discussion---------------------------------------------------------------------------------------- 13 Conclusion --------------------------------------------------------------------------------------- 13 References ---------------------------------------------------------------------------------------14 Appendix ---------------------------------------------------------------------------------------15

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Abstract The objective of this lab is to learn how to use laboratory electronics instruments and to build an integral circuit, and a derivative circuit. [1] A LM324 Operational Amplifier from Texas Instrument (TI) was used in this experiment to accomplish the goal. The output signals of both circuits with 1Hz, 10Hz, 100Hz input frequencies with 1k, 5K, 10K adjusted resistances were compared. An Agilent oscilloscope provides AC voltage with an amplitude of 1 V and simultaneously measures the output voltages of both circuits.

Introduction The integral circuits will provide signal at the output which takes an integral of the input signal; and the derivative circuit will provide signal at the output which takes a derivative of input signal. The derivative circuit diagrams were given for this experiment. (Figure #1, Figure #2) [1]

Integral circuit (Integrator) [1] Derivative circuit (Differentiator) [1]

The Operational Amplifier used in this experiment is LM 324 from Texas Instrument (TI). This device Consist of four independent high-gain frequencies-compensated operational amplifiers (Figure #3) that are designed specifically to operate from a single supply of split supply over a wide range of voltages. [2] A DC power supply was sued to provide +/- 12 DC voltage to the LM324. ½ of LM324 was used in this experiment.

Figure #3 LM324 Top View [2]

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The Integrator: [3] The fundamental (ideal) integrator circuit (Figure #4) is constructed by placing a capacitor C, in the feedback loop of an inverting amplifier.

Figure #4 The Integrator (theory) [3]

Assuming an ideal op-amp, current conservation at the indicated node gives d V out V¿ I R=I C =−C ( ) R dt Eq. 01 Rearranging Equation (Eq. 01) and integrating from 0 to t, we obtain t d V ¿ (t ) −1 Eq. 02 d V =− dt V ( t ) = ∫ out ∫ RC ∫ V ( t ) dt+V out ( 0 ) out RC 0 ¿ The output voltage is thus the integral of the input. The voltage Vout (0) is the constant of integration and corresponds to the capacitor voltage at time t = 0. The frequency domain analysis is obtained by expressing the impedance of the feedback components in the complex plane. The transfer function may thus be written as −1 V out −Z c j c Eq. 03 j = = = V¿ ZR R c∗R The expression (Eq. 03) indicates that there is a 90o phase difference between the input and the output signals. This 90o phase shift occurs at all frequencies. The gain of the amplifier V out j given by the modulus is also a function of frequency. [3] = V ¿ cR Since the equivalent circuit of a capacitor for ω=0 is an open circuit, the feedback path is opened. This lack of feedback results in a drift (cumulative summing) of the output voltage due to the presence of small dc offset voltages at the input. This problem may be overcome by connecting a resistor, RF, in parallel with the feedback capacitor C (Figure #5, as well as figure #1) [3]

Figure #5 The Integrator (Enhanced) [3]

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Since RF limits the frequency gain, this circuit acts as an integrator when the frequency of the AC 1 voltage source is above fc = and as an inverting amplifier when the frequency of the 2 πRFC AC voltage source is less than fc. The Vout can be derived as follows.

IR = IC + IRF V ¿ 0−V out 0−V out = + , S= j 2 πf R RF 1 SC −V out V¿ 1 =−V out SC + = ( j2 πfCRF +1 ) R RF RF V out −R 1 = V ¿ RF 1+ j2 πfCRF [4] The Differentiator:

(

)

(

)

Figure #6 Differentiator (Theory) [4]

The current through a capacitor is given by the following equation: dv dt Eq. 04 dV ∈ ¿ =V out dt ¿ dV ∈ =−V out∗R f ⇒−R∗C ¿ dt Since iC=−i R f ⇒ C ¿ Since this differentiator works a capacitor in its input path, its performance will be subject to some effects of frequency. The equation when t=0 (time) is, iC=C

V ∈¿+f (0)=V out dV ∈ ¿ =V out ⇒ RC ¿ dt RC ¿

Eq. 05

V ∈¿=RCjω −Z R V out V ∈¿= ⇒ ¿ Zc V out ¿

Eq. 06

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An extra resistor is necessary to compensate for the monotonic increase in gain at high frequencies, (Figure #7) Since it needs to have a moderately level gain applied by the differentiator. [4]

Figure #7 Differentiator (Enhanced) [4]

R2 1 , for frequency > Eq. 07 [4] R1 R 1C 1 V out ∣ ¿ The cut off frequency, fc, is regulated by R1. When the input frequency is less than fc, the circuit acts as a differentiator and operates as an amplifier when the input frequency is greater than fc.

V ∈¿∣ ≈

1k Ohms, 5K Ohms, and 10K Ohms resistor with the 1Hz, 10Hz and 100Hz input frequencies were used in this experiment.

The input voltage is a function of time V(t)= Asin(wt) Integrator: Vout =

−1 Vi dt RC ∫

=

1 ∗A∗cos ⁡( wt ) RCw

Differentiator:

dVi = −RCwcos (wt ) dt Procedure 1) Obtain a LM324, a 10K potentiometer, a 100K resistor, two 10uF capacitors from delta kit.

2) Built the circuits as shown on the Figure #1 and Figure #2 with 1K adjusted resistance 3) Set and connect the 12V DC power supply to the LM324 where +12V at Pin Vcc and -12V at Pin GND at the LM324 (Figure #3) 4) Set the AC power supply to 1Vpp at 1Hz 5) Connect the Ch.01 of the oscilloscope to the AC power supply.

6) Connect the Ch.02 of the oscilloscope to the output of the Integrator.

7) Connect the Ch.03 of the oscilloscope to the output of the Differentiator.

8) Power on the oscilloscope and open the UltraScope application on the PC. 9) Save .CSV files to desktop. (*Detail of using UltraScope application on Appendix page. 15) Vout = -RC

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10) Repeat the procedure 9) with the frequency of 10Hz and 100Hz. 11) Turn off the power supplies and replace the Ri1 and Rd2 to 5K and Repeat the procedure 8) to 10) 12) Turn off the power supplies and replace the Ri1 and Rd2 to 10K and Repeat the procedure 8) to 10) Results CH 1Input 1 Vpp CH 2 Integrator Output CH3 Differentiator Output

1K Ohms 1Hz Integrator 1.00E+01 8.00E+00 6.00E+00 4.00E+00 2.00E+00 0.00E+00 -2.00E+00 -4.00E+00 -6.00E+00 -8.00E+00 -1.00E+01 INTEGRATOR VOLT

Vin VOLT

THEROY OUT VOLT

Figure⁡8⁡

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1K Ohms 1Hz Differentiator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 DIFFERENTIATOR VOLT

Vin VOLT

THEORY OUT VOLT

Figure⁡9

1K Ohms 10Hz Integrator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH2 VOLT

Vin Volt

Theory Out Volt

Figure⁡10

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1K Ohms 10Hz Differentiator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH3 VOLT

Vin Volt

Theory Out Volt

Figure⁡11

1K Ohms 100Hz Integrator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH2 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡12

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1K Ohms 100Hz Differentiator 2.50E+00 2.00E+00 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 -2.00E+00 -2.50E+00 CH3 VOLT

Vin Volt

THEROY OUT Volt

Figure⁡13

5K Ohms 1Hz Integrator 2.00E+00 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 -2.00E+00 CH2 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡14

5K ohms 1Hz Differentiator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH3 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡15

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5K Ohms 10Hz Integrator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH2 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡16

5K Ohms 10Hz DIfferentiator 2.00E+00 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 -2.00E+00 CH3 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡17

5K Ohms 100Hz Integrator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH2 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡18

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5K Ohms 100Hz Differentiator 1.50E+01 1.00E+01 5.00E+00 0.00E+00 -5.00E+00 -1.00E+01 -1.50E+01 CH3 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡19

10K Ohms 1Hz Integrator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH2 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡20

10K Ohms 1Hz Differentiator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH3 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡21

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10K Ohms 10Hz Integrator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH2 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡22

10K Ohms 10Hz Differentiator 4.00E+00 3.00E+00 2.00E+00 1.00E+00 0.00E+00 -1.00E+00 -2.00E+00 -3.00E+00 -4.00E+00 CH3 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡23

10K Ohms 100Hz integrator 1.50E+00 1.00E+00 5.00E-01 0.00E+00 -5.00E-01 -1.00E+00 -1.50E+00 CH2 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡24

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10K Ohms 100Hz Differentiator 2.00E+01 1.50E+01 1.00E+01 5.00E+00 0.00E+00 -5.00E+00 -1.00E+01 -1.50E+01 -2.00E+01 CH3 VOLT

Vin Volt

THEORY OUT Volt

Figure⁡25

The cut off frequency fc for the integrator circuit is 1 fc= =¿ 0.16Hz, 2 πRFC so the integrator will act as an integrator when the input frequency is 1Hz, 10Hz, and 100Hz. In the differentiator, 1 fc= =¿ 159.16Hz, 2 πR 1C so the differentiator will act as an differentiator when the input frequency is 1Hz, 10Hz, and 100Hz. Discussion For an integrator to be useful it must be allowed to be reset to zero. Since the output is stored in the charge of the feedback capacitor, all we need to do is to short out the capacitor in order to reset the integrator. Integrators are very sensitive to DC drift; small offsets lead to a steady accumulation of charge in the capacitor until the op amp output saturates. [3] From figure 25,

Conclusion   

Vout of the integrator is inversely proportional to frequency and resistance.

Vout of the differentiator is directly proportional to frequency and resistance.

Integrators and differentiators can operate as amplifiers depending on the input frequency.

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References [1] Noah D. Machtay, Ph.D., P.E, Lab Manual: MEC 220 Lab #8, Spring 2018 [2] LMx24, LMx24x, LMx24xx, LM2902, LM2902x, LM2902xx, LM2902xxx Quadruple Operational Amplifiers Data Sheet, Texas Instrument, March 2015 [3] Chaniotakis and Cory. Operational Amplifier Circuits, 6.071 Spring 2006 [4] Nash Reilly, More First Order Op-amp Circuits - Differentiators, September 07, 2015

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Appendix [1] How to acquire data from the Rigol oscilloscope using the PC 1. Turn on and log into the computer 2. Turn on the oscilloscope 3. Start the UltraScope software 4. Select “Search for Instrument” 5. If a message prompt states that “Ultra Sigma is not running” click “OK” 6. Click “Start Ultra Sigma” 7. Identify the “DS1054Z . . .” instrument, right click on it, and select “UltraScope” from the menu 8. Conduct your experimental work as necessary 9. To capture a data set, press the RUN/STOP button on the oscilloscope so that it turns red; the data displayed on the osilloscope is now frozen as of the moment you pressed the button.

10. To record a data set to the computer, right click on the on-screen output display in the UltraScope application and select “Save Channel . . .” from the menu a. Select the channels you would like to include in your data set b. Select the comma seperated variable file format (.CSV); spreadsheet programs like Excel know how to open these, making your life easier. c. Enter a file name, and save your data. d. Review your data to ensure it is what you were trying to record before you move on.

[1] Noah D. Machtay, Ph.D., P.E, Lab Manual: MEC 220 Lab #8, Spring 2018 16

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