Tutorial on Second Order Systems

Capacitor Charging and Discharging

Introduction:
Capacitors are electrical components that store energy in the form of an electric field. This lab demonstrates the charging and discharging pattern of a capacitor.





First, we calculated expressions for a non-ideal charging/discharging capacitor circuit.

 calculated for the component values of an ideal capacitor circuit such that the lab's parameters were met.

2.5 mJ of  Energy into the capacitor.

The variable resistance box has a max power output of 1W, which is more than enough for how we are using it.

Because the oscilloscope was set on continuous recording, we couldn't acquire the charging graph, so instead we used a stopwatch and a voltmeter to measure the voltage of the charging capacitor within 20 seconds.

Materials 1 Voltmeter and 1 stopwatch (optional), 1 oscilloscope, 2 variable resistors, 1 33 microfarad capacitor, cables.

The circuit setup, we found that the voltage of the charging time at around 20 seconds to be 11 volts.
The time taken for the capacitor to discharge took about two seconds which was expected.



Leakage Resistance:

Error calculation:

The charging and discharging graphs, how it should appear if the oscilloscope read once and not continuous.

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Conclusion: The experiment sucessfully proved the validity of the equations used for the charging and discharging of a capacitor. The leakage resistance was found to be roughly 10 times greater than the charging resistor, which is expected due to the leakage resistor being parallel to the capacitor.

Integrating and Differentiating OP Amps

Introduction: 

We processed signals through a series of circuits with capacitors, resistors, and OP amps.

Integrating OP amp, Yellow = output Red = input voltage. Note the decrease in amplitude and similar frequencies.  

 

Saturation. The voltage applied to the OP amp wasn't enough to apply the necessary change, so the output became a square wave with a capacitor-like exponential decay. 

Differentiating OP amp, not the increase in amplitude and similar frequency. The phase is noticeably shifted.

Practical signal conditioning

Objectives: Build a circuit that is able to show conversion of temperature from degrees Fahrenheit to degrees Celsius, and also get more familiar with the use of op amps.

Procedure:

Step 1. Testing: We will connect everything to check that our semiconductor LM35(Temperature sensor) is working properly assuming 1C equals 10mV.

We observe that it is working properly and now we can move ahead to the actual experiment.

The function that lets us convert from degrees f to degrees c and vice-versa is the following and we will use it to proceed with our calculations.

    Tf=1.8 Tc+32

This is the function transformed in therms of resistance and voltages where vf is the temperature in fahrenheit and vc is the temperature in Celsius. We calculate our resistances from a ratio of 4/5 so that means than R2 has to be 0.8 times smaller than R1.

Step 2: Determine R1 and R2

We decided to use 8.2k ohms for R2 and 10K ohms for R1 since we were not able to find exactly the ratio mentioned before. 

Now we have to set up the following circuit and also we found Vref to be -0.4V 

Step 3: Set up the circuit

And this is how we did it. We later had to do a voltage divider because Vref is too small for us to provide to the system and we ended up getting Vc =24mV and Vc=76.2mV

Step 4: Calculations

After that we find the percent to see how far is our circuit from being very accurate and efficient

Observations: Our circuit shows a 1.33% error which is pretty good. We consider it to have successfully achieved the main objective. We surely became more familiar with op amps and also with the semiconductor LM35 and surely understand how useful are these two for applications in real life.

Operational Amplifiers I

Objective:
-Understand the use of operational amplifiers
-Using our understanding of operational amplifiers to design a circuit with certain specifications

Specifications or assumptions:

-Sensor with an output ranging from 0-1V

-Signal conditioning circuit must have an output range between 0 to -10V

-No more than 1mA current from the sensor

-Op Amp power supplies should supply no more than 30mW

Procedure:

1. Visualize this circuit and do the following predictions 

Since our Rf has to be 10 times larger than Ri so through calculations we found Ri to be 1k ohms and Rf to be 10k ohms.

Then we find the resistances that help us satisfy the requirements previously stated

Since R Thevenin has to be at least a factor of 20 times smaller than the resistant for R1, we will have to change the values since our calculations do not meet this requirement 

Since we have to use resistors that are available in our lab, our resistors numbers have to change

Step 2: Build the circuit using a breadboard 

And the following picture is a closer look at the breadboard

 Step 3: Change the resistance to get different values for your voltage in

Step 4: We find the power delivered and the percent error

Observations:
We had a 85.9% which is not good at all. This means that the operational amplifier is not perfect; Therefore we must not assume that the internal resistance is infinite. 

Thevenin equivalents

Objectives:

- Understand the usefulness of Thevenin equivalents

- Identify the percent error between using the complete circuit or Thevenins equivalent method

Procedure:

Step 1: do calculations. Find R Thevenin 

 And then find V Thevenin 

Step 2: Use nodal analysis to find R Thevenin 

Step 3: Determine the smallest permissible R_L2 using a. Voltage divider b. Short circuit c. Open circuit by inspection

Step 4: Now we set up the circuit and record the data 

Data: 

Observations: We see that the percent error is significantly small which lets us come with the conclusion that by using Thevenins equivalent we are able to find accurate and precise answers.