Bandwidth and Signal Analysis

In an AC powered RLC circuit the power delivered to a resistor will be maximum at some resonant frequency. This property of RLC circuits allows people to tune into specific radio stations without picking up on other frequencies that are being broadcast simultaneously.

In this experiment our goal is to solve theoretically for the resonant frequency for a circuit of known elements, and compare with measured results. We will do this two times with different elements in the circuit for each trial.

To find the bandwidth experimentally we will adjust the frequency until we get a current reading Im/squrt 2

Trial 1

R = 10 ohm

C =  1 micoF

L = 2.2 mH

Bandwidth threshold is between 3393 +/- ~360 Hz

Percent error in bandwidth (1420-723)/723 *100 = 97%

Trial 2

R=100

C = 100 micro F

Percent error in Bandwidth (8070 - 7230)/7230*1100 = 11.6%

Conclusion: 

We were able to obtain experimental results that matched reasonably well to theoretical values.

Frequency Response and Filters

Mission (if you choose to except it): Calculate the gain across a capacitor (low pass filter frequency response) and resistor (high pass filter frequency response) and then compare it to the experimental results to verify the theory.

Low pass circuit design:

Note: we did not use an oscilloscope, but rather used two multimeters to measure the voltage across the voltage supply (Vin) and the voltage across the capacitor (Vout)

Set up: measuring voltage across capacitor

Set up: measuring voltage of the source (Thanks Micheal for being our hand model)

Experimental data measuring V(out) and Frequency

we maintained the Voltage of the source to be 5 Vrms

Theoretical Gain magnitudes (at 50Hz 320Hz 1280Hz 10,000Hz) 

Vout (40Hz) = 4.94 V => dividing by 5 we get 0.97 ..Hurray!

Vout (320Hz) / Vin = 2.3/5 = 0.46 ..life is good!

Vout (1280Hz) / Vin = .61/5 = 0.122 ..awesome sauce!

Vout (10,000) / Vin = 0.078/5 = 0.156 ..Epic!

We like graphs so lets plot this stuff

 High pass Filter Experimental Results

Note: The graph of this data appears to be bell shaped were the gain at low frequencies is small and you will reach a peak between 640Hz and 1280Hz and will begin to drop off as frequency in increased.

Plot of High pass data:

Conclusion:

The measured data fits the theoretical values extraordinarily well. The largest percent error we obtained was 2.5 % at the highest frequency of 10,000Hz. (0.016-0.0156)/0.016 *100 = 2.5% error. 

Unknown Capacitance in AC circuit

Freq / V

  We assume that the circuit acts like a DC source at 30 Hz

Our calculations found that the capacitor was about 3.9 micro Farads 

DC Charge and Discharge of a Capacitor

Goal: Calculate the charge and discharge times for a capacitor circuit. Then conduct an experiment that shows agreement between theory and practice.

Task 1

Rth = (Rc)(Rl)/(Rl+Rc)

Vth = Vs/(Rc/Rl+1)

Step 1

 Step 2A

We found the Value for the Capacitor to be 525 micro Farads

 Step 2B

The resistor will be able to handle this max power because it is rated for 1 W and the max power is only 19 mW

Conducting the experiment

Blurry Picture of Set up

Time to charge

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It took approximately 20 secs for the capacitor to charge to 11 volts and 2 seconds to discharge which we calculated.

Leakage Resistance

Graph of Charge/Discharge

Conclusion:

The experiment was successful. Our calculations showed to be consistent with the experimental findings of charging and discharging the capacitor

Practical Signal Conditioning

Goal:
We were given a circuit component LM35 that delivered a voltage equal to that of the temperature in Celsius. However we are American we like to do things differently so we must change the temperature reading from the Celsius scale to Fahrenheit. We want to use and op amp to perform the operation converting Celsius to Fahrenheit. F = 1.8C + 32

LM 35 

Circuit diagram

Here we calculate R1 and R2 

Complete circuit with a voltage that is .782 Volts which is roughly what we were expecting

Since the resistors we used did not have the same exact values that we found we needed in our calculations our conversion was slightly off. 

Imaginary Number Computer Calculations

Computing imaginary numbers by hand can be tedious, so we like to use tools to make life more enjoyable.

Happy days, no need to rip hair out. Here is a sample calculation done on a calculator.

RLC lab

Online Lab

Practical Intergrator

Objectives:
1. Sketch the input and output waveforms of 1kHz sine wave,
2. Triangle wave,
3. Square wave inputs.
4. Explain what it  is there for (10M ohm R)
5. Determine what would happen if it is removed.

Diagram of Circuit

4. A 10 M ohm resistor is used to dampen any feedback voltage from the voltage source.

5. If this resistor was to be removed the sine waves would no longer look like a sine waves! the peaks would be off the charts which is not good.

1. Sine wave

Note: if a small DC component was present in the input waveform

 3. kind of a square wave.

 2. Triangle waves?

Conclusion:

Integrating op amps create substantial gains and are much easier to work with than differentiating op amps because the op amp's tendencies to saturate.