04/26 - Lab 19: Inverting Differentiator

Lab 19: Inverting Differentiator

In this lab, we explored another mathematical operation that can be performed by op amps; differentiation. Essentially, you can set up the op amp as an inverting amplifier, but replace input resistor with a capacitor. Essentially, if you feed a time-varying signal into the differentiator, it will output the derivative of the signal function with respect to time.

We used a 1.0 uF capacitor, and were instructed to choose a resistor that would give us a gain of -1. Since gain = -RC, we initially chose R to be 1.0 MOhm, but for some reason the op amp would saturate, so we kept swapping it for lower and lower resistors until we achieved a circuit that did not saturate. We also had to turn down the input voltage, but eventually we found that a 1 kOhm resistor and a 100 mV supply did the trick. The circuit, as well as the WaveForms oscilloscope, are pictured below:

Actual Resistor and Capacitor values

Photo of the actual circuit

Sinusoidal wave being fed into the circuit. f = 1 kHz, V = 100 mV

Yellow = Input Voltage, Blue = Output Voltage

f = 1 kHz

 f = 2 kHz

f = 500 Hz

In Class Examples

1. Sketch the input and output waveforms for 1kHz sine wave, triangle wave, and square wave inputs.

2. Given vC(t) = [5 u(t) +6 r(t)] V, find iC(t) for the circuit shown.

3. Find the step response v(t) and i(t) to vs = 5u(t)V in the circuit below:

4. Example: Find v0(t) for t>0 in the circuit below.

04/14 - Lab 17: Passive RC Circuit Natural Response, and Lab 18: Passive RL Natural Circuit Response

Lab 17: Passive RC Circuit Natural Response

In this experiment, we explored how capacitors change a circuits response to a sudden change in voltage. Capacitors store energy in an electric field, which prevents sudden voltage drop throughout the circuit. We set up a simple RC circuit and test manual switching, as well as sending a square wave signal through it and measured the voltage drop across the capacitor.

Diagram of the circuit modeled

Before setting up the circuit, we ran some calculations to determine the theoretical time constant, Tao. We used R_1 = 1 kOhm, R_2 = 2.2 kOhm, and C = 20 uF. 

Actual resistor values used

Actual breadboard circuit

Charge voltage across the capacitor using trigger

Discharge voltage across the capacitor using trigger

Voltage over time across the capacitor when implementing a square wave

The discharge time of a capacitor is typically ~ 5 time constants, which we calculated to be 15 ms. In other words, the theoretical charge/discharge time should have been approximately 75 ms. The graph shows that the total charge/discharge time of the capacitor in the circuit was nearly spot on.

Lab 18: Passive RL Natural Circuit Response

In this experiment, we explored how inductors change a circuits response to a sudden change in current. We ran essentially the exact same lab as the one above, but we swapped the capacitor for an inductor and used Ohm's law to calculate the change in current across the circuit when the voltage supply is switched on and off.

Diagram of the circuit

Calculations to determine the time constant,Tao

Actual breadboard circuit

Inductor discharge using manual trigger

Voltage across inductor using a square wave

In the calculations above, the time constant was found to be 1.45 microseconds, and thus, the time for discharge should be ~ 6 - 7 microseconds. The manual trigger showed this result exactly, but the square wave did not, likely because the frequency was slightly too high.

In Class Examples

1. Given the circuit below, find Vc(T) and Ic(t) given that Vc(0) = 10V.

2. The switch in the circuit shown below has been closed for a long time. At t = 0, the switch is opened. Calculate i(t) for t > 0.

04/12 - Lab 16: Capacitor Voltage Current Relations

Lab 16: Capacitor Voltage-Current Relations

The purpose of this lab was to gain an understanding of the effect a capacitor has on the change in voltage of a circuit. Capacitors store energy in an electric field, and are used to prevent voltage from instantaneously changing along a wire. In this lab, we will build a simple LC circuit in order to observe how the voltage is affected with a capacitor in the circuit.

For the pre-lab, we determined the voltage-current relationship of an LC circuit with a sinusoidal voltage input, and a triangle wave input.

We built a very simple circuit, containing a 100 Ohm resistor in series with a 1 uF capacitor.

We used the waveform output to feed the circuit a 1 kHz and 2 kHz sinusoidal wave, as well as a 100 Hz triangular wave. We used the oscilloscope built into the WaveForms program in order to see the voltage input versus the voltage across the capacitor.

2 kHz sinusoidal wave

1 kHz sinusoidal wave

100 Hz triangular wave

In Class Examples

1. Obtain the energy stored in each capacitor shown below under DC conditions.

2. Find the equivalent capacitance for the collection of capacitors shown:

04/07 - Lab 15: Temperature Measurement System Design

Lab 15: Temperature Measurement System Design

A cascaded op amp set up is a head-to-tail arrangement of two or more op amps in a single circuit. The output voltage from the first op amp becomes the input voltage for the second, etc. In this lab, we designed a temperature measurement system which implements a wheatstone bridge, pictured below:

A thermistor was used as R + delta R, since it its a resistor that changes with change in temperature. A poteniometer was used to match the initial resistance of the thermistor, which balanced out the bridge, causing V_ab = 0. As the thermistor gained heat, there becomes a difference in resistance, and thus a potential difference between a and b. This voltage will be directed into an op amp, which was used to tone down the sensitivity of the wheatstone bridge.

We began by drawing a diagram of our proposed design and figuring out what the gain needed to be in order to have our temperature measuring system range between 0 and >2 V:

In order to achieve the appropriate voltage range, we determined that the desired gain should be 4. This means that the lowest output voltage should be 0, and the highest output voltage should be ~2.3V. The actual circuit is pictured below:

Wheatstone Bridge

Full Circuit with balanced wheatstone bridge

Output voltage achieved was 2.64 V, near the expected 2.3 V

In Class Examples

1. Practice: Find io as a function for vs.

04/05 - Lab 14: Summing Amplifier

Lab 14: Summing Amplifier

In this lab, we used the OP-27 as a summing amplifier. Op amps can be used to perform various mathematical operations. In the previous lab, we used it for multiplying the input voltage by a constant. In this lab, we will use them to sum multiple input voltages.

For this summing amplifier setup, the output voltage should follow the equation derived below (but with two terms instead of three):

In this picture are the actual resistance values recorded, along with a diagram of the setup we chose for this lab.

Actual circuit setup is shown below:

We set V_b to 1 V, and used the wavegen from the analog discovery to vary the voltage of V_a. We chose R_1 = R_2 = 3.6 kOhm, and Rf to be half that value, or R_f = 1.8 kOhm, so that the sum of the two voltages is cut in half in order to prevent the op amp from saturating. In mathematical terms, this circuit should yield the following voltage: V_o = -1/2(V_a +1). The values of V_a and V_o are recorded in the table and corresponding graph below.

In Class Examples:

1. Predict: The OP27 is powered at 9V with V- connected to ground. A -100 – 100mV square wave is input. R1 ~2.5K and R2~7.5K. Make a quantitative sketch of the output.

2. For the circuit shown, Calculate vo if vs = 0:

3. Design a difference amplifier to have a gain of 2 and a common mode input resistance of 10 k Ω at each input.

4.