Day 14 Passive RC Circuit Natural Response

Chapter 7 of our textbook discusses the natural response of an RC circuit as a capacitor charges and discharges. In order to observe this, we built a circuit in which we can switch the voltage from on to off and use a trigger in our Analog Discovery oscilloscope to capture the plot of the discharge/charge.

Our pre-Lab below shows two different step function scenarios. The first is a circuit in which the voltage source is disconnected. The second shows the voltage source being turned off by switching. We compare the two different effects of each scenario.

The first time constant we solve for below is for scenario A where the source is simply removed. This creates a circuit where all we have is R2 and C because no current will flow through resistor R1 since it is now an open circuit.

The next time constant we solve for is when the voltage source is switched from one wire to the next. This leaves resistor R1 as a passive element in the circuit, rather than an open circuit. The effects on time constant Tau are shown below.

It is evident that scenario A will discharge faster because the load resistance will absorb more of the energy stored than it will in scenario B where we have two resistors in parallel for a lesser resistance.
Now that we calculated the theoretical values of Tau we are ready to move on.

Below is the build of the circuit:

Placing 3 capacitors in parallel allowed us to have an equivalent capacitance of roughly 22 uF. Switching the voltage from on to off results in the discharge of the equivalent capacitance.

Below is the oscilloscope readings and a plot of the data, including the voltage value after one time constant tau.

I extracted the critical data from our Diligent Oscilloscope and plotted it using Matlab, placing the red asterisk at our measured value of Voltage after one time constant.

The Time constant from our measurement was roughly .020 seconds. Our theoretical value for tau was calculated to be .015 seconds. Our measurements were off by 33.3%. This can be directly a cause of experimental errors, inconsistencies in our resistor and capacitor values, and also in the measurement of tau.

I do not have any pictures or data from the switching of the second scenario. However, below is a plot of the step response of introducing a voltage source to the same capacitor in the circuit.

While our data and circuit elements were not ideal, the goal of the project was satisfied: to observe and understand the natural response of an RC circuit.

Day 13 Capacitor Voltage-Current Relations

The purpose of today's lab is to measure the relationship between capacitor voltage and capacitor current. The voltage applied will be three different types of time-varying signals: sine wave, triangle wave, and square wave voltages will be applied. Because our analog discoveries do not measure current we will use a math function to calculate the current.

From our studies we know that


Ic = C dv/dt

Below is a sketch of our pre-lab activity. Given a sinusoid and triangle wave input voltage, draw the resulting current. Current in a capacitor is proportional to the derivative of the voltage and our sketches reflect that below.

Lab :

The components we used in this lab were simply a 1uF capacitor and a 100 Ohm resistor connected in series. Using the oscilloscope function of the Analog Discovery we measure the output voltage of the resistor and the capacitor. By adding a Math Channel of (VR/100(ohms)) we can also see the plot of the current. It mirrors what we predicted.

Applying a 1kHz sine wave to the circuit results in a capacitor current that is proportional to its derivative. Vout for the capacitor is the plot of C2 above, and the current is the plot of M1. As expected, we see a plot of a sine and cosine wave.

Below is a 2kHz plot of the same:

Below is our successful circuit.

We were able to successfully measure, observe, and apply current- voltage relationship through a capacitor.

Day 12 Temperature Measurement Design System

The objective of this lab was to design a simple temperature measurement system which outputs a DC voltage that indicates temperature. We again used a thermistor to indicate the temperature based on the variable resistance provided by the thermistor as temperature changes. Today, we introduced a Wheatstone Bridge circuit that allows us to take a change in resistance and turn it into a change in voltage. The final stage of this circuit will include a difference amplifier (OP 27) to increase the overall sensitivity of the temperature measurement system.

Pre Lab: Balancing the Wheatstone Bridge

The following is a drawing of our Wheatstone Bridge circuit. Like mentioned before, this circuit is useful for taking a change in resistance and outputting it as a voltage. Resistances R2 and R3 are equivalent. We have a Potentiometer labeled Pot and our thermistor's room temperature resistance as Rth. The first step of this lab was to balance the bridge so that there is no voltage measured at our output nodes across a and b. To balance it, we attached our multimeter to measure Vab, and adjusted the resistance of our potentiometer until there was no potential difference.

In practice, our 10K potentiometer did not have a high enough resistance to balance out the circuit. We were required to add another resistance in series with it to simulate a higher POT value. Below is our finished and balanced Wheatstone Bridge circuit. We were able to balance it to 0.02 V of output.

Difference Amplifier

The second half of our temperature measurement design system is an op amp. Here we utilized a difference amplifier in order to measure the change in voltage from our input voltage to the change in voltage due to our thermistor and Wheatstone bridge.

Above is a scheme for our difference amplifier. Having 10 k Ohm resistors for R1-R4 simplifies the output of the voltage measured: the ratio of Vout to the input voltages is simply,

Vout = Vb - Va

Now we are ready to put the entire circuit together.

Temperature Measurement System
Together, the Wheatstone Bridge and the Difference Amplifier work in conjunction to form a simplified temperature measurement system. The complete circuit is pictured below.

The output voltage from our Wheatstone Bridge (+Vab and -Vab) are the voltage supplies for the difference amplifier. We have three different voltages throughout this circuit: our original supply, our Wheatstone Bridge Output, and then the output voltage from our difference op amp.
At this point, this temperature measurement system is the most practical use of op amps we have seen so far.

Below is our system in action.

Our design objective of an increase of a minimum of 2V was not entirely achieved. Instead, we observed a change in our voltage closer to 1.3 V. I'm still uncertain about what could be changed to achieve the minimum of 2V. Perhaps our resistance values could be changed. All in all, it was a new and challenging experience learning about the Wheatstone bridge and the Difference Amplifier.

Day 11 Summing and Difference Amplifiers

Today we continued our journey with operational amplifiers. We've learned so far that depending on how you connect resistors and voltage sources in circuit with an op amp, you will achieve different mathematical operations out of them.

Summing Amplifier

The objective of the first portion of our lab is to model a summing amplifier and to analyze it in an actual circuit. Ideal summing amplifiers (which we will consider today) have a relationship between V-in and V-out as follows for the circuit below:

Reducing this equation using nodal analysis we get:

For our pre lab we were instructed to design a circuit that would fulfill the parameters: we would have a constant 1-V supply for Vb and a variable supply for Va. Our goal was to be able to plot the V-out while remaining in the linear region without hitting either +- saturation. Below was our circuit. 

As always, the funnest part of the lab was building the circuit. Below is our successful build. 

From supplying variable input voltages and recording the consequent output voltages, we collected the following data for the summing amplifier:

Below is a plot of that data. Our measured value is graphed in blue and our theoretical in red. 

Clearly, we satisfied the parameters of the lab. Between -4 and 5 volts, our summing amplifier never reached saturation, + or -.

Difference Amplifiers

This lab aimed at using an op amp to perform subtraction between two voltages. For ideal op amps we know the relationship between input and output voltage is as follows:

Since (roughly) R1 = R2 and R3 = R4:

VO = V2 - V1

Below is the difference amplifier circuit that we constructed. 

Below is our measured data for the voltage out, as well as the theoretical values that we expected. This data has a larger discrepancy from our theoretical than our summing amplifier had. This is seen below in the graph and also in our plot.

Below is the plot:

In conclusion, we successfully were able to model a difference amplifier using the OP 27 op amp. Our measurements stayed within the linear range before reaching saturation, which was the aim of the lab. There are many ways to utilize a single op amp depending on the surrounding circuit elements.

Day 10 Inverting Voltage Amplifier

Today is our first introduction to operational amplifiers, or op amps for short. Our first lab is centered on an inverting op amp, or an op amp that performs multiplication by a negative constant. Below is our circuit diagram that includes the OP 27 and its representation in the circuit. We have designed the circuit so that the Vout = -2 * Vin. The blue numbers around the Op Amp represent the pin terminals on the actual op amp. We have a supply rail from -5 to +5 that is powering the op amp.

Below are the actual measured value of the feedback resistor and the input resistor, respectively. Our ratio, therefore, is not exactly Vo = -2Vi, but very close.

Below is our successfully designed circuit. We have all of the resistors, input voltages, and supply rail voltages connected at the right pins. We had to connect the op amp across the bridge of the breadboard because otherwise the pins would be connected to each other which would make the op amp inoperable for what we needed.

The last step was to measure and plot our measurements for Vin vs. Vout, using Matlab. The graph of the data below demonstrates the properties of the Op Amp very well because we see that the amp saturates when approaching the supply rail voltages, and is inverted from the input voltage.

Op Amps have been some of the most interesting circuit elements to work with so far. They operate the same internally, but depending on how you design the circuit around it is what changes the operation of the op amp. Looking forward to learning more about the Op Amp.

Day 9 Maximum Power Transfer and Non-Ideal Power Sources

Max Power Transfer

Today we explored the idea of Maximum Power Transfer and Non-ideal Power Sources. Maximum power transfer occurs when the resistances inside the voltage source is 0, or close to it. That is what we would call an Ideal power source. If it were non-deal, then there would be a value for an internal resistance.

Our goal is to maximize the power delivered to our load resistor.

Our measured value of P is 2.82 mW.

We can see that using the simplified equation for ideal power sources, P = V^2/4Rth, our theoretical and measured values are almost identical.

Non-Ideal Power Sources

Because we don't live in a perfectly, ideal world, we have to practice our circuit analysis with more real life conditions: this means that our power sources are not going to be ideal. This lab aimed at experimentally exploring the behavior of non-ideal power sources.

Day 8 Thevenin's Theorem

Today's lab experimentally investigated Thevenin's Theorem. As a pre-lab, we wanted to determine the Thevenin Equivalent circuit of our given circuit diagram. We then will measure the load voltage in the original circuit, and then measure the voltage of the Thevenin equivalent circuit.

The semi-complex circuit of resistors and voltage supplies is below. Thevenin's theorem will allow us to simplify the circuit with a variable load, R thevenin, and V thevenin.

Our next task was to measure the current that would flow through our thevanin equivalent circuit, or the resistor R_L which we would measure with two different resistors. Theoretically our value was 54 micro-Amps. Experimentally our value was 55 micro-Amps.

Our second value of R_L was 8.1 k Ohms. Again, or theoretical values and experimental values were very close in magnitude.

For the last section of the lab we were required to create a circuit with two potentiometers, one that was fixed. Below is that circuit.

We measured the Voltage as the resistance was varied using the variable potentiometer. Below are those measurements.

Power as a function of load resistance is graphed below.