04-14-2016

04-14-2016: Passive RC and RL Circuit Natural Response

I. Introduction:

Today, we spent most of our time to discuss the RC and RL. We also reviewed the first differential equation to find out the relationship between current and voltage across the inductor.

Given this simple circuit, our goal is to find out the equivalent inductor. For the inductor, a rule of thumb here is to follow the same as the equivalent resistors. For my notation, + indicated they are in series, not adding.

II. Formula Derivation:

Here is our derivation of the voltage across the capacitor over the course of time. Notice that when the switch is closed and DC current, the capacitor acted like an open circuit. If the switched is open, the capacitor will be fully charged and has its own voltage, ideally a straight line, but in real world, the voltage will be dropped over the course of time

Here is our derivation to find out what time the voltage across the capacitor would be left with 1% of the original voltage, which means it would take 5tau. tau=R*C

After we got the voltage, the current would be voltage divided by the resistor, the power is the multiplication of both voltage and current

Given a simple circuit, our goal is to find out the V(t) and i(t). First of all, we have V(0)=10 V already. We then had to find tau and got our V equation, I=V/R

Here we need to find the smallest tau and the largest tau, since tau is directly proportional to RC, we picked the lowest values for RC and largest values for RC respectively

II. Prelab:

Given a simple circuit,we had to tau, we first find the Rth in the circuit, then use that to find out tau

II. Voltage and current in capacitor and inductor:
1. The capacitor:
For this lab, we were trying to measure the current and voltage of capacitor and inductor over the course of time. We change the voltage by two ways: mechanically, which means that we had to basically take in take out the voltage supply power; or we can use the square wave voltage built in feature of the analog discovery

Here is how the graph looks like when we did it manually, the region we concerned about is the going downslope of the graph

Here is what it should look like if we the built-in feature of square wave voltage instead of doing it mechanically. 

Here we used triggering and single acquisition to find out the slope of the curve

Here is our data. For part A (mechanically), we found the percent to be 35.88%, which was not really favorable in terms of acceptable range. It could due to that the value we were trying measure to small therefore a small change of number could cause a great impact to our percent error

III. Some More derivation

Here we derived the formula to find the current across the inductor

After we got the formula of the current. We know that V=l*di//dt. We took the integral to find the formula for the voltage

Here is our setup for this lab, this setup applied for both the capacitor and the inductor 

The percent error is about 35% for the manual experiment

Here is the portion we need for the inductor. The percent error is about 28% in this case

here is the setup for the inductor circuit

III. Analysis:
Since we don't have an ideally inductor or capacitor, the current or voltage are not perfectly trapped in the circuit, over the course of time, those values would be dissipated into the resistors, causing a great percent error in our lab. Moreover, since we dealt with small value of time division, it could make our measurement really sensitive to the percent error
IV. Conclusion:
For this lab, we spent our time discussing the behaviors of voltage and current in capacitor and inductor circuit. We derived many equations using derivative and integral to the review of what we have learned in our 4b class. In the lab, we verified our knowledge about natural response of the passive RC and RL circuit. The percent error is great due to the time is too small to measure. For the rule of thumb, the capacitor acts like the open circuit with DC, and the inductor acts like short-circuit.

04-12-2016

04/12/2016

Review of Capacitor and Inductor Lab

I. Introduction:

Today we spent most of our time reviewing all the knowledge we had learned in 4B class,specifically the capacitors and inductors, their relationships with currents, resistors, and voltage.

Professor Mason introduced us to rage bridge, which was the complex combination of capacitors. Due to the fact that the bridge had a large spark because the mechanical engineer designed the bridge, not an electrical engineer.

 2. Design of capacitor:

Here is our simple design of capacitor. We believed that we could design a variable capacitor by pumping in and out the dielectric oil; however, the downs side is that the dielectric oil would be dried out eventually

We knew that I=cdV/dt. P=VI=Cvdv/dt. We then take the integral with respect to t to find the total energy

Given the current graph with respect to time of a capacitor. We had to plot a graph of voltage with respect to time. Our approach was to use the formula I=c*dV/dt. therefore, V=Ict

Given the circuit with a capacitor on the top. Our goal is to find the current of i2. Since we already knew the current in the loop 1, we used mesh analysis to find the current in loop 2, therefore, we could find the voltage across resistor R2

We were given some more sinusoidal graph of voltage and triangular wave of voltage. Our goal is to find the graphs of current over time, using the fact that I=cdV/dt. We could find the corresponding graph of current over time

 II. Lab: Capacitor and Voltage in the circuit
1. Introduction:
 Our goal is to find out the relationship between the capacitor and the current pass through that capacitor over time. Using the two graphs we had derived previously for the prelab, we expected our experimental labs should look the same

The blue graph measures the voltage across the capacitor. The orange graph measures the voltage across the resistor, notice that the orange graph is a sin function, whereas the blue graph is a cos function, which indicates the voltage across the resistor has a phase shift of pi/2 

Here is the same setup but we used the 2kHz for the frequency, these are the graphs of voltage and current across the capacitor

Here is the picture of capacitor voltage waveform. We input 100Hz triangular voltage here

Here is our simple setup for the lab

Our data for the lab, we checked with the other group to make sure our data is correct. Notice that for part A and part B, the voltage across C1 is closely followed by the voltage across C2

 III. Capacitor explosion:
Professor Mason demonstrated a capacitor explosion, what he did basically applying  a large voltage a capacitor, a capacitor would be exploded

what was left after the explosion

Here is a picture of the polar capacitor.

 IV. Equivalent capacitor:
Here is a rule of thumb: equivalent capacitor is the opposite of equivalent resistor, which means that we have to add when they are in parallel and inverse when they are in series

Given a simple of capacitor, we had to find the  equivalent capacitor, which in this case we approached right to left.

Here is a short summary of all the formulas we should know

 V. Summary:
For today lab, we spent most of our time reviewed all the knowledge we had learned in 4B class. We did derive some equation using derivative and integral to get  the current and voltage across the capacitor. For the lab part, we applied different voltage sources to observe the change in current using the built in scope of the analog discovery

04/07/2016

04/08/2016: Cascaded Op ams and Temperature Measurement System Design

I. Introduction:

Today we spent most of our time studying op ams, which indicated the connection between many op ams in one circuit. The system in return would be more complex but not impossible to analyze using nodal analysis. For today lesson, we would have to use many combinations of summing amplifier op ams, inverting and so on

Here is a picture of inverting cascard op ams.

Professor Mason asked us to determine the type of op ams circuit  he put on the screen. We chose inverting for a and b due to their terminal, c is summing because has more than one op am

Here is a picture of cascaded op ams. We found the V1 using the formula we nodal analysis, and using the same approach to find V_2, notice that they were all determined by Vs and those resistors

We plugged the numbers into the formulas we had derive previously solving for V_2 and i_0

II. Temperature Measurement System Design
1. Introduction:
For this lab, we used the wheatstone bridge circuit to convert resistance to a voltage change. The change in resistance was determined by the thermistor. The change in voltage was the change Voutput on the side of the op am

For this picture, my lab partner Carlos derived the expression for Vab, 

Here is the drawing for our circuit, it took us 15 mins to decide our values for the resistors, we also went around to check with other groups

The last equation is the most important equation for this lab, for this part, we used Maple to solve for Vo symbolically

Here is my setup for the wheatstone. 

Here is the direct view of my wheatstone design

d

Last but not least, here Professor Mason introduced use to use bit representation for the resistors, except that the power would be negative instead of positive, made them a fraction

IV. Summary:

After many experiments, we believe nodal analysis worked best compared to other techniques that we have learned so far. For today lab, we spent most of our time understanding the cascaded circuit. We derived many formulas for summing amplifiers, difference op ams, and so on. For the lab, we only did the wheatstone lab, but it took us a lot of time to figure out the right design, the good thing about the wheatstone is that we could use it to convert the change in resistance to the change in voltage

04/05/2016

04/05/2016: Phase Shift and Continue with Op Ams

I. Introduction:

Today, we started out with a simple graph of sinusoidal wave. The question is how would the function change if shift the graph to stay totally positive. We spent part of our time to do some op am circuit, including the inverting, non-inverting, summing, and difference.

First, I wrote down the general equation. I missed the sin there so it should be Asin(kx+wt). The second graph is exactly the first one except its amplitude is 25 and it shifts up 2.5 unit.

Next, we were given a simple op am circuit, Professor Mason to find the ratio between Vout/Vin, in this case, we have inverting op am circuit, therefore, the ratio would be R2/R1=-3

Here we were given another inverting op am circuit. Since Vs is connected to the ground, the voltage of the negative input would be equal to the Vs=0. Using nodal analysis, the would be able to find the Va and Vo. Vo is always greater than or equal to Vin for the inverting op ams

Here is the formula for non-inverting op am cicruit, we have V1=V2=Vi, using that application, we simplified it to Vo/Vi=1+Rf/Ri

Here is the formula for the summing amplifier circuit, Notice that as we add more branch of voltage, we simply need to add up more terms into the formula

II. Experiment with the inverting and summing amplifier op am circuit:

The op am circuit here is actually the combination of the summing amplifier and inverting op am circuit. Notice that the input is given the summing amplifier and the output is given by the inverting op am circuit.

Here is our set up for this lab, next time we should utilize more short wire to make it easier to debug. A good matching color with the analog discovery is also a good practice to follow

Here is our result, notice that since  we have inverting op am circuit, the output would be the opposite sign of the input we choose R1=R2 to simplify our work

For this problem, the output is more complicated than we thought. It was actually one of the mastering engineering homework assigned on that day. R4/R3 due to the fact that

Here we were given a difference amplifying circuit. By applying the nodal analysis, we actually be able to find the gain of 30.33

V. Summary:
For today lab, we spent most of the time understanding more about three more types of op ams: non-inverting, summing amplifier, and difference. With the nodal analysis approach, we were able to derive formula for all those types of ams. We also spent some of our time do the summing amplifier lab at the end to verify that our derivation is correct