05-26-2016

Lab :Signals with Multiple Frequency Components

I. Introduction:

For today lecture, we learned about the transfer function of the current.

To find the gain, we get the ration between Io/Iw. To find zeros, we find the roots of the numerator, to find poles, we find the roots of denominator

With the given graph, we have to find the ratio between Vo and I

Follow Professor Mason codes, we were able to graph the transfer function using log scale 

II. Signals with Multiple Frequency Components:
In this lab project, we will calculate the magnitude response of an electrical circuit and use this information to infer the effect of the circuit on some relatively complex input signals

We used AWG editors to type in custom graph 20*(sin(2Pi*x)+sin(4pi*x)+sin(40*pi*x))

The graph here is created using sinusoidal sweep

Here is our derivation for the pre-lab. The gain of V0/Vin is about 5.06*10^-4 -0.0159j

III. Conclusion:
For today lab, we learned about the transform function of electric circuit, we also did some matlab experiment to plot the gain with zeros and poles using log scale. For the lab, we did signals with multiple frequency components which we learned about sinusoidal sweep function in the waveform software

05-24-2016

Lab: Maximum power transfer in AC circuit

I. Introduction:

We had learned how to find the maximum power transfer we learned back then in DC current using the Thevenin equivalent. Given the definition of eff( rms ) value, how could we solve the maximum power transfer problem

Given the equation, we have to solve for I_eff which is also its own definition.

To find Vrms, in this case we simply need to divided by square root of 2

Rule of thumb: ELI the ICE man. ELI: voltage lead current in the inductor, ICE: current lead inductor in the capacitor load

Here we solved for the complex power of the circuit, which is the combination of average power and reactive power

Here we solve for transparent power by just taking the absolute value of the complex power

III. Phase Shift Lab:
For this lab, we are going to observe voltage lead current for the inductor, and current leads voltage for the capacitor

Here is our calculation of the lab

In this case the yellow graph is the graph of the power supply, where as the blue is the voltage across the capacitor

Here we find the apparent power in the pre-lab using the given information

We converted everything to frequency domain then we used that to do our calculations

The yellow is the voltage of the supply, where as the the blue is the voltage across the inductor.

III. Conclusion:
For today lecture, we learned more about how to find the complex power based on the relationship between the average power and the reactive power. The complex power is a short way of represenation of the combination of those two. We also verified our knowledge about ELI the ICE man, which tells us that the voltage leads current in the inductor and current leads voltage in the capacitor.

05-19-2016

Lab: Op amp Relaxation Oscillator

I. Introduction:

With our lessons learned from the previous lecture, today we first reviewed how to convert between the wave length and frequency, assume our waves have speed of light 

Given the wavelength, we have to calculate the frequency

II. Op amp relaxation oscillator:
A relaxation oscillator is constructed using some type of device that act as a switch when a certain voltage is applied to one of its terminals. They must meet the following criteria:
1. The overall phaseshift must be zero
2. The unity must be one or greater (to be compensated by the amplifying device)

Here the period is about 8ms,the percent error of experiment is about 7%

Here is the setup of our circuit

Here is our derivation to find the gain beta which is half given that R1 and R2 the same 

Using trig identities to derive an equation for power

We used every circuit to make sure that we have correct graph in the oscillator graph of analog discovery

We were given some problem to find the equivalent impedance in the circuit. in this case, we took away the wire in the middle because there is nothing between it

III. Conclusion:
Today we did some more practice about the mesh and nodal analysis in AC circuit. For the lab, we did the op amp relaxation oscillator. Oscillator is a setup acting like a switch when a certain voltage is applied to one of its terminals. We also find the average power Pave=0.5 Vm*Im*cos(theta v-theta i)

05-17-2016

Lecture: Review learned technique within AC circuit:

I. Introduction:

Given the new concept of the impedance, we were asked to list out many techniques we had learned in DC circuit. The good news is that even though with imaginary, all the techniques we learned in the past still hold true

Here we listed out as many techniques as we had learned before

II. Practice:

Given the circuit, we had to use nodal analysis to find out the two equation of V2 and V1

Solving system of imaginary equation requires a lot of techniques, it could be easier if we have mathlab comes in handy

Here we applied mesh analysis to find i1, i2, and i3. After we found out the new 3 equations, we let mathlab solved for our 3 unknowns

III. Phase shift

Here we compared the two voltage sources. The blue one is from our power supply and the red one is the voltage across the capacitor, which indicates it has some lagging

Here is our data analysis, we let the omega=R/L to get the initial base frequency, the second, third, fourth is just multiplied by 10,100,1000.

Simple conversion from imaginary to poplar to time domain

Some more practice with mesh analysis

Using the source transformation to find the Vth of circuit

III. Conclusion:
Today lab, we did some more experiments with the inductor in series within a circuit. We reviewed the old techniques we learned back then in DC current with the only thing changed is that the frequency could affect the outcome complex resistance of the capacitor and the inductor

05-12-2016

Lecture:The Impedance of inductor and capacitor

I. Introduction:

For today lecture, we will continue our studying the with impedance of capacitor and inductor in AC circuit

The impedance have two parts, the real part is called pure resistance, whereas the imaginary is called the reactive part.

Impedance can be known as the resistor in AC circuit, using that knowledge, omh law still holds for voltage and current, with one more imaginary term

Here we convert everything back to frequency domain to do the calculation, after we finished we converted everything back to time domain

II. Impedance Lab:
The purpose of this lab is to check the understanding of impedance. The frequency will affect the ratio of Vo/Vin in this lab. 
For this lab, the frequencies we used are 1kHZ,5kHZ,10kHZ

Here is our data collection, only the impedance of capacitor and inductor affected by the frequency The gain of overall circuit will not be affected by the frequency.

Graph of 1kHz

Here is our setup

I=V/Z=0.036cis-34.4,
Vout/Vin=X/(R+jXL)=0.56cis55.6

Here there is some lagging between the inductor and the capacitor

Here is our derivation to find out the Zequi in the circuit

since the angle is 45, which tells us the gain is 45 degree. The phase between Vo and Vin is 45

III. Conclusion:
For this lab, we learned how to apply phasor transformation to solve for many problems within frequency domain. With phasor converts everything to the impedance, we now can use the techniques we had learned before, namely, nodal analysis, mesh current and so on. The only thing that difference is that the frequency could affect the impedance of capacitor and the inductor

05-10-2016

Lecture: Introduction to phase shifting, phasor representation in frequency domain

I. Introduction to Phase shifting:
For today lecture (no lab today), we will learn about how to modify the orginal graph using the idea of pre-cal, which is known as vertical shift and horizontal shift, but for engineering, horizontal shift would be phase shift.

We have to parametrized the two graphs. The graph y(t) is really close to the orignal which only had the vertical shift;therefore we chose to do it first; whereas the green one also had the horizontal shift

II. Phasor representation:

The idea is genius, it is really hard to add the trig identities together. We assumed that voltage and current had the same frequency, therefore, we could use the phasor with polar and imaginary part to simplify our calculation

Given the original time domain function, we could convert it to the polar coordination to multiply and division operation, or to imaginary representation to do the add or subtracting equations

In trig, to find the phase angle, simply take the acrtan of y.x

We practice some operations between polar and imaginary part of phasor representation

For adding, imaginary part will be a better choice

Some more practice for phasors, recall that to to convert from sin to cos , we subtract the angle by 90.

Here given the time domain voltages, we have to combine them into one new voltage. We performed many steps of phasors to get the answer

We find the current with the given impedance of the inductor

III. Conclusion:
Today we spent most of our time to understand the phasor representation of the trig in the frequency domain. For phasors,  we have to use polar to divide or multiply, and imaginary to subtract and add. The capacitor and inductor act like a complex resistors in the AC circuits.