Final Project Blog

Everycircuit Model 

First prototype on breadboard using microphone with opamp gain, we couldn't get a clear sound broadcasted on our channel

We changed our approach by hooking up our  breadbard to waveform and applied a sinusoidal function at 1 kHz and we hear a pitch at our channel, so we figure out that there is something wrong with our microphone.

We then moved to our perforated board stage, we created 3 different prototypes, one uses an input of aux cable, one with 2 hooks up cable to waveform and one with an electret microphone

Final Product using the electret microphone, we still couldn't get a clear enough signal due to our antenna, so we decided to put a more rigid structure for our antenna

Final Product with better antenna, we are able to broadcast our station acorss our house

week 15 day 2*

Lecture 

Calculating Resonance frequency(⍵0), Half power Frequencies (ɷ1,ɷ2), Bandwidth and quality factor of RLC circuit in parallel :

 Finding Transfer Function of a filter, Cutoff frequency of Filter is equal to the half power frequency of  RLC circuit:

Lab

For our Pre-lab, we calculated the Theoretical Cutoff frequency of lowpass and highpass filter by finding the transfer function of RL circuit on the resistor and inductor respectively.   inspecting the behavior of transfer function (H) when the frequency is approaching zero and infinity we observed that we the circuit behave as a low pass filter when we took Output Voltage at the inductor and high pass Filter if we took the output voltage at the resistor.

We measured A 99.6-ohm resistance and 1 mH inductance with a theoretical Cutoff frequency of 15915 HZ. We use our oscilloscope to measure the voltage of our inductor and resistor. We also created a custom channel by adding the 2 voltage to measure our input voltage.The yellow line represents the voltage of resistor, blue represent the voltage at the inductor and red is our input voltage.  We measured the amplitude and phase shift of our voltage out from both the inductor and resistor

Oscilloscope at 1591 Hz

Oscilloscope at 159150 Hz

 Values of the amplitude and phase shift at 8 different frequency

Summary

Our circuit acted as a lowpass filter when we put our output voltage across inductor and highpass filter when we put our output voltage across the resistor. Our experimental gain at the cutoff frequency is 0.67 and 0.695 at lowpass and high pass respectively, which is close to our theoretical value of 0.707.

week 15 day 1

Lecture 

Sketching Bode Magnitude plot with one zero  at the origin and 2 simple poles: 

Bode magnitude plot and Bode Phase Plot with one simple zero,  one simple pole and one pole at the origin>

Sketching Bode magnitude plot and Bode Phase Plot with a constant shift :

 Translating a Bode Plot into Transfer Function

Derivation of Half power frequency of RLC circuit in series. Half power frequencies are the 2 frequency that provided half the power dissipated through resistor at resonance frequency Algebraically there are four solutions obtained from our equation, but since half power frequency is always greater than zero( positive) we can eliminate two of the four solutions

 Calculating resonance frequency and half power frequency of RLC circuit with 2-ohm resistor, 1 mH inductor, and 0.4 pF capacitor

Summary 

Bode Plot are a semi-log plot that helps us to predict the behavior of transfer function in relations with frequency by using linear functions to approximate the behavior of poles and zeroes. Resonance Frequency is the frequency where the circuit is purely resistive. Resonance frequency of RLC circuit in series is equal to one over the square root of the product of inductance and capacitance.

week 14 day 1*

Lecture 

Calculating transfer function of current gain across the capacitor, we used current divider formula to help us find Io /Ii :

 

calculating Transfer function of  Impedance across the inductor, we applied current divider formula and the fact that the voltage across the inductor is equal to the current times the impedance of the inductor. we also sketch the transfer function versus frequency graph 

 Calculating voltage gain transfer function, we applied voltage divider formula to solve the voltage out. we then sketched the transfer function versus frequency.

LAB

Pre-Lab :

we calculated the theoretical voltage gain transfer function across the resistor. we use voltage divider formula of 680-ohm resistor and the impedance equivalent of 0.1 micro Farad and 680-ohm resistor in parallel with each other. Inspecting the transfer function we observe that as frequency approaches 0 the transfer function is equal to 1/2( voltage out is half the input voltage ), while when frequency is approaching infinity ( high frequency) the gain is approaching 0 (voltage  out is approaching zero)

we then use excel sheet to calculate the gains across 680-ohm resistor in parallel with 0.1 microfarad inductor when the frequency is equal to 500 Hz,1000Hz and 10 KHz

From the Calculation, we calculated a theoretical gain of 0.49 when the frequency is equal to 500 Hz,  0.48 when the frequency is equal to 1KHz and  0.21 gain on 10 KHZ frequency. We started our experiment by measuring our resistors and inductors. we then build the circuit on the breadboard.

Measuring Resistor

Breadboard circuit:

For our first experiment, we measure both input voltage and the output voltage of a custom waveform that includes 3 sinusoidal functions with 500 Hz, 1000 Hz, and 10000Hz frequencies.  

The yellow line represents the input voltage and the blue line represents our output voltage.we observe half gain on the bigger sinusoid (low frequency) and1/3 gain on sinusoidal with higher frequency. Our next activity is to apply an  AC voltage sweep with frequency going from 100 Hz to 10 KHz in 20 ms.

We observe a decrease in amplitude of the output voltage as the frequency is going larger.

Summary
Transfer function can be used to measure voltage gain, current gain, impedance, or admittance in respect of changing frequency. In our experiment, we are comparing our theoretical voltage gain with 3 different frequency using transfer function, and experimental voltage gain by using voltage sweep and 3 sinusoidal functions stacked on top of each other. In our theoretical gain, we observe that the amplitude of our voltage out will decrease to zero as the frequency gets larger.By setting frequency approaches zero we also observe that our theoretical gain is equal to 1/2. By Observing the Voltage sweep we can see clearly that the ratio of our experimental gain is close to 1/2 at low frequency (100 HZ) and decreases as frequency increases to 10 KHz. Looking at our custom stacked sinusoidal waveform we observe a gain of 0.5 on the larger sinusoidal. and 0.3 gain on the higher 10 KHZ frequency sinusoid.

week 13 day 2

Lecture 

Calculating Effective Current (I RMS) and Effective Voltage (V RMS):

 Calculating V RMS of  voltage graph with an offset of amplitude (Vp)

 Calculating Apparent Power and power factor using V RMS and I-RMS:

Calculating Complex Power: 

Apparent power and Power Factor Lab

Calculation of Vrms, I rms, Apparent power, Power factor, and Average power : 

Graph between input Voltage( Blue line), load voltage (yellow) and current across the load ( red )

RL=10 Ω

RL=47Ω

Rl= 100Ω

Below are the values of our calculated  and measured values:

RL 10 Ω

Rl=47Ω

Rl=100Ω

Summary 

Because voltage and current of AC circuits are sinusoids, it's more effective to describe voltage and current as it's effective voltage (Vrms) and effective current ( Irms) we can represent power as it's complex power Phasor. Complex power consists of average power as it's real part and reactive power as it's imaginary part. Average power is the product of apparent power( S) and power factor. apparent power can be calculated by finding the product of the magnitude of effective current and the effective voltage. In our, we obtain a low percent difference between calculated and measured effective voltage and current. by comparing the Power dissipated by resistor and load and taking the ratio,  we observe that the average power comes purely from the real part of the load. The ratio between the 2 power is identical with the ratio between two resistors.  therefore we can assume that all the power dissipated through the load is dissipated only by the two resistors. we also observe that power factor( P avg load divided by S load) is equal to the cosine of phase shift  angle between voltage and current)

Week 13 day 1

Lecture 

Using Nodal Analysis to calculate the output voltage of inverting op-amp, note that we have to convert all the capacitor into their phasor form first :

An op-amp with no feedback: when the voltage across the inverting and non-inverting input are equal, then the output voltage is equal to zero

 Instantaneous Power equation, given  sinusoidal current and voltage, applying Sum Trig identities  we can separate power to time independent average power

Op-Amp Relaxation Oscillator 

Pre Lab: 

Designing Op AMP relaxation oscillator  to produce 777 Hz  frequency  using 100 nF capacitance

we calculated a theoretical resistance of  7.32

We measured the  capacitance of 90 nF acorss the capacitor 

TO obtain resistance close to 7.5 K connected 5.49 K and  1K resistor in series

To check our circuit we run the model in every circuit, here we get a frequency of  769 Hz, which is really close to our desired frequency.  Don't forget to shake the op amp inorder to jump start !!

The final step we run our circuit with a +/- 5 V saturation:

view of our Circuit:

first, we scope the feedback resistor to compare our graph with every circuit graph:

Our frequency across the feedback resistor is 812 Hz, that's 4.5% difference from our desired Frequency 

Lastly, we scoped the voltage across the resistor and the output voltage of the OP AMP :

The result is identical with what our theoretical graph predicted, the output voltage of op amp is a step function, while voltage across the capacitor is a combination between an inverse exponential and exponential graph 

Summary 

In an AC OP  AMP circuit, we can apply nodal analysis to calculate output voltage by using phasor and impedance.  In our OP AMP Relaxation oscillation Lab, we use saturation of op amp as a way to create a periodical function. We designed a circuit that gives 777 Hz frequency, we then tested our design using both every circuit and oscilloscope. Our final design are able to produce oscillator with 813 Hz frequency. We obtained a 4.5% difference, these difference in values is caused by our inability to find a precise resistor. 

Week 10 day 1

Lecture

Finding Initial Value Problem of an RLC circuit:

Our Initial Value Problem Solution: After switch is turned on for a long time Capacitor acted as an open circuit and Inductor acted as a short circuit.

Second order RLC formula derivation  in series :

Overdamped RLC Problem:

Parallel RLC problem :

Series RLC Step Response Lab

Pre Lab: Calculating damping ratio (1/𝛼) , natural frequency (𝝎o), damped frequency (𝝎d) and overshoot (

We calculated a damping ratio of 0.002 and a damped frequency 99999 with an overshoot of 4 V.
We measured the value of capacitor and resistor :

Since the value of the resistor is relatively small, our DMM couldn't measure the proper resistance :

View of our Breadboard :

WE used a unit step response of 2 V with zero offset

scope of  Voltage across the capacitor:

Graph of voltage across the capacitor in response with the unit step function we can measure the damping period by dividing 2𝝿 with the distance across peak to peak of the sinusoidal function.

We can measure the overshoot of the graph by finding the distance between the highest peak and the equilibrium line . we obtain an overshoot of 5.75 V and damping frequency of 101341.

Summary 

 To Determine the  total step response of a second order circuit we need to calculate both a steady state response and transient response of the second order circuit. Steady state response of a second order circuit is the value of either voltage or current across a component after a long time (t approaching infinity). To calculate transient response of a second order circuit we turned off independent sources and uses KCL and KVL to calculate our characteristic root. In our lab today we tried to measure the response of voltage across capacitor in response of a unit step function with 2 V amplitude with zero offset( the input voltage is switching between 2 V and -2V). Our calculated damoing frequency is 9999, 1/s while  our measured damping frequency is 101341 1/s  We obtained a 1.34 %  difference in theoretical and our measured value of damping frequency.