Thursday, April 30, 2015

2nd order circuit continue...

We continue to learn more about 2nd order  circuits.

Step response of series RLC circuit: the step response is obtained by the sudden application of a dc source. It has the source free equation with an addition term for the Voltage supply. The solution to the differential equation above has two components: the natural response v n (t) and the forced response v f (t).
We did following problem of series step response RLC circuit.

The natural response is the solution when we set V s = 0 and is the same as the one obtained for the source free circuit.
v n (t) = Vs + A 1 es1t + A 2 es2t  (Overdamped)
v n (t) = Vs +  (A 1 + A 2 t)e−αt     (Critically damped)
v n (t) = Vs + (A 1 cos ω d t + A 2 sin ω d t)e−αt    (Underdamped)


The values of the constants A 1 and A 2 are obtained from the initial conditions: v(0) and dv(0)/dt.

RLC Circuit Response LAB:  we tested and analyzed step response of the following circuit shown in the picture. 

Then we learned about parallel step response RLC circuit. It  is pretty much the same process. Natural response is the same as what we did before.
Following is an example we did in class for parallel Step Response RLC circuit.



From the above differential equation we estimated the damping ratio and natural frequency of the circuit. R1= 4.7 ohm and R2= 1.1 ohm 



Our % Difference was 81.9% for omega and 42.8% for alpha. This % error could have caused by the Resistors and capacitor we used. We might not have used correct decimal points to get exact answer.


FollowinWe input 500Hz square wave at 2VDC into the circuit,and above graph is the outputg is the graph of the circuit from the oscilloscope. The circuit is underdamped. 


Data obtained from the graph  shows the higher and lower peak, we calculated period, omega, and alpha



Summary: 
In this lab we obtained large percent error between theoretical and experimental values. Which could have caused from not measuring the data correctly and inputting the values correctly. 

Tuesday, April 28, 2015

2nd order circuits

Today we started class of with learning about 2nd order circuit. For second order differential equation we need additional boundary values.
V(0)=v(0-)
Capacitor voltage and inductor current are variables that cannot change abruptly.

Source free series RLC Circuit: the circuit is being excited by the energy initially stored in the capacitor and inductor.

Source free circuits do not have voltage and current. 


Series RLC Circuit Step Response: In this lab we modeled and tested a series RLC second order circuit by analyzing step response of a ciruit and comparing with expected values on the dampling ration and natural frequency.
The initial value of i is given as I0 across the inductor


Pre lab: we wrote differential equation relating Vout and Vin for the RLC circuit.
We tested step response of a given circuit. Then we compared the measured response with expected or calculated values based on the damping ration and natural frequency of the circuit.  

The roots s1 and s2 are called natural frequencies, measured in nepers per second (Np/s), because they are associated with the natural response of the circuit; ω 0 is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second (rad/s); and α is the neper frequency or the damping factor, expressed in nepers per second.

1.  If α > ω 0 , we have the overdamped case.
2.  If α = ω 0 , we have the critically damped case.

3.  If α < ω 0 , we have the underdamped case.
The critically damped case is the borderline between the underdamped and overdamped cases and it decays the fastest. And overdamped case takes longer to settle.

Damping is the gradual loss of the initial stored energy, as evidenced by the continuous decrease in the amplitude of the response. The damping effect is due to the presence of resistance R. The damping factor α determines the rate at which the response is damped.

Following is the expected graph of RLC series circuit step response. It is Underdamped circuit and
α < ω 0 )

Following is the graph of underdamped circuit we captured on waveform. 

Percent Error: we calculated 19.5% error as calculated in above few pictures. The percent error might be caused from the capacitor and inductor that we used are non ideal. The inductor has non negligible resistance. 



Above is the output graph fromt the capacitor.
This graph yields a damping formula of V=0.43e^(-2340t), meaning that our experimental alpha is 2340, which is about 0.4% of our theoretical alpha value of 550000. 


Summary: 
Possible source of error is from our input variables, the way we measured the circuit, or maybe did wrong calculation for theoretical value.



Tuesday, April 21, 2015

1st order circuits/ op amp

we started class with learning about Integrator:  an op amp circuit whose output is proportional to the integral of the input signal.
If the feedback resistor Rf in the familiar inverting ampliflier of a is replaced by a capacitor and we obtain an ideal integrator.

We draw following graphs of vin vs t and vout vs t. It’s graph of integrator op amp. They saturate really quickly. If Vin is sinusoid then Vout would look like something in the picture.


Differentiator: is an op amp circuit whose output is proportional to the rate of change of the input signal. If the input resistor is replaced by a capacitor , the resulting circuit is a differentiator. iR=iC
Vo = -RC (dvi/dt)

Differentiator circuits are electronically unstable because any noise within the circuit is exaggerated by the differentiator.


above picture also shows graph of op amp differentiator. 
LAB: Inverting Differentiator: In this lab we study the forced response of a circuit which performs a differentiation. The circuit output is the derivative with respect to time of the input to the circuit.

PreLAB: we did following equation to solve for Vout as a function of the circuit input.  W=2f The frequency f has units of hz 


We constructed the circuit shown below using R=470 ohm C= 470 nF . we used oscilloscope to measure both the input and output voltages .
Then we applied sinusoidal input voltage with frequency = 1kHz A= 1V and offset=0V.
Below is image of oscilloscope window showing the wave forms and their measured amplitudes.

Measured A= 1.12 and calculated was  1.39. % Error was 19.4%

Following is the output signal at 1Khz


Following is the input signal graph



Then we applied sinusoidal input voltage with frequency = 500Hz A= 1V and offset=0V.
Below is image of oscilloscope window showing the wave forms and their measured amplitudes.
% Error was 15.9% when compared with calculated values of 0.69V. measured values were 0.58               

 output sine wave signal of 1V at 2KHz


Then we applied sinusoidal input voltage with frequency = 2kHz A= 1V and offset=0V.
Below is image of oscilloscope window showing the wave forms and their measured amplitudes.

Measured A= 2.21 and calculated was A= 2.78. % Error was 20.5%

Output signal graph 

Input signal graph


Error source: op amp has really big saturation which caused big % error.
 Below is our calculations and measured values showing % error.. 



Step Response of an RC Circuit:  its behavior when the excitation is the step function, which may be a voltage or current source.
v = v f + v n
v f = V and  vn = (V0 − Vs )e− t/T



Vn is the natural response of the circuit.

The natural response or transient response is the circuit’s temporary response that will die out with time. The forced response or steady state response is the behavior of the circuit a long time after an external excitation is applied. 

Below is problem we did in class to get step response circuit


More problems done in class..


Functions are either discontinuous or have discontinuous derivatives. Types include unit step, unit impulse, and unit ramp functions:

Summary: we learned about op amps and how they integrate in RC circuits. we did lab to see how  inverting differentiator and signularity functions work. 

Thursday, April 16, 2015

1st order circuits


We started class with learning about series and parallel inductors.  The equivalent inductance of series connected inductors is the sum of the individual inductances. Inductors in series are combined in exactly the same way as resistors in series.

The equivalent inductance of parallel inductors is the reciprocal of the sum of the reciprocals of the individual inductances.  Inductors in parallel are combined in the same way as resistors in paralle.

We did following example of finding value of the equivalent inductance.


1st order circuits:  circuits that contain various combinations of two or three of the passive elements. 

Source free RC circuit:  occurs when its DC source is suddenly disconnected. The energy stored in the capacitor is released to the resistors.  
At time t = 0 , the initial voltage is v(0)=V0
Energy sotred is E - 1/2 CV^2
V(t) = V0 * e^(-t/RC)

Capacitor is fully discharged or charged after five time constants. For every time interval of Taho the voltage is reduced by 36.8% of its previous value regardless of the value of t.

We did following source free circuit example by finding time constant.

LAB: Passive RC Circuit Natural REsponse

In this lab we examine the natural response of a simple RC Circuit. we used manual switching operation and a square wave voltage source to create our circuit's natural response. Following is the set up of our circuit.


Following is the schematic of our RC circuit. We did calculated time constant since we were given R and C values. We estimated intital capacitor voltage and time constant for the circuit as shown below. 


Following is an image of oscilloscope window showing the capacitor voltage response for the circuit shown in previous image where V+ is used as the voltage source. % Error was 20.6% between the calculated values and measured values.


Following is an oscilloscope window showing capacitor voltage response when applying 5V. Percent error was 5.9% . Percent Error is shown in above calculation.


Following is a graph of capacitor voltage response with sinusoidal. Percent Error was 7.56% when comparing with the calculated values.


Error source: Multimeter was not functioning correct at first. some connections were not strong


We did following example of RL circuit with switch been closed for a long time. at t<0 wswitch is closed and the inductor acts as a short circuit to DC.  At t>0 the switch is open and the voltage source is disconnected.


LAB : Passive RL Circuit Natural Response
Professor Mason did this lab as a demo since we ran out of time. Following is he graph of the inductor voltage response. The energy stored in the inductor eventually dissipated in resistor.

Voltage graph generated by the RL circuit.

Tuesday, April 14, 2015

capacitors

Today we started class with learning about capacitors. capacitor consists of two conducting plates separated by an insulator or dielectric.
Capacitors stores energy. They act completely opposite of resisters.
C= q/v (capacitor is measure of how much charge per volt stored on a plate


We did following RC circuit to obtain E stored in Capacitors by using source transformation and open circuiting capacitors.



LAB: Capacitor voltage current relations
In this lab we measured the relationship between the voltage difference across a capacitor and the current passing through it and we compared it through graphs displayed below. 

Procedure: we applied several types of time varying signals to a series combination of a resistor and a capacitor. The voltage difference across the resistor in conjuction with ohm's law provided an estimate of the current through the capacitor .

Below was set up of our RC circuit in series so that the current through the capacitor was the same as the current through the resistor. 




Pre LAB:  we sketched capacior voltage and capacitor current sinusoidal wave and triangular wave  using the capacitor voltage current relations. 
As you can see in both sketch that pink color is current and black is voltage.  In both graph when current is positive , voltage is negative and vise versa. 


Triangular input: As you can see in below sketch that red line which indicates the current is almos in square shape. 

This graph shows that when the voltage doesn’t change across the capacitor, current doesn’t flow; to have current flow, the voltage must change.

Graph of voltage is sine graph and current is cosine grpah which shows that they are 90 degree out of phase.


the current going through a capacitor and the voltage across the capacitor are 90 degrees out of phase
The current leads the voltage by 90 degree

capacitors in parallel combine in the same manner as resistors in series and in series capacitors combine in the same manner as resistors in parallel.

Below is an example we did in class to get Ceq.

Inductors:  consists of a coil of wire. inductor exhibits opposition to the change of current flowing through it. Its units are henrys (H)

It acts like a short circuit to DC.

Below is the comparison of capacitors and inductors which shows that capacitors are completely opposite of inductors.


LAB : Inductor Voltage current relations . This lab was demonstrated by Mr. Mason. 
Below is the graph of Inductor voltage current. The current going through an inductor and the voltage across the inductor are 90 degrees out of phase. Here the voltage leads the current by 90 degrees

Thursday, April 9, 2015

Cascaded op amp

We started class with cascaded op amp circuits by drawing ideal op amp .
A cascade connection is a head to tail arrangement of two or more op amp circuits such that the output of one is the input of the next.

We did following problem of cascaded op amp



LAB: Temperature Measurement Design and Appendix A -Wheatstone Bridge Circuits

LAB: Temperature measurement system design
We started this lab with building wheatstone bridge circuit first. Wheatstone bridge circuits are most often used to convert vaiations in resistance to variations in voltages. It is used in measurement system as a number of common sensors provide a resistance variations in response to some external influence.
Thermistors change resistance in response to temperature change. Straing gages change resistance in response to deformations.

Then we balanced our wheatsone bridge circuit so that output voltage is zero when the variable resistance is at its nominal value.

Following is the set up of our wheatstone brdige.


Following is the video showing the bridge working. 


Difference ampliflier design. Difference ampliflier circuit is shown in above picture. The output of the circuit is proportional to the difference between the two inputs Va and Vb.

R1=R2=R3=R4

vout= R2/R1 (Vb-Va)