Physics 241 Lab – Phasorshttp://bohr.physics.arizona.edu/~leone/ua_spring_2009/phys241lab.htmlName:____________________________ from “King of Pain”There's a little black spot on the sun todayIt's the same old thing as yesterdayThere's a black cat caught in a high tree topThere's a flag-pole rag and the wind won't stopI have stood here before inside the pouring rainWith the world turning circles running 'round my brainI guess I'm always hoping that you'll end this reignBut it's my destiny to be the king of pain-The PoliceImportant:- In this course, every student has an equal opportunity to learn and succeed.- How smart you are at physics depends on how hard you work. Work problems daily.- Form study groups and meet as often as possible.. - Join professional organizations. - Physicists help people: science => technology => jobs.Section 1: 1. In a circuit where an inductor, resistor and capacitor (RLC) are connected in series and driven by anAC source, the voltage across each of the components varies with time reaching a maximum and aminimum at regular intervals. The properties of the RC circuit and RL circuit studied previouslycombine in a straightforward manner. Combining the results obtained in previous labs we can measurethe voltage across each component with respect to time as done below:The voltage across each component oscillates at the same frequency as the driving frequency ofthe source, -drive. The properties of the inductor and capacitor are frequency dependent, which makesthe circuit respond differently to different driving frequencies.1.1 ( )tIRtVdriveamplituderesistorcos)( ω⋅=( )sourcedriveamplitudesourcesourcecos)( φω += tVtV ⎟⎠⎞⎜⎝⎛−⋅=2cos)(driveamplitudecapacitorπωχ tItVC ⎟⎠⎞⎜⎝⎛+⋅=2cos)(driveamplitudeinductorπωχ tItVLWhere: € Iamplitude=Vsource amplitudeZ, € Z = R2+ χL− χC( )2, € χC=1ωdriveC, € χL= ωdriveL,€ φsource= tan−1χL− χCR ⎛ ⎝ ⎜ ⎞ ⎠ ⎟. Analyze the above equations and you will find that the capacitor lags behind the resistor while the inductor leads the resistor. Please note that in the above equations we take the resistor as the reference point and hence we will also have a phase shift in the source voltage. Not a question. This graph is useful in showing the relations between the voltages across the resistor, inductor,and capacitor.1.2. If R = 20 -, L = 15 H, C = 10 F, Vsource amplitude = 10 volts and fdrive = 50 Hz, find the voltageamplitudes of the inductor and capacitor. Your work and answers:1.3. A few practice problems. Vsource amplitude= 10V , fdrive= 700Hz, R= 1000Ω, C = 10μF, L = 100mH.Calculate each of the basic RLC circuit parameters in SI units though not necessarily in the ordergiven.. Write the numerical value with correct SI units for each listed parameter:-C-LZ€ VR,amplitude€ VC,amplitude€ VL,amplitude-source€ IamplitudeSection 2:Section 2: A phase vector (phasor) is a representation of a sine wave whose amplitude, phase, and frequency are time invariant. Phasors are complex numbers used to represent a time varying voltage. Complex numbers are an extension to real numbers obtained by adjoining an imaginary part denoted by i. Every complex number can be written in the form a + bi. One of the properties of i is: i2 = -1.All imaginary numbers can be plotted in the Complex plane where the y axis is the imaginary term, and the x axis is the real term as shown by this figure: A complex number z can be viewed as a point or position vector in a 2-dimensional Cartesian Coordinate system called the complex plane. If z = a +bi, the angle formed by the vector is tan-1(b/a). Also, z = r(cosθ + i·sinθ).The concept of phasors is applied to RLC circuits which are sinusoidally driven as described above. Recall Euler’s Formula shows a relationship between the trigonometric functions and the complex exponential function and is represented by the formula:eiθ= cos(θ) + i·sin(θ)Therefore, we conclude that z = reiθ.We will use this formula to convert all 4 time dependent equations from above into phasors.Section 3:Using the explanation above, this can now be applied to the time dependent voltage equations.€ V (t)resistor=RZVsourceamplitudesin ωdrivet( ) VR,0 · Re{cos(ωDt) + i·sin(ωDt)} (Note: If the sine function in the original time dependent equations is changed to the cosine function, all relations remain the same with respect to each other.)Taking the new equation from above, you can see that the part in brackets is exactly from Euler’s Formula and therefore can be written as eiθ (where θ= ωDt). If we ignore the imaginary part, we get back the original formula for the time dependent resistor voltage which indicates the equations are equal.This same method used above can be implemented for all other components in the RLC circuit.3.1. Convert the other 2 components, VC(t) and VL(t), to phasors.3.2. Graph of VR(t) in Complex-Plane.3.3. Keeping this as a reference, the other graphs for the capacitor and inductor voltages are as follows.Given VL,0= 20V , VC,0= 30V, and VR,0= 10V, and f = 50Hz and t=2sec. Draw a new graph with similar pattern as above to correctly represent these given values. 3.4. A single graph can be drawn that caters to the conservation of energy. Conservation of energy states that VC(t) + VR(t) + VL(t) = VS(t). Based on this graph, identify where φs would be.3.5. Given the same values from section 3.3, identify the value of VS and
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