Analyzing Circuits

SOLUTION AT My Australian Academy the many parameters that describe RLC circuits that are driven by an AC Source. Wewant to pick certain R, L, and C values and fix them for the rest of the lab. We will be doing 3 ‘data runs’, onewith a relatively low frequency that gives us a ‘capacitive’ circuit (where the current leads the voltage), a runwith a relatively higher frequency that will be ‘inductive’ (where the voltage peaks before the current). The thirdand final data run will be at the frequency closest to what we calculate to be the resonant frequency for thecircuit.The following calculations will give us the resonant frequency.The angular resonant frequency is given by:ωr=1L⋅C√ ,where L is the inductance and C is the capacitance. Note that the angular frequency will be in units of radiansper second. We now use:fr=ωr2⋅π to get the resonant frequency in hertz. On the third data run, set the driving frequency (of the source)as close as possible to this resonant frequency.On all three data runs, we need to obtain the following data:From the circuit picture we will obtain f, the frequency of the ac source, R, the resistance, L, the inductance, C,the capacitance and, Vtotrms, irms, VRrms, VCrms and VLrms. The last five variables are the rms values forthe total voltage, the current (which is the same throughout the circuit), the voltage drop across the resistor, thevoltage drop across the capacitor, and the voltage drop across the inductor, respectively. You may use thepictures of the circuit during each data run to record this data instead of putting all of it in a table.From the graph below the circuit, we need to also obtain VRm, VLm, VCm, Vtotm. These are the amplitudes ofthe voltage drops over the resistor, inductor, capacitor and all components, respectively. Furthermore, we needto read from the graph, im, the amplitude of the current throughout the circuit and ϕ. The way we obtain phi isto first find tVRPeak and tVtotPeak. which are the time when the resistor voltage first peaks (remember that thevoltage curve peaks the same time as the current peaks) and the first time when the total voltage peaks. To getthe phase angle, note that:ΔtT=ϕexp360o=Δt⋅fwhere T is the period of oscillations, ϕexp is (the experimental value of) the phase angle between the totalvoltage and the total current, f is the frequency of oscillations and:Δt=tVRPeak−tVtotPeakwhere tVRPeak is the time when the resistance voltage peaks and tVtotPeak is the time when the total voltagepeaks. Note that this will be positive if the total voltage peaks before the current and it will be negative if thetotal voltage peaks after the current. We can now find the experimental value of the phase difference using:ϕexp=f⋅Δt⋅360oWe should obtain a ϕexp for all three data runs. Even though we had to do a calculation to find it, we treat thisas the experimentally measured phase value.The following computations are to be done for every data run.We first check to see whether our experimentally obtained amplitudes correspond with their respective rmsvalues:VRmth=2–√⋅VRrms5/1/2021 Order 343859736 3/5Does this agree with VRm, the amplitude for the resistor voltage that was read from the graph?We now repeat this comparison for VLmth, VCmth, Vtotmth, and imth, where the ‘th’ stands for the theoreticalvalue. In each of these cases, we obtain the theoretical value by multiplying all of the corresponding rms valuesby 2–√ . We then compare these theoretical values with VLm, VCm, Vtotm and im, the amplitudes of thesevoltages and current as they were read from the graph. If the theoretical and experimental values agree, thenwe can assure ourselves that we are indeed working working with rms values for the voltages and current asthey appear in the circuit diagram.Next, let’s check to see if our rms values are consistent with Ohm’s Law (for the resistor, the reactances, andfor the impedance).VRrmsth=irmsR.Does our theoretical rms value for the voltage drop across the resistor agree with the VRrms given on thecircuit diagram?Compute the reactances:XL=ω⋅L,XC=1ω⋅C ,and the impedance,Z=R2+(XL−XC)2−−−−−−−−−−−−−−√ .Now we find the following rms values:VLrmsth=irms⋅XL.Does our theoretical rms value for the voltage drop across the inductor agree with the VLrms given on thecircuit diagram?VCrmsth=irms⋅XC.Does our theoretical rms value for the voltage drop across the capacitor agree with the VCrms given on thecircuit diagram?irmsth=VtotrmsZ .How does irms compare with irmsth?Next, we look at the phase angle.ϕth=tan−1(XL−XCR) . How does this compare with ϕexp, found above? Make sure to express the phaseangles in degrees when doing the comparisons.Compute the power dissipated in this circuit using two different methods:5/1/2021 Order 343859736 4/5P1=i2rms⋅R ,where irms and R may be read off of the circuit diagram, andP2=irmsVtotrms⋅cos(ϕth) ,where irms and Vtotrms may be read off of the circuit diagram, but use the theoretical value for the phasedifference. How do P1 and P2 compare with each other?This ends the computations that must be done for every data run.Once you have finished the above computations for the resonance data run, note how XL compares with XC;and how Z compares with R. Does the value for ϕ make sense for resonance?Write the lab report using the standard format. When you get to the Raw Data Section, note that some of yourdata is given in the screenshots of the apparatus, but make sure to note the amplitude of the current curve inthe Raw data section (unless you included a picture of the current curves in your data graphs. You should alsoplace all of your graphs in the Raw Data Section. Be clear about which graph corresponds to which picture inthe Apparatus section. Do not forget to label the components in the Apparatus section.In the Conclusions section, state whether the theoretical values agreed closely with the correspondingexperimental values. Also state here any conclusions you draw from looking at the resonance curves.
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