PhET Simulation: This lab uses the Circuit Construction Kit AC simulation from PhET Interactive Simulations at University of Colorado Boulder. After launching the simulation, please select R-L-C window. Leave two significant figures in your calculated values. Theory: The capacitive reactance and the inductive reactance of a capacitor and an inductor, respectively, are defined as: Xc = 1/(WC) and XL = wh, where w = 2If, is the angular frequency in rad/s, and f is the frequency in Hz. RC circuit and RL circuit. R R v(t)= Vo cos(wt) v(t)= Vo cos(wt) C Page 1 of 6If the current in the R-C circuit is i(t) = / cos(wt), then the maximum voltage across each element and the entire circuit is: VR = lexBe Kc= loka, and Vo= lez, where XR is simply the resistance R of the resistor, and Z = R2 + X2 is the impedance of the circuit. The relation between the maximum voltage across each circuit component and the entire circuit is Vo = VR + V2 . (Eq.1) For an R-L circuit, similarly, the voltage across the elements and the entire circuit is: VR = lexBe VL = IXL, and Vo= lez, with the impedance of the circuit being Z = VR2 + X? . The relation between the voltages then is Vo = VVR + V2 . (Eq. 2) R-L-C circuit R W v(t)= Vo cos(wt) triangle Figure 3 In the case of a resistor, an inductor, and a capacitor connected in series to an AC source (R-L- C circuit), as shown on the diagram above, the impedance of the circuit is equal to: Z = VR2 + (Xc - XL)Z . (Eq. 3) (for Xc > XL) If the current in the circuit is given by i (t) = I. cos cos (wt) , the voltage across the circuit as a function of time is then given by v(t) = V. cos (wt+), where o is called a phase angle and for the RLC circuit above is equal to: tan (p) = 2c-XL R (Eq.4) Experimentally, the phase angle between the current and the voltage in the circuit can be determined by measuring the voltage value at the time when the current is zero. If the current i (t) = locos (wt ) = 0 at some time t,then (may = 0 at that time. Using a trigonometric relation for cos (0+ [3), we can write the voltage v(t) as: 170:) = 14,1205 (mt + (35) = V0 [cos (out) cos (dJ) sin (wt) sin ()] When cos (133(th = 0 , then potential difference across RLC, v(t);=o = Vacos ((15) The phase angle (,5, then, can be determined as: ab = arccos [1'63\"] (Eq.5) 0 Activity 1: Voltage across the elements in the RC circuit. (10 points) |. Connect a resistor and a capacitor in series and connect them to an AC source as shown in gure 1. Use standard symbols to build the circuit. i. Click on a resistor, a capacitor and an AC source and drag them into the circuit board. ii. Drag wires and connect the elements in series to form an RC circuit. Two elements are connected when their ends overlap. To disconnect two elements, place the cursor of the mouse over the joint and press the left mouse button. The scissor button appears on the screen and select it. To remove an element, place the cursor of the mouse over the element and press the left mouse button. Trash bin button appears on the screen. click the trash bin icon to delete it. iii. To change the values for an element, place the cursor of the mouse over that element and select the left mouse button. Window appears on the screen to change the value. iv. Adjust following values for elements in the circuit. Resistance, R Capacitance, C Power source voltage Frequency of the power max, V0 source, f 1-0 Hz v. Check the values box on the top right corner. Take a screenshot of your circuit and add it here. l|. Perform the m I. Measuring the maximum voltage using the voltmeter. If you choose the voltmeter, you can pause the simulation and go step by step over the different voltage values. Record the maximum value shown in the voltmeter. ii. Measuring the maximum voltage on the Voltage Chart. Click on the Voltage Chart. Measure the amplitude of the voltage on the chart (it's half the height from a through to a crest) iii. Taking measurements. Measure the voltage across the resistor, VR, the capacitor, Va, and across both elements, VRc. iv. Take extra care to take the measurements accurately. \"I. Write down the measure value for V3; , Vc and VRC in the following table. Calculate the value for Vc and (VR2 + V3) and include the calculated value in the table. Then calculate the % difference of VRZC and (V5? + ch) 0 -VRM W W) m ------- IV. Results: Is the sum of the square of the maximum voltages across R and C, VR2+V02. equal to the square of voltage across both elements, Vac;2 in agreement with Eq.1? Explain Activity 2: Voltage across the elements in RL circuit (15 points) |. Connect a resistor and an inductor in series and connect them to an AC source as shown in the gure 2. Use standard symbols to build the circuit. i. Click on a resistor, inductor and AC source and drag them into the circuit board. ii. Drag wires and connect the elements in series to form an RL circuit. Two elements are connected when their ends overlap. To disconnect two elements, place the cursor of the mouse over the joint and press the left mouse button. The scissor button appears on the screen. Select it. To remove an element, place the cursor of the mouse over the element and press the left mouse button. Trash bin button appears on the screen. click the trash bin icon to delete it. iii. To change the values for an element, right-click over that element and select change value. iv. Adjust following values for elements in the circuit. Power source Frequency of the voltage max, V0 power source, f Resistance, R Self-Inductance, L v. Check the values box on the top right corner. Take a screenshot of your circuit and add it here. ll. Perform the experiment: i. Taking measurements. Measure the maximum voltage across the resistor, VR, the inductor, VL, and across both elements, VRL. lll. Write down the measured values in the following table. Calculate the value for VEL and (V + V3) and include the calculated value in the table. IV. Then calculate the % difference of V5.2L and (V1? + VLZ) Frequency 2 2 (VRZ'PVLZL --W W) m 9/2) V. Results i. Is the sum of the square of the maximum voltages across R and L, VR2+VL2, equal to the square of voltage across both elements, VRL2 in accordance with Eq.2? Explain Activity 3: Impedance of an RLC circuit. (15 points) |. Set up the RLC circuit using standard symbols. i. Click on a resistor, an inductor, a capacitor, and an AC source and drag them into the circuit board. ii. Select an ammeter and voltmeter from the Tools Menu on the right. Connect the Ammeter in series to the RLC circuit. iii. Drag wires and connect the elements in series to form an RLC circuit with an ammeter and AC source. iv. Change the value of the elements to the values given in the table below. Frequency,f _Se|f-lnductance,L Capacitance,C v. Check the values box on the top right corner. Take a screenshot of your circuit and add it here. l|. Perform the experiment: i. Measure the maximum voltage across the entire RLC circuit, V0. ii. Measure the maximum current in the circuit, 10 write it down on the table. iii. Measure the value for the potential difference across RLC, when the current in the circuit is zero, v(t);=0 III. Data and Results. i. For the experimental values of the impedance (Z) and the phase angle (o), use the relation ngp = 1/0/10 and Eq.5, respectively. ii. For the theoretical value of the impedance and the phase angle, use Eqs.3-4. iii. Show your calculation for Z and (p values right below the table. % % IVCt)i=o| - exp '1'\