Lab2_Fall2023

.pdf

School

Edmonds Community College *

*We aren’t endorsed by this school

Course

352

Subject

Electrical Engineering

Date

Dec 6, 2023

Type

pdf

Pages

Uploaded by chackben001

Lab Assignment 2: State Variable Modeling and Mutual Inductance Revision: January 24, 2023 1 © Washington State University School of EECS Summary State variable modeling is compact and powerful technique commonly used in advanced engineering systems. The state variable model may initially sound complex, but in reality it simplifies the analysis and design of complex systems with multiple inputs and multiple outputs (MIMO). In this lab, we will explore the state variable technique to model a mutual inductance circuit and an underdamped RLC circuit. In the first experiment, we will use a simplified model for the mutual inductance, and experimental measurements will be collected to estimate the mutual inductance (M) and the coupling coefficient k. Once those parameters are determined experimentally, the circuit is modeled as a state variable and then simulated using MATLAB. You will be able to compare your experimental measurement with your simulated results. In the second experiment, you will model an underdamped RLC circuit with state variables, and compare the experimental measurements with MATLAB and PSPICE simulations. Learning Outcomes: After completing this lab, you should be able to: • Experimentally estimate the mutual inductance M and the coupling coefficient k. • Experimentally locate the dots and the sign conventions of the mutual inductances. • Model electric circuits using state variables and simulate them using MATLAB. • Plot the 2-D state trajectories for simulated and experimental measurements. Required Equipment • EE352 analog parts kit • Breadboard • Function generator, oscilloscope, DMM • Two inductors that can be coupled by placing them within close proximity.

Lab Assignment 2: State Variable Modeling and Mutual Inductance 2 © Washington State University School of EECS I. Mutually Coupled Transformer Circuit When two inductors are placed in close proximity, they share a magnetic flux that causes a mutual interaction among them such that the current in one inductor creates a flux which induces a voltage across the second inductor, and vice versa. The mutual interaction is a function of many variables such as the permeance of the medium, the flux linkage, the number of turns of each inductor, the self-inductances and the distance between the two inductors. The induced voltage across the second inductor (v 2 ) is caused by the rate of change of the current in first inductor times the mutual inductance (M), and vice versa, in addition to the voltage caused by the self-inductance. This can be expressed as 𝑣𝑣 1 = 𝐿𝐿 1 𝑑𝑑𝑖𝑖 1 𝑑𝑑𝑑𝑑 + 𝑀𝑀 𝑑𝑑𝑖𝑖 2 𝑑𝑑𝑑𝑑 (1 𝑎𝑎 ) 𝑣𝑣 2 = 𝑀𝑀 𝑑𝑑𝑖𝑖 1 𝑑𝑑𝑑𝑑 + 𝐿𝐿 2 𝑑𝑑𝑖𝑖 2 𝑑𝑑𝑑𝑑 . (1 𝑏𝑏 ) One approach to estimate the mutual inductance experimentally is to connect the two inductors in a transformer setting as shown in Figure 1. Then, by applying a time varying signal u(t) at the input, an induced voltage is observed at the output. If we model a very large secondary output resistance by setting R o =∞ (open circuit ), then the secondary current i 2 =0 and its derivative is zero, so that v 2 becomes a function of i 1 . R 1 is useful in measuring and estimating i 1 . We will ignore the effect of R L1 (the coil winding resistance of inductor L 1 ), as it will be assumed to be much less than R 1 . Two types of signals can be easily applied to this circuit to find an analytic solution. The first is a sinusoidal signal which allows us to use phasor analysis. Hence equation (1) becomes 𝑉𝑉 1 = 𝑗𝑗𝑗𝑗𝐿𝐿 1 𝐼𝐼 1 + 𝑗𝑗𝑗𝑗𝑀𝑀𝐼𝐼 2 (2 𝑎𝑎 ) 𝑉𝑉 2 = 𝑗𝑗𝑗𝑗𝑀𝑀𝐼𝐼 1 + 𝑗𝑗𝑗𝑗𝐿𝐿 2 𝐼𝐼 2 , (2 𝑏𝑏 ) where 𝑉𝑉 1 , 𝑉𝑉 2 , 𝐼𝐼 1 and 𝐼𝐼 2 are phasors. The second is a triangular pulse which allows us to estimate the derivatives using the linear approx imation of ΔI/Δt , and at a given frequency f the derivatives of the current are approximated as L 2 M L 1 R L1 R 1 u(t) v 1 v R1 R L2 R o v 2 V o I 1 I 2 Figure 1. Mutual inductance in a transformer

Lab Assignment 2: State Variable Modeling and Mutual Inductance 3 © Washington State University School of EECS 𝑑𝑑𝑖𝑖 1 𝑑𝑑𝑑𝑑 = 2 𝑓𝑓∆𝐼𝐼 1 and 𝑑𝑑𝑖𝑖 2 𝑑𝑑𝑑𝑑 = 2 𝑓𝑓∆𝐼𝐼 2 . (3) The objective of the experiment is to estimate the mutual inductance M using phasor domain analysis for a sinusoidal input, and using the time domain analysis with a triangular pulse at the input. Pre-lab: (a) Let the self-inductance L 1 = L 2 = 200 mH, R o =∞ and R L1 = R L2 = 5 0 Ω. If input u(t) is a sinusoidal signal at 4 00Hz (ω=2513.3 rad/sec) evaluate the impedance across the inductors L 1 and L 2 . If we choose R 1 >>R L1 , then justify that R L1 can be ignored. If we choose R 1 =10 kΩ, f ind an expression to estimate the mutual inductance M when u(t) = 5sin(800πt) and v 2 is known (measured) such that v 2 (t) = V m sin(800πt +θ) . (b) From the analysis in (a) you may assume that v R1 (t) ≈ u(t). If you apply a triangular pulse with 10 V peak-to-peak at 400 Hz, find an expression for the 𝑑𝑑𝑖𝑖 1 𝑑𝑑𝑑𝑑 and sketch 𝑖𝑖 1 ( 𝑑𝑑 ) , 𝑑𝑑𝑖𝑖 1 𝑑𝑑𝑑𝑑 and 𝑣𝑣 2 ( 𝑑𝑑 ) . Find the expression for M if 𝑣𝑣 2 ( 𝑑𝑑 ) is known. Lab Procedures: 1. Pick two bulky inductors (available in the lab cabinets); measure their values R L1 , L 1 , R L2 , L 2 using the RLC meter and record them in your lab notebook. 2. Construct the transformer circuit as shown in Figure 1, using 10 k Ω for R 1 . 3. Apply a triangular pulse with 10 V peak-to-peak centered at zero at 400Hz. 4. Connect CH1 of the scope to the input signal, CH2 across L 1 and CH 3 to the output. Use the x1 probe for CH3 (with 1 MΩ internal resistance), and assume R o = R scope ≈ ∞ . 5. To reduce noise on CH3, select the ‘Acquire’ menu, then select ‘Average’. Set the number of signals averaged to 128. To disable averaging, select ‘Sample’ from the ‘Acquire’ menu. 6. Estimate the current I 1 by measuring the voltage v R1 across R 1 . To measure voltage v R1 , use the scope’s Math function CH1 − CH2. You should get a triangular pulse waveform which is approximately equal to the input u(t); sketch this waveform in your lab notebook, and save it on your thumb drive to be submitted in your lab report. 7. Measure the output voltage; you should get a square wave. Sketch the measured output voltage on your lab notebook and save the measured waveform on your thumb drive to be submitted in your report. Estimate the mutual inductance according to equation (1b) and your solution to question (b) of the pre-lab. 8. Demonstrate your circuit with the triangular wave to the TA and have him/her initial your checklist. 9. Apply a 10 V peak-peak sine wave at 400 Hz. Measure the output voltage and the voltage v R1 across R 1 using CH1-CH2 math function. Sketch measured plots on your notebook. Save these two scope signals on your thumb drive to be submitted in your report. 10. Estimate the mutual inductance M using the sinusoidal wave measurements and your solution to question (a) of the pre-lab. 11. Demonstrate your circuit with the sinusoidal wave to the TA and have him/her initial your checklist. 12. Get the average M of the two estimated cases, the triangular pulse and sinusoidal wave, and then estimate the coupling coefficient k from M.

Lab Assignment 2: State Variable Modeling and Mutual Inductance 4 © Washington State University School of EECS 13. From your measurements (either triangular or sinusoidal) determine the location of the dot on the secondary winding assuming that the dot is located on the R 1 side of the primary winding. Show that the dot convention changes when you change the relative direction of the coil windings. Demonstrate the dot convention to the TA and have him/her initial your check list. Post-Lab Exercise: Appendix A of this assignment derives a state variable model of the transformer circuit of Figure 1. Appendix B contains a MATLAB m-file which simulates the transformer’s response to sinusoidal and triangular input waveforms, using the state space model of Appendix A. The MATLAB m-file requires values for the oscilloscope resistance R o (estimated during lab assignment 1) and the mutual inductance M (estimated from the experimental data collected in this lab assignment). (a) Set the MATLAB m-file variables R1, Ro, L1, L2, M, RL1 and RL2 (on the first line of code in the file) to their measured or estimated values . Do not use the default values of these variables supplied in the m-file. (b) Check that the dot convention used in the state space model is consistent with the dot convention of the circuit you constructed in the lab. If not, insert negative signs at the appropriate places in Step 2 of the m-file. (c) Run the MATLAB m-file to simulate the responses to both the sinusoidal and triangular inputs. Include the MATLAB simulated responses in your lab report. (d) Compare the MATLAB simulated responses to the responses you measured in the lab. Note: The control systems toolbox is necessary in order to run the MATLAB m-file in Appendix B. Type “help lsim” at the command prompt in MATLAB to determine whether the version of MATLAB you are using includes the control systems toolbox. II. State Variable Modeling of RLC circuit You will create a state variable model of the RLC circuit shown in Figure 4. Then you will compare the measured response of your circuit with a MATLAB simulation of the circuit. State variables are intermediate signals within the circuit. If the state variable signals are known, then you can solve for any voltage and current in the circuit. As a rule of thumb, use capacitor voltages and inductor currents as state variables because their derivatives are continuous; hence, they can be used to define the state equations. To measure the inductor current, you will use R 2 in series with L to sense the inductor current. R 2 is selected to be 1.1 Ω ≈1Ω so that V R2 ≈ I L . You should assign the first state variable x 1 (t) = v C (t) and the second state variable x 2 (t) = i L (t). The outputs of this circuit are v C (t) and v R2 (t). You will plot the simulated and measured state trajectories of the circuit. The state trajectory is a graph that plots x 1 (t) vs. x 2 (t) at every t. To obtain accurate measurements, the internal resistance R L of the inductor shown in Figure 3 must to be included, and the internal 50Ω resistance of the signal generator mu st also be included. Hence, given the ideal circuit shown in Figure 2, the actual circuit to be simulated is shown in Figure 4.

Lab Assignment 2: State Variable Modeling and Mutual Inductance 5 © Washington State University School of EECS Pre-lab: (a) Determine a state variable model for the circuit shown in Figure 4. Use the states shown in Figure 4 ( x 1 , x 2 ) as your state variables. Your model should output both system states. You need to include the 50Ω internal resistance R s of the function generator and the internal resistance R L of the inductor, which is about 2 Ω . (Note: The solution to this question is provided in Appendix C below.) Lab Procedures: 1. Record actual values for R 1 , R 2 , L, R L and C in your lab notebook and construct the circuit in Figure 4. 2. Before connecting the input voltage to the circuit, apply a 1 V (pp) square wave from the function generator to emulate a unit step input function. Connect the input signal using a Tee directly to CH1 of the scope and measure the open circuit input signal. Adjust the offset voltage so that the unit step signal starts from 0V to 1V. L (a) L R L (b) Figure 3. (a) Ideal inductor. (b) Practical inductor with series resistance. Figure 2. Ideal RLC circuit. V C I L R 2 1.1Ω R 1 = 68Ω u(t) L 1 mH v R2 C 10 µF Figure 4. Actual RLC circuit R 2 1.1Ω v R2 x 1 C 10 µF R 1 = 68Ω x 2 R L ≈ 2Ω L 1 mH u(t) R s = 50Ω I 1

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version