Assignment: Introduction to Modeling of Engineering Systems

Question 1. A battery can be modeled as an ideal source with an internal resistance. The Thevenin model includes a constant voltage source in series with a resistor. The negative battery terminal corresponds to the inlet to the voltage source while the point AFTER the resistor corresponds to the positive battery terminal. The Norton model has a constant current source with a parallel resistor. Here the battery terminals span across the resistor. These models exhibit a drop in terminal voltage as the tocurrent provided by the battery increases. Apply this model for the situations described below. Start with a diagram showing the source model and an unspecified load. Write equations using KVL, KCL and Ohm's law as needed to allow you to determine the model parameters (Vs or I, and Rs).

a) When measured with open circuit (no load - zero current draw) the terminals of a battery show a voltage of 13.8 volts. When delivering 10. amps to an external load, the voltage drops to 11.4 volts. What is the value of the constant voltage source (Vs) and the internal resistance (Rs)?

b) The voltage of a battery is measured under 2 different loads and yields the following results: Load 1, 5.0 amps. measured terminal voltage: 13.6 Volts.

Load 2, 50. amps, measured terminal voltage: 9.6 Volts

Develop models for the battery of these 2 types:

i. a constant voltage source with source resistance (find values for Vs and Rs)
ii. a constant current source with source resistance (find values for Vs and Rs)

Question 2. An automobile battery is modeled as a 13V ideal voltage source in series with a 0.03 Ohm internal resistance. It is to be charged from a 15 V source (the alternator). The charging current must be limited to 2.5 Amps by placing a resistor in series with the 15 volt source. What resistance will achieve the desired current? (Note when charging a battery the "larger" source must drive current into the battery.)

Question 3. The data sheet for a size D alkaline battery shows the following characteristics after 2 hours of usage.

Develop a model for this battery using a constant voltage source (Vs) with an internal resistance (Rs) which duplicates these characteristics by performing the following steps:

• Sketch the circuit diagram for the load test showing the battery (Vs in series with Rs) connected to an external load
• Write the KVL equation that would apply for the test.
• Substitute VT (terminal voltage) for the voltage drop across the load resistance to obtain a linear equation for VT as a function of current.
• Use regression methods to get the best fit values for Vs and Rs.
• Prepare a plot that shows how the experimental values in the table compare to the predicted values from your model (use best practices for your plot from the plot guidelines)

Use the source transformation method to obtain a battery model that uses a constant current source with associated resistance. Show a diagram for this model and find the values for I, and R.