Problems With Assistance Module 6 – Problem 1

1. The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. Problems With AssistanceModule 6 – Problem 1 Filename: PWA_Mod06_Prob01.ppt Go straight to the First Step Go straight to the Problem Statement Next slide

2. Overview of this Problem In this problem, we will use the following concepts: • Defining Equation for Inductors • Defining Equations for Capacitors Go straight to the First Step Go straight to the Problem Statement Next slide

3. Textbook Coverage The material for this problem is covered in your textbook in the following sections: • Circuits by Carlson: Sections #.# • Electric Circuits 6th Ed. by Nilsson and Riedel: Sections #.# • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Section #.# • Fundamentals of Electric Circuits by Alexander and Sadiku: Sections #.# • Introduction to Electric Circuits 2nd Ed. by Dorf: Sections #-# Next slide

4. Coverage in this Module The material for this problem is covered in this module in the following presentation: • DPKC_Mod06_Part01 Next slide

5. Problem Statement • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. Next slide

6. Solution – First Step – Where to Start? • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. How should we start this problem? What is the first step? Next slide

7. Problem Solution – First Step • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. • How should we start this problem? What is the first step? • Use Ohm’s Law to find the voltage across the resistor. • Use source transformations to convert the current source and resistor to a voltage source and resistor. • Define the inductive voltage. • Define the capacitive current. • Define the inductive current. • Define the capacitive voltage.

8. Your choice for First Step –Use Ohm’s Law to find the voltage across the resistor • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is not a good choice for the first step. We can certainly find the voltage across the resistor. However, it will not help us much, since we would also need to know the voltage across the current source to be able to find vX. To find the voltage across the current source, we need to solve the rest of the circuit, which would make this approach unnecessary. Finding the voltage across the resistor will not help much. Go back and try again.

9. Your choice for First Step –Use source transformations to convert the current source and resistor to a voltage source and resistor • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is not a good choice. The current source and resistor are in series, and not in parallel. Therefore, it would be a mistake to try to use source transformations here. We might recognize that the resistor is going to have no effect on anything here, except for the voltage across the current source, which we don’t need. Please go back and try again.

10. Your choice for First Step –Define the inductive voltage • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is a good choice for the first step, and the one that we will choose here. Our goal here will be to find the inductive voltage and capacitive voltage, and use them with KVL to get vX. The capacitive voltage is already defined. Let’s go ahead and define the inductive voltage.

11. Your choice for First Step –Define the capacitive current • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This can be done, but it is not a good choice for the first step. We are sophisticated enough about circuit analysis by this point to recognize that the current through the capacitor is equal to the source current. We don’t need to define a new current here. Therefore, we recommend that you go back and try again.

12. Your choice for First Step –Define the inductive current • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This can be done, but it is not a good choice for the first step. We are sophisticated enough about circuit analysis by this point to recognize that the current through the inductor is equal to the source current. We don’t need to define a new current here. Therefore, we recommend that you go back and try again.

13. Your choice for First Step –Define the capacitive voltage • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is not a good choice. The capacitive voltage has already been defined. There is no need to define it again. Please go back and try again.

14. Defining the Inductive Voltage • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. • We have defined the inductive voltage. What should the second step be? • Find the inductive voltage. • Find the capacitive voltage. • Find the resistive voltage. • Find the voltage across the current source.

15. Your choice for Second Step –Find the inductive voltage • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is a good choice for the second step, and the one that we will choose here. It would have been just as good a choice to start with the capacitive voltage. However, just by arbitrary choice, we have chosen to find the inductive voltage first. Let’s go ahead and find the inductive voltage.

16. Your choice for Second Step –Find the capacitive voltage • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is a good choice for the second step, but it is not the one that we will choose here. It is a reasonable choice to start with the capacitive voltage. However, just by arbitrary choice, we have chosen to find the inductive voltage first. Let’s go ahead and find the inductive voltage.

17. Your choice for Second Step –Find the resistive voltage • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is not a good choice for the second step. If we were to find the resistive voltage, it would be because we were going to take KVL around Loop A shown in this diagram. That would mean we would also need to find the voltage across the current source. The only way to find the voltage across the current source is to find the voltage across the resistor and vL and vC. If we found vL and vC, we could just use Loop B, and not need the voltage across the resistor and the current source. Please go back and try again.

18. Your choice for Second Step –Find the voltage across the current source • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. This is not a good choice for the second step. If we were to find the voltage across the current source, we would need to find the voltage across the resistor and vL and vC. If we found vL and vC, we could just use Loop B, and not need the voltage across the resistor and the current source. Please go back and try again.

19. Finding the Inductive Voltage – 1 • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. To find the inductive voltage, we use the defining equation for the inductor. This equation is given here, as a reminder. Be careful about the sign convention. Here, since vL and iS are in the passive sign convention for the inductor, the equation has a positive sign. We have Next step

20. Finding the Inductive Voltage – 2 • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. We can substitute in, and get Note that this will come out with units of [Volts], since we have used [Henries], [s], and [A] for the other units. Now, we can perform the differentiation, and get the result in the next slide.

21. Finding the Inductive Voltage – 3 • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. Performing the differentiation, we get The next step is to find an expression for the capacitive voltage vC(t), for t > 0.

22. Finding the Capacitive Voltage – 1 • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. To find the capacitive voltage, we use the defining equation for the capacitor. This equation is given here, as a reminder. Be careful about the sign convention. Here, since vC and iS are in the passive sign convention for the capacitor, the equation has a positive sign. We have Next step

23. Finding the Capacitive Voltage – 2 • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. We can substitute in, to get Note that this will come out with units of [Volts], since we have used [Henries], [s], and [A] for the other units. Now, we can perform the integration, and get the result in the next slide.

24. Finding the Capacitive Voltage – 3 • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. Performing the integration, we get Now, we can write KVL.

25. Writing KVL • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. Writing KVL around Loop B, we get Now, we can get the solution to part a).

26. Solution to Part a) • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. Simplifying, we get the solution Next, we do part b).

27. Solution to Part b) • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. At this point, part b) involves simply plugging in the value for t, or Now for part c).

28. Solution to Part c) • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. To get the energy stored in the inductor, we use the formula for this, Now for part d).

29. Solution to Part d) • The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. • Find vX(t) for t > 0. • Find vX(0.1[ms]). • Find the energy stored in the inductor at t = 0.1[ms]. • Find the energy stored in the capacitor at t = 0.1[ms]. To get the energy stored in the capacitor, we use the formula for this, Go to Comments Slide

30. Do I have to remember how to integrate? • Yes, I am afraid that we need to remember our calculus pretty well at this point. We are going to use it many different ways. Reviewing calculus concepts is a good plan at several stages during our training as engineers. • Hopefully this will be fun. Calculus is a very powerful tool. While it can be confusing at first (or at second, or at third), for most people, with time, it becomes a easy to use tool to solve key problems. Go back to Overviewslide.