The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t >

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

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

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

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

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

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

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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms]. How should we start this problem? What is the first step? a)Use Ohm’s Law to find the voltage across the resistor.Use Ohm’s Law to find the voltage across the resistor b)Use source transformations to convert the current source and resistor to a voltage source and resistor.Use source transformations to convert the current source and resistor to a voltage source and resistor c)Define the inductive voltage.Define the inductive voltage d)Define the capacitive current.Define the capacitive current e)Define the inductive current.Define the inductive current f)Define the capacitive voltage.Define the capacitive voltage

Your choice for First Step – Use Ohm’s Law to find the voltage across the resistor 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 v X. 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.try again The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for First Step – Use source transformations to convert the current source and resistor to a voltage source and resistor 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.try again The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for First Step – Define the inductive voltage 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 v X. The capacitive voltage is already defined. Let’s go ahead and define the inductive voltage.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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for First Step – Define the capacitive current 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.try again The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for First Step – Define the inductive current 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.try again The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for First Step – Define the capacitive voltage 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.try again The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Defining the Inductive Voltage We have defined the inductive voltage. What should the second step be? a)Find the inductive voltage.Find the inductive voltage b)Find the capacitive voltage.Find the capacitive voltage c)Find the resistive voltage.Find the resistive voltage d)Find the voltage across the current source.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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for Second Step – Find the inductive voltage 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.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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for Second Step – Find the capacitive voltage 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.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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for Second Step – Find the resistive voltage 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 v L and v C. If we found v L and v C, we could just use Loop B, and not need the voltage across the resistor and the current source. Please go back and try again.try again The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Your choice for Second Step – Find the voltage across the current source 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 v L and v C. If we found v L and v C, we could just use Loop B, and not need the voltage across the resistor and the current source. Please go back and try again.try again The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Finding the Inductive Voltage – 1 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 v L and i S are in the passive sign convention for the inductor, the equation has a positive sign. We have Next 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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Finding the Inductive Voltage – 2 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.next slide The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Finding the Inductive Voltage – 3 Performing the differentiation, we get The next step is to find an expression for the capacitive voltage v C (t), for t > 0.next 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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Finding the Capacitive Voltage – 1 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 v C and i S are in the passive sign convention for the capacitor, the equation has a positive sign. We have Next 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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Finding the Capacitive Voltage – 2 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.next slide The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Finding the Capacitive Voltage – 3 Performing the integration, we get Now, we can write KVL.write 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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Writing KVL Writing KVL around Loop B, we get Now, we can get the solution to part a). 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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Solution to Part a) Simplifying, we get the solution Next, we do part b).do 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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Solution to Part b) At this point, part b) involves simply plugging in the value for t, or Now for part c).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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

Solution to Part c) To get the energy stored in the inductor, we use the formula for this, Now for part d).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. a) Find v X (t) for t > 0. b) Find v X (0.1[ms]). c) Find the energy stored in the inductor at t = 0.1[ms]. d) Find the energy stored in the capacitor at t = 0.1[ms].

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

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 Overview slide. Overview

The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t >

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The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t >

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Presentation on theme: "The value for the current source in the circuit shown is given below. The voltage across the capacitor at t = 0 is also given. a) Find v X (t) for t >"— Presentation transcript:

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