Problems With Assistance Module 1 – Problem 1

Problems With AssistanceModule 1 – Problem 1 Filename: PWA_Mod01_Prob01.ppt Go straight to the First Step Go straight to the Problem Statement Next slide

In this problem, we will use the following concepts: • Power and energy • Sign conventions • Changing directions of energy flow Overview of this Problem Next slide

The material for this problem is covered in your textbook in the following sections: • Circuits by Carlson: Sections 1.1 & 1.2 • Electric Circuits 6th Ed. by Nilsson and Riedel: Sections 1.5 & 1.6 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Sections 1.2 & 1.3 • Fundamentals of Electric Circuits by Alexander and Sadiku: Sections 1.3 through 1.6 • Introduction to Electric Circuits 2nd Ed. by Dorf: Sections 1-2, 1-4 & 1-5 Textbook Coverage Next slide

The material for this problem is covered in this module in the following presentations: • DPKC_Mod01_Part01 • DPKC_Mod01_Part02 Coverage in this Module Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. A blender Problem Statement Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. The coefficient of time in the argument of a sinusoid is typically assumed to have units of [rad/s]. This is usually not shown, but is given here for clarity. Problem Statement (Preliminary Notes 1) Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. I have enclosed all units in square brackets. This is not a convention, and you are not required to do so. However, it sets off units, and thus adds to clarity for these problems. Problem Statement (Preliminary Notes 2) Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. Solution – First Step – Where to Start? How should we start this problem? What is the first step? Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. Problem Solution – First Step • How should we start this problem? What is the first step? • Integrate the voltage over one period. • Integrate the current over one period. • Find an expression for the power with the right sign. • Find the period of the sinusoids.

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. Problem Solution – First Step -- Integrate the voltage over one period. This is not a good choice for a first step. While we have to integrate power to get energy, integrating voltage does not give us anything we can use in this problem. Go back to Problem Solution – First Step.

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. Problem Solution – First Step -- Integrate the current over one period. This is not a good choice for a first step. While we have to integrate power to get energy, integrating current, while it does give us total charge, does not give us anything we can use in this problem. Go back to Problem Solution – First Step.

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. Problem Solution – First Step -- Find the period of the sinusoid. It would be useful to know the period of the sinusoid at some point in this problem, but not at the beginning. In Part d) we will use this, but it is not crucial at the beginning. It would be better to start someplace else. Go back to Problem Solution – First Step.

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. Problem Solution – First Step -- Find an expression for the power with the right sign. This is a good choice for a place to start. Once we have this expression, we can start finding things of interest. Now, what should the correct expression be? Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. • Find the power absorbed by the blender at t = 4.3[ms]. • Find the power absorbed by the blender at t = 8.6[ms]. • Find the energy absorbed by the blender over one cycle of the sinusoid. Power – Using the Proper Sign We can write a power expression for either the blender or for the wall socket. It makes sense to write the expression for the blender, since all the questions relate to the blender. We would write: Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. For Part a), the solution is obtained by substituting in t = 0 in this expression, and evaluating. Go ahead and do this, and write down your answer. Solution for Part a)—Step 1 Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. For Part a), the solution is obtained by substituting in t = 0 in this expression, and evaluating. Go ahead and do this, and write down your answer. Solution for Part a) — Evaluating For your answer, did you get: Answer 1: Answer 2: Something else?

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. For Part a), the solution is obtained by substituting in t = 0 in this expression, and evaluating. Go ahead and do this, and write down your answer. Your Solution for Part a) was 750[W] If your answer was then you have made an error. You have neglected to find the cos (-5o) when you evaluated, or made some error getting it. Try again.

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. For Part a), the solution is obtained by substituting in t = 0 in this expression, and evaluating. Go ahead and do this, and write down your answer. Your Solution for Part a) was Something Else If your answer was not then you have made an error. You could try again, or you can look at the solution.

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 0. For Part a), the solution is obtained by substituting in t = 0 in this expression, and evaluating. If your answer was then you have the correct solution. What you did was Your Solution for Part a) was 747[W] Next part (b)

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 4.3[ms]. For Part b), the solution is obtained by substituting in t = 4.3[ms] in this expression, and evaluating. Go ahead and do this, and write down your answer. Solution for Part b) — Step 1 Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 4.3[ms]. For Part b), the solution is obtained by substituting in t = 4.3[ms] in this expression, and evaluating. Go ahead and do this, and write down your answer. Solution for Part b) — Evaluating For your answer, did you get: Answer 1: Answer 2: Something else?

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 4.3[ms]. For Part b), the solution is obtained by substituting in t = 4.3[ms] in this expression, and evaluating. Go ahead and do this, and write down your answer. If your answer was then you have made an error. You have neglected to convert radians to degrees. Note that when you plug in 4.3[ms] for t, and multiply by 377, the result will have units of [rad]. You need to convert to degrees, or use radians as your units on your calculator. Note that you will also need to convert 5 degrees to radians if you take this approach. Try again. Your Solution for Part b) was 748[W]

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 4.3[ms]. For Part b), the solution is obtained by substituting in t = 4.3[ms] in this expression, and evaluating. Go ahead and do this, and write down your answer. Your Solution for Part b) was Something Else If your answer was not then you have made an error. You could try again, or you can look at the solution.

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 4.3[ms]. For Part b), the solution is obtained by substituting in t = 4.3[ms] in this expression, and evaluating. If your answer was then you have the correct solution. What you did was Your Solution for Part b) was –1.39[W] Next part (c)

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 8.6[ms]. For Part c), the solution is obtained by substituting in t = 8.6[ms] in this expression, and evaluating. Go ahead and do this, and write down your answer. Solution for Part c) — Step 1 Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 8.6[ms]. For Part c), the solution is obtained by substituting in t = 8.6[ms] in this expression, and evaluating. Go ahead and do this, and write down your answer. Solution for Part c) — Evaluating For your answer, did you get: Answer: Something else?

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 8.6[ms]. For Part b), the solution is obtained by substituting in t = 8.6[ms] in this expression, and evaluating. Go ahead and do this, and write down your answer. Your Solution for Part c) was Something Else If your answer was not then you have made an error. You could try again, or you could look at the solution for Part b) which is very similar, or you can look at the solution for Part c).

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the power absorbed by the blender at t = 8.6[ms]. For Part c), the solution is obtained by substituting in t = 8.6[ms] in this expression, and evaluating. If your answer was then you have the correct solution. What you did was Your Solution for Part c) was 746[W] Next part (d)

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the energy absorbed by the blender over one cycle of the sinusoid. For Part d), the solution is obtained by integrating the power over a single period. The power was Thus, we want to integrate this over one period. Any period will be fine. Generally, we want to use the easiest possible period. This is usually the period from t=0 to t equal to one period. Let T be the period, and we write To do this, we need to find the period, T. Solution for Part d) — Step 1 Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the energy absorbed by the blender over one cycle of the sinusoid. We need to find the period, T. Note that we have the frequency in terms of [rad/s]. This is called the angular frequency. To get the period from this, we convert to frequency in [cycles/s], or [Hertz], or [Hz]. Then we take the inverse. Or, to help us remember, we can set it up as a series of unit conversions. We can write Note that the [cycles] are usually left out of this result. Symbolically, we can write Solution for Part d) — Finding the Period Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the energy absorbed by the blender over one cycle of the sinusoid. So, for Part d), we need to integrate the power over a single period. We have the integral To do this integral, we have many choices. It is complicated enough so that I am going to choose to use a computer mathematics tool. There are many, but I am going to use Mathcad, a Math Soft product. Solution for Part d) — The Integral Next slide

We have been given the following information. We want to integrate these over a period. To make this easier, we convert 5[degrees] to radians. We use the fact that p[radians]=180[degrees] to write the following. Next we define the variables for Mathcad, and then evaluate. Here, we have gone ahead and checked our calculations for Parts a), b), and c), and then Part d). answer for part a) answer for part b) answer for part c) answer for part d) This is shown as an example only. It involves several kinds of notation that are special for Mathcad. The point is that you should find some tool to perform mathematical computations. This could be a computer program, an advanced calculator, or fundamental calculus skills. Mathcad Evaluation Next slide

The voltage in a wall socket has been measured, and found to be equal to A blender is connected to this wall socket, and the current in the blender is measured and found to be The diagram shows the voltage and current polarities in this connection. • Find the energy absorbed by the blender over one cycle of the sinusoid. Thus, when we integrate the power over a single period, we get The answer is wabs,bl = 6.224[J]. Solution for Part d) – Completed Some notes

In this problem, the blender is absorbing positive power most of the time, but is actually delivering positive power to the wall socket part of the time. • This is possible because of the inductive nature of the motor in the blender. As we shall see later, energy can be stored in an inductor, and delivered back to the circuit later. • However, energy flow in inductors is not an important concept right now. What is important right now is that we need to be very careful to use the correct signs in our power and energy expressions. Note 1: Why was the sign of the power changing? Next note Go back to Overviewslide.

In this problem, the integral is complicated enough so that it might not be included in the integral table in your textbook. However, most mathematicians would suggest that it is not that complicated, as integrals go. • Whatever. The key here is that with present tools, even a complicated integral can be solved relatively easily, so there is no reason to fear them. • In addition, later in this subject you may be introduced to another technique for solving this kind of problem. The technique is called complex power. Since these kinds of problems are common with sinusoidal power sources, like wall sockets, engineers have come up with this powerful approach that makes these problems easier. This can be something to look forward to. Note 2: Are all the integrals this hard? Next note Go back to Overviewslide.

In this problem, one of the major complications was that the coefficient of time was given in [rad/s], and the phase of the sinusoid was given in [degrees]. If the phase had been given in [rad] from the beginning, the problem would have been easier. • True enough. However, in later techniques in the circuits area, it is convenient to use the phase in degrees, since it is a common unit for angles. The technique is called phasor analysis. In any case, this is a recurring problem in all of engineering. • The only real long-term solution is to get in the habit of paying attention to units all the time. In 1999, a very expensive space probe that was headed to Mars was lost because of insufficient attention to units. One must be alert to this problem. If your instructor is emphasizing the importance of units in your course, this is at least one reason why. Note 3: Why was the phase of the current given in degrees, and not radians? Go back to Overviewslide.

Problems With Assistance Module 1 – Problem 1