Problems With Assistance Module 1 – Problem 2

Problems With AssistanceModule 1 – Problem 2 Filename: PWA_Mod01_Prob02.ppt This problem is adapted from: Problem 1.16, page 37 in Circuits by A. Bruce Carlson Brooks/Cole Thomson Learning 2000 ISBN: 0-534-37097-7 Go straight to the First Step Go straight to the Problem Statement Next slide

In this problem, we will use the following concepts: • Voltage • Current • Definitions of ideal sources Go straight to the First Step Overview of this Problem Go straight to the Problem Statement Next slide

The material for this problem is covered in your textbook in the following sections: • Circuits by Carlson: Section 1.2 • Electric Circuits 6th Ed. by Nilsson and Riedel: Section 2.1 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Section 1.3 • Fundamentals of Electric Circuits by Alexander and Sadiku: Section 1.6 • Introduction to Electric Circuits 2nd Ed. by Dorf: Sections 2-2, 2-3 & 2-4 Textbook Coverage Next slide

The material for this problem is covered in this module in the following presentation: • DPKC_Mod01_Part03 Coverage in this Module Next slide

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. Problem Statement Next slide

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. Solution – First Step – Where to Start? How should we start this problem? What is the first step? Next slide

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. • How should we start this problem? What is the first step? • Plot the current versus the voltage. • Plot the voltage versus the current. • Calculate the current. • Calculate the voltage. • State the definition of an “ideal current source”. Problem Solution – First Step

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. “Calculating the current” is not the best first step. It is possible to plug in values for voltage and find the current that results. However, it is not clear what the results will tell you. It is better to establish first what the goal is. If you knew the goal already, then this might give you some insight. However, we recommend that you try again. You Said the First Step Was to Calculate the Current

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. “Calculating the voltage” is not the best first step. It is possible, but not easy, to plug in values for voltage and find the current that results. However, it is not clear what the results will tell you. It is better to establish first what the goal is. If you knew the goal already, then this might give you some insight. However, we recommend that you try again. You Said the First Step Was to Calculate the Voltage

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. Making a plot at this point is not that bad an idea. You might gain some insight from the plot. We would suggest that, while this is a reasonable first step, there is a more fundamental first step. We will suggest that you try this step again after the first step. In this case, though, we recommend that you try again. You Said the First Step Was to Plot the Current versus Voltage, or to Plot the Voltage versus the Current.

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. Next slide In order to defend the statement that this source acts like an ideal current source, we need to know what that would mean. We would suggest that the most fundamental first step is to state the definition of an “ideal current source”. Definition: A current source is a two terminal circuit element that maintains a current through its terminals. The value of the current is the defining characteristic of the current source. Any voltage can be across the current source, in either polarity. You Said the First Step Was to State the Definition of an “Ideal Current Source”. The key here is that an ideal current source would have a constant current independent of voltage.

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. An ideal current source is a source where the current is established, independent of the voltage. In other words, we can show this, if we can show that the current is not a strong function of voltage for The Definition of an “Ideal Current Source”. Next slide

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. Skip plot and go direct to an analytical solution. So, is the current a strong function of voltage for This would be a good time to try plotting either current vs. voltage, or voltage versus current. Here, it appears that current is a function of voltage, so let’s plot current vs. voltage. Among many tools for this Microsoft Excel. Suggestion: Try Making a Plot of this Formula

Plot of Current vs. Voltage The Excel spreadsheet used to make this plot should be included with these materials. The file name is: Plot_for_PWA_Mod01_Prob02.xls Go to the analytical solution. What we see here is that the change in current with voltage is fairly small in range marked in green, which is the specified range. An ideal current source would be a horizontal line. The plot in this range is not horizontal, but does have a limited variation.

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when So, is the current a strong function of voltage for Next slide An analytical approach is to consider the formula itself. The term under the radical, (400+v2), can be examined for its behavior as a function of v. Note that if v is small enough, the term (400+v2) is not a strong function of v. Why? Here, we make the engineering assumption that if we add A + B, and if A >>B, then A + B » A. Of course, there is some error in this. But, if A>>B, then it is approximately true. Try this yourself with some numbers. Is 1000 + 1 = 1000? No, but we only make a 0.1% error if we make this approximation! This is called an engineering approximation. An Analytical Approach: Is the Current Independent of Voltage?

A certain source produces a current i, with where v is given in [V]. • Justify the assertion that it acts like an ideal current source when So, is the current a strong function of voltage for The term under the radical, (400+v2), varies in the range from 400 to 500 in the range of v from –10[V] < v < 10[V]. Plugging these values into the expression gives a variation in i in the expression from 25[mA] to 22.36[mA]. This is a variation of less than 11%. If this variation is acceptable, then we can say that the source behaves like an ideal current source, within these error bounds. The Current is Independent of Voltage, Within an Error of 11% Go to Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Let’s start with the number 400. What must the units be for this number for the dimensions to be consistent under the radical? Solution for Part b) — Step 1 Next slide

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. • For Part b), the key is that the dimensions must come out properly. • Let’s start with the number 400. What must the units be for this number for the dimensions to be consistent under the radical? • [V] • [V]2 • [V]1/2 • [A/V] • [V/A] Solution for Part b) — Step 1What is Your Answer?

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Let’s start with the number 400. What must the units be for this number for the dimensions to be consistent under the radical? This answer is not correct. Since we are given that v is in [V], and is squared, then v2 must have units of [V]2. To be consistent with this, 400 must have units of [V]2. Your Solution for Part b) was [V] Go to Step 2 of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Let’s start with the number 400. What must the units be for this number for the dimensions to be consistent under the radical? This answer is not correct. Since we are given that v is in [V], and is squared, then v2 must have units of [V]2. To be consistent with this, 400 must have units of [V]2. Your Solution for Part b) was [V]1/2 Go to Step 2 of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Let’s start with the number 400. What must the units be for this number for the dimensions to be consistent under the radical? This answer is not correct. Since we are given that v is in [V], and is squared, then v2 must have units of [V]2. To be consistent with this, 400 must have units of [V]2. Your Solution for Part b) was [V/A] Go to Step 2 of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Let’s start with the number 400. What must the units be for this number for the dimensions to be consistent under the radical? This answer is not correct. Since we are given that v is in [V], and is squared, then v2 must have units of [V]2. To be consistent with this, 400 must have units of [V]2. Your Solution for Part b) was [A/V] Go to Step 2 of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Let’s start with the number 400. What must the units be for this number for the dimensions to be consistent under the radical? This answer is correct. Since we are given that v is in [V], and is squared, then v2 must have units of [V]2. To be consistent with this, 400 must have units of [V]2. Your Solution for Part b) was [V]2 Go to Step 2 of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. • For Part b), the key is that the dimensions must come out properly. • Now, let’s consider the number 0.5. What must the units be for this number for the dimensions for the entire expression to come out in [A]? • [V] • [V]2 • [V]1/2 • [A/V] • [V/A] • [W] Solution for Part b) — Step 2What is Your Answer?

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Now, let’s consider the number 0.5. What must the units be for this number for the dimensions for the entire expression to come out in [A]? This answer is not correct. Since the units under the radical are [V]2, then when we take the square root, the units in the denominator must be [V]. When 0.5 is divided by some number of [V], the answer must come out in [A]. The units of 0.5 must be [V] times [A], or [W]. Your Solution – 2nd step, Part b) was [V] Go to Solution of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Now, let’s consider the number 0.5. What must the units be for this number for the dimensions for the entire expression to come out in [A]? This answer is not correct. Since the units under the radical are [V]2, then when we take the square root, the units in the denominator must be [V]. When 0.5 is divided by some number of [V], the answer must come out in [A]. The units of 0.5 must be [V] times [A], or [W]. Your Solution – 2nd step, Part b) was [V]2 Go to Solution of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Now, let’s consider the number 0.5. What must the units be for this number for the dimensions for the entire expression to come out in [A]? This answer is not correct. Since the units under the radical are [V]2, then when we take the square root, the units in the denominator must be [V]. When 0.5 is divided by some number of [V], the answer must come out in [A]. The units of 0.5 must be [V] times [A], or [W]. Your Solution – 2nd step, Part b) was [V/A] Go to Solution of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Now, let’s consider the number 0.5. What must the units be for this number for the dimensions for the entire expression to come out in [A]? This answer is not correct. Since the units under the radical are [V]2, then when we take the square root, the units in the denominator must be [V]. When 0.5 is divided by some number of [V], the answer must come out in [A]. The units of 0.5 must be [V] times [A], or [W]. Your Solution – 2nd step, Part b) was [A/V] Go to Solution of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Now, let’s consider the number 0.5. What must the units be for this number for the dimensions for the entire expression to come out in [A]? This answer is not correct. Since the units under the radical are [V]2, then when we take the square root, the units in the denominator must be [V]. When 0.5 is divided by some number of [V], the answer must come out in [A]. The units of 0.5 must be [V] times [A], or [W]. Your Solution – 2nd step, Part b) was [V]1/2 Go to Solution of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. For Part b), the key is that the dimensions must come out properly. Now, let’s consider the number 0.5. What must the units be for this number for the dimensions for the entire expression to come out in [A]? This answer is correct. Since the units under the radical are [V]2, then when we take the square root, the units in the denominator must be [V]. When 0.5 is divided by some number of [V], the answer must come out in [A]. The units of 0.5 must be [V] times [A], or [W]. Your Solution – 2nd step, Part b) was [W] Go to Solution of Part b)

A certain source produces a current i, with where v is given in [V]. • Give the units that must be associated with the numbers 0.5 and 400 in the first expression, to get the units to come out in [A] as indicated. Go to Notes For Part b), the key is that the dimensions must come out properly. We have found that the expression for the current could have just as easily been given as When we use this format, the units for 400 tell us to use [V] for v, and when we do the unit analysis, the answer comes out in [A]. The Solution for Part b)

When we work in a single text, or a single class, or a single area, we often agree to use the same units all the time. In these situations, it is not that important to show units. • However, in any situation where the units being used is not very clear, it is good practice to show them. • A recent Mars mission failed because two groups used different units and conversions were not made. This is only one of many, many errors that have happened because of the failure to communicate about units. Show units. When in doubt, show units. When in doubt about whether you are in doubt, show units. Why is showing the units so important? Go back to Overviewslide. Go to Next Note

In this problem, we made the engineering assumption that if we add A + B, and if A >>B, then A + B » A. This is the kind of thing that engineers do to simplify problems, so that we can use a simpler method to solve the problem. The key is that if we have an accurate enough answer to the problem, it doesn’t matter whether our answer is exact. Judging when to approximate, and when not to approximate, is a part of engineering. In this problem, we are not given any firm guidelines. In engineering practice, the guidelines will depend on the application. What is this engineering approximation stuff? Go back to Overviewslide.

Problems With Assistance Module 1 – Problem 2