Here are a couple of hints: Follow-up question: why is there a negative sign in the equation? Derivatives are a bit easier for most people to understand, so these are generally presented before integrals in calculus courses. www.epemag.co.uk ISBN 13: 978-0-75-068071-4 ISBN 10: 0-75-068071-7 For information on all Newnes publications visit our web site at books.elsevier.com Typeset by Cepha Ltd Printed and bound in Great Britain 0708091011 10987654321 Substituting algebraically: Review question: Rogowski coils are rated in terms of their mutual inductance (M). %%EOF Challenge question: can you think of a way we could exploit the similarity of capacitive voltage/current integration to simulate the behavior of a water tank’s filling, or any other physical process described by the same mathematical relationship? Calculus is a branch of mathematics that originated with scientific questions concerning rates of change. What physical variable does the differentiator output signal represent? The problem is, none of the electronic sensors on board the rocket has the ability to directly measure velocity. Find all elements to solve the func. This race for speed causes problems for the power supply circuitry of computers, though, because of the current “surges” (technically called transients) created in the conductors carrying power from the supply to the logic circuits. Follow-up question: manipulate this equation to solve for the other two variables ([di/dt] = … ; L = …). One of the notations used to express a derivative (rate of change) appears as a fraction. News the global electronics community can trust. Don't show me this again. The integrator’s function is just the opposite. %PDF-1.5 %���� Another way of saying this is that velocity is the rate of position change over time, and that acceleration is the rate of velocity change over time. One way I like to think of these three variables is as a verbal sequence: Arranged as shown, differentiation is the process of stepping to the right (measuring the rate of change of the previous variable). This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in Mathematics, Statistics, Engineering, Pharmacy, etc. Thankfully, there are more familiar physical systems which also manifest the process of integration, making it easier to comprehend. If students have access to either a graphing calculator or computer software capable of drawing 2-dimensional graphs, encourage them to plot the functions using these technological resources. (ex) 4. Whenever you as an instructor can help bridge difficult conceptual leaps by appeal to common experience, do so! It is very important to your students’ comprehension of this concept to be able to verbally describe how the derivative works in each of these formulae. The studies of electricity and electronics are rich in mathematical context, so exploit it whenever possible! Your task is to determine which variable in the water tank scenario would have to be measured so we could electronically predict the other variable using an integrator circuit. Thus, when we say that velocity (v) is a measure of how fast the object’s position (x) is changing over time, what we are really saying is that velocity is the “time-derivative” of position. Just as addition is the inverse operation of subtraction, and multiplication is the inverse operation of division, a calculus concept known as integration is the inverse function of differentiation. Don't have an AAC account? ... An Engineers Quick Calculus Integrals Reference. Being air-core devices, they lack the potential for saturation, hysteresis, and other nonlinearities which may corrupt the measured current signal. As switches, these circuits have but two states: on and off, which represent the binary states of 1 and 0, respectively. This question not only tests students’ comprehension of the Rogowski coil and its associated calculus (differentiating the power conductor current, as well as the need to integrate its output voltage signal), but it also tests students’ quantitative comprehension of integrator circuit operation and problem-solving technique. Velocity is a measure of how fast its position is changing over time. What would the output of this differentiator circuit then represent with respect to the automobile, position or acceleration? The rate of the changing output voltage is directly proportional to the magnitude of the input voltage: A symbolic way of expressing this input/output relationship is by using the concept of the derivative in calculus (a rate of change of one variable compared to another). Differential calculus arises from the study of the limit of a quotient. Even if your students are not ready to explore calculus, it is still a good idea to discuss how the relationship between current and voltage for a capacitance involves time. So, if the integrator stage follows the differentiator stage, there may be a DC bias added to the output that is not present in the input (or visa-versa!). Normally transformers are considered AC-only devices, because electromagnetic induction requires a changing magnetic field ([(d φ)/dt]) to induce voltage in a conductor. Symbolically, integration is represented by a long “S”-shaped symbol called the integrand: To be truthful, there is a bit more to this reciprocal relationship than what is shown above, but the basic idea you need to grasp is that integration “un-does” differentiation, and visa-versa. ... AC Motor Control and Electrical Vehicle Applications Seconds Edition by Kwang Hee Nam PDF Free Download. Here, I ask students to relate the instantaneous rate-of-change of the voltage waveform to the instantaneous amplitude of the current waveform. It was submitted to the Free Digital Textbook Initiative in California and will remain unchanged for at least two years. Capsule Calculus by Ira Ritow PPD Free Dpwnload. Also, what does the expression [de/dt] mean? Not only is this figure realistic, though, it is also low by some estimates (see IEEE Spectrum, magazine, July 2003, Volume 40, Number 7, in the article “Putting Passives In Their Place”). To illustrate this electronically, we may connect a differentiator circuit to the output of an integrator circuit and (ideally) get the exact same signal out that we put in: Based on what you know about differentiation and differentiator circuits, what must the signal look like in between the integrator and differentiator circuits to produce a final square-wave output? endstream endobj 987 0 obj <>/Metadata 39 0 R/Pages 984 0 R/StructTreeRoot 52 0 R/Type/Catalog>> endobj 988 0 obj <>/MediaBox[0 0 612 792]/Parent 984 0 R/Resources<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 989 0 obj <>stream A familiar context in which to apply and understand basic principles of calculus is the motion of an object, in terms of position (x), velocity (v), and acceleration (a). Like the water tank, electrical inductance also exhibits the phenomenon of integration with respect to time. Potentiometers are very useful devices in the field of robotics, because they allow us to represent the position of a machine part in terms of a voltage. Basic Mathematics for Electronics with Calculus by Nelson Cooke (1989-01-01) 4.4 out of 5 stars 9. �]�o�P~��e�'ØY�ͮ�� S�ე��^���}�GBi��. The purpose of this question is to have students apply the concepts of time-integration and time-differentiation to the variables associated with moving objects. For each of the following cases, determine whether we would need to use an integrator circuit or a differentiator circuit to convert the first type of motion signal into the second: Also, draw the schematic diagrams for these two different circuits. That is, it contains an amplifier (an “active” device). For so many people, math is an abstract and confusing subject, which may be understood only in the context of real-life application. Welcome! The integration required to obtain the answer is commonly found in calculus-based physics textbooks, and is an easy (power rule) integration. Significant voltage drops can occur along the length of these conductors due to their parasitic inductance: Suppose a logic gate circuit creates transient currents of 175 amps per nanosecond (175 A/ns) when switching from the “off” state to the “on” state. Quite a bit! In calculus, we have a special word to describe rates of change: derivative. We may calculate the energy stored in an inductance by integrating the product of inductor voltage and inductor current (P = IV) over time, since we know that power is the rate at which work (W) is done, and the amount of work done to an inductor taking it from zero current to some non-zero amount of current constitutes energy stored (U): Find a way to substitute inductance (L) and current (I) into the integrand so you may integrate to find an equation describing the amount of energy stored in an inductor for any given inductance and current values. The differentiator circuit’s output signal represents the angular velocity of the robotic arm, according to the following equation: Follow-up question: what type of signal will we obtain if we differentiate the position signal twice (i.e. Advanced answer: the proper way to express the derivative of each of these plots is [dv/di]. I have found that the topics of capacitance and inductance are excellent contexts in which to introduce fundamental principles of calculus to students. The differentiator’s output signal would be proportional to the automobile’s acceleration, while the integrator’s output signal would be proportional to the automobile’s position. A passive differentiator circuit would have to possess an infinite time constant (τ = ∞) in order to generate this ramping output bias These two complementary operations may be seen at work on the instrument panel of every automobile: On this one instrument, two measurements are given: speed in miles per hour, and distance traveled in miles. Follow-up question: manipulate this equation to solve for the other two variables ([de/dt] = … ; C = …). In other words, if we were to connect an oscilloscope in between these two circuits, what sort of signal would it show us? h�bbd```b``: "k���d^"Y��$�5X��*���4�����9$TK���߿ � Usually introduced at the beginning of lectures on transformers and quickly forgotten, the principle of mutual inductance is at the heart of every Rogowski coil: the coefficient relating instantaneous current change through one conductor to the voltage induced in an adjacent conductor (magnetically linked). The easiest rates of change for most people to understand are those dealing with time. 0 One possible solution is to use an electronic integrator circuit to derive a velocity measurement from the accelerometer’s signal. This question introduces students to the concept of integration, following their prior familiarity with differentiation. Besides, it gives some practical context to integrator circuits! Incidentally, the following values work well for a demonstration circuit: If this is not apparent to you, I suggest performing Superposition analysis on a passive integrator (consider AC, then consider DC separately), and verify that VDC(out) = VDC(in). Calculus: Differential Calculus, Integral Calculus, Centroids and Moments of Inertia, Vector Calculus. Being able to differentiate one signal in terms of another, although equally useful in physics, is not so easy to accomplish with opamps. Explain how the derivatives of these functions relate to real electrical quantities. h�b```�pf�OB cB� Gaw-c���BO8�N}���ī�ص�� View All Tools. Students should also be familiar with matrices, and be able to compute a three-by-three determinant. the level of an introductory college calculus course. Special Honors. We know that the output of an integrator circuit is proportional to the time-integral of the input voltage: But how do we turn this proportionality into an exact equality, so that it accounts for the values of R and C? Ohm’s Law tells us that the amount of current through a fixed resistance may be calculated as such: We could also express this relationship in terms of conductance rather than resistance, knowing that G = 1/R: However, the relationship between current and voltage for a fixed capacitance is quite different. Hint: this circuit will make use of differentiators. Calculus for Engineering Students: Fundamentals, Real Problems, and Computers insists that mathematics cannot be separated from chemistry, mechanics, electricity, electronics, automation, and other disciplines. The easiest rates of change for most people to understand are those dealing with time. Thus, a differentiator circuit connected to a tachogenerator measuring the speed of something provides a voltage output representing acceleration. When we determine the integral of a function, we are figuring out what other function, when differentiated, would result in the given function. The calculus relationships between position, velocity, and acceleration are fantastic examples of how time-differentiation and time-integration works, primarily because everyone has first-hand, tangible experience with all three. It is perfectly accurate to say that differentiation undoes integration, so that [d/dt] ∫x dt = x, but to say that integration undoes differentiation is not entirely true because indefinite integration always leaves a constant C that may very well be non-zero, so that ∫[dx/dt] dt = x C rather than simply being x. Follow-up question: what electronic device could perform the function of a “current-to-voltage converter” so we could use an oscilloscope to measure capacitor current? What I’m interested in here is the shape of each current waveform! Follow-up question: draw the schematic diagrams for these two circuits (differentiator and integrator). A Rogowski coil has a mutual inductance rating of 5 μH. Voltage remaining at logic gate terminals during current transient = 3.338 V, Students will likely marvel at the [di/dt] rate of 175 amps per nanosecond, which equates to 175 billion amps per second. Ohm’s Law tells us that the amount of voltage dropped by a fixed resistance may be calculated as such: However, the relationship between voltage and current for a fixed inductance is quite different. So, we could say that for simple resistor circuits, the instantaneous rate-of-change for a voltage/current function is the resistance of the circuit. Underline all numbers and functions 2. This question provides a great opportunity to review Faraday’s Law of electromagnetic induction, and also to apply simple calculus concepts to a practical problem. As anyone with calculus background knows, integration introduces an arbitrary constant of integration. That is, one quantity (flow) dictates the rate-of-change over time of another quantity (height). For example, a student watching their savings account dwindle over time as they pay for tuition and other expenses is very concerned with rates of change (dollars per year being spent). These laws are straightforward, but when you’re trying to solve for one variable or another, it is easy to get them confused. The integrator circuit produces an output voltage changing at a rate proportional to the input voltage magnitude ([(dvout)/dt] ∝ vin). The subject of Rogowski coils also provides a great opportunity to review what mutual inductance is. We could use a passive integrator circuit instead to condition the output signal of the Rogowski coil, but only if the measured current is purely AC. Lower-case variables represent instantaneous values, as opposed to average values. 1004 0 obj <>stream A question such as this one highlights a practical use of calculus (the “chain rule”), where the differentiator circuit’s natural function is exploited to achieve a more advanced function. Speed is the derivative of distance; distance is the integral of speed. The d letters represent a calculus concept known as a differential, and a quotient of two d terms is called a derivative. Given that the function here is piecewise and not continuous, one could argue that it is not differentiable at the points of interest. Chapter 1 Introduction 1.1Themes1 From its beginnings in the late nineteenth century, electrical engineering has blossomed from focusing on electrical circuits for power, telegraphy and telephony to focusing on a much broader range of disciplines. “175 billion amps per second” is not the same thing as “175 billion amps”. What relationship is there between the amount of resistance and the nature of the voltage/current function as it appears on the graph? One of the fundamental principles of calculus is a process called integration. Velocity is nothing more than rate-of-change of position over time, and acceleration is nothing more than rate-of-change of velocity over time: Illustrating this in such a way that shows differentiation as a process: Given that you know integration is the inverse-function of differentiation, show how position, velocity, and acceleration are related by integration. Suppose we had an oscilloscope capable of directly measuring current, or at least a current-to-voltage converter that we could attach to one of the probe inputs to allow direct measurement of current on one channel. Suppose, though, that instead of the bank providing the student with a statement every month showing the account balance on different dates, the bank were to provide the student with a statement every month showing the rates of change of the balance over time, in dollars per day, calculated at the end of each day: Explain how the Isaac Newton Credit Union calculates the derivative ([dS/dt]) from the regular account balance numbers (S in the Humongous Savings & Loan statement), and then explain how the student who banks at Isaac Newton Credit Union could figure out how much money is in their account at any given time. I show the solution steps for you here because it is a neat application of differentiation (and substitution) to solve a real-world problem: Now, we manipulate the original equation to obtain a definition for IS e40 V in terms of current, for the sake of substitution: Substituting this expression into the derivative: Reciprocating to get voltage over current (the proper form for resistance): Now we may get rid of the saturation current term, because it is negligibly small: The constant of 25 millivolts is not set in stone, by any means. Ask your students to come to the front of the class and draw their integrator and differentiator circuits. In case you wish to demonstrate this principle “live” in the classroom, I suggest you bring a signal generator and oscilloscope to the class, and build the following circuit on a breadboard: The output is not a perfect square wave, given the loading effects of the differentiator circuit on the integrator circuit, and also the imperfections of each operation (being passive rather than active integrator and differentiator circuits). Follow-up question: what do the schematic diagrams of passive integrator and differentiator circuits look like? However, we may measure any current (DC or AC) using a Rogowski coil if its output signal feeds into an integrator circuit as shown: Connected as such, the output of the integrator circuit will be a direct representation of the amount of current going through the wire. Hopefully the opening scenario of a dwindling savings account is something they can relate to! If time permits, this would be an excellent point of departure to other realms of physics, where op-amp signal conditioning circuits can be used to “undo” the calculus functions inherent to certain physical measurements (acceleration vs. velocity vs. position, for example). What is available is an altimeter, which infers the rocket’s altitude (it position away from ground) by measuring ambient air pressure; and also an accelerometer, which infers acceleration (rate-of-change of velocity) by measuring the inertial force exerted by a small mass. Find what is the main question (ex) Max. PDF Version. A voltmeter connected between the potentiometer wiper and ground will then indicate arm position. 994 0 obj <>/Filter/FlateDecode/ID[<324F30EE97162449A171AB4AFAF5E3C8><7B514E89B26865408FA98FF643AD567D>]/Index[986 19]/Info 985 0 R/Length 65/Prev 666753/Root 987 0 R/Size 1005/Type/XRef/W[1 3 1]>>stream Download PDF Download Image Integrals. Mathematics in electronics. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve. 986 0 obj <> endobj That integration and differentiation are inverse functions will probably be obvious already to your more mathematically inclined students. Challenge question: can you think of a way we could exploit the similarity of inductive voltage/current integration to simulate the behavior of a water tank’s filling, or any other physical process described by the same mathematical relationship? I have found it a good habit to “sneak” mathematical concepts into physical science courses whenever possible. It is the difference between saying “1500 miles per hour” and “1500 miles”. In addition, a number of more advanced topics have Download PDF Differential Calculus Basics. Some of your students may be very skeptical of this figure, not willing to believe that ä computer power supply is capable of outputting 175 billion amps?!”. 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