This question asks students to relate the concept of time-differentiation to physical motion, as well as giving them a very practical example of how a passive differentiator circuit could be used. Follow-up question: the operation of a Rogowski coil (and the integrator circuit) is probably easiest to comprehend if one imagines the measured current starting at 0 amps and linearly increasing over time. 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. It is the opposite (inverse) function of differentiation. In this case, the derivative of the function y = x2 is [dy/dx] = 2x. 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 Find what is the main question (ex) Max. The coil produces a voltage proportional to the conductor current’s rate of change over time (vcoil = M [di/dt]). Besides, it gives some practical context to integrator circuits! 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. Underline all numbers and functions 2. Determine what the response will be to a constant DC voltage applied at the input of these (ideal) circuits: Ask your students to frame their answers in a practical context, such as speed and distance for a moving object (where speed is the time-derivative of distance and distance is the time-integral of speed). CY - New York City. The time you spend discussing this question and questions like it will vary according to your students’ mathematical abilities. Position, of course, is nothing more than a measure of how far the object has traveled from its starting point. If we connect the potentiometer’s output to a differentiator circuit, we will obtain another signal representing something else about the robotic arm’s action. Definition of an Integral Properties Common Integrals Integration by Subs. Discrete Semiconductor Devices and Circuits, What You Should Know About Organic Light-Emitting Diode (OLED) Technology, Predicting Battery Degradation with a Trinket M0 and Python Software Algorithms, Evaluating the Performance of RF Assemblies Controlled by a MIPI-RFFE Interface with an Oscilloscope, Common Analog, Digital, and Mixed-Signal Integrated Circuits (ICs). Which electrical quantity (voltage or current) dictates the rate-of-change over time of which other quantity (voltage or current) in a capacitance? Acceleration is a measure of how fast the velocity is changing over time. We know that velocity is the time-derivative of position (v = [dx/dt]) and that acceleration is the time-derivative of velocity (a = [dv/dt]). Hints: saturation current (IS) is a very small constant for most diodes, and the final equation should express dynamic resistance in terms of thermal voltage (25 mV) and diode current (I). Thankfully, there are more familiar physical systems which also manifest the process of integration, making it easier to comprehend. In a capacitance, voltage is the time-integral of current. As switches, these circuits have but two states: on and off, which represent the binary states of 1 and 0, respectively. 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?!”. Its value varies with temperature, and is sometimes given as 26 millivolts or even 30 millivolts. What would a positive [dS/dt] represent in real life? Don't forget unit of the answer. To others, it may be a revelation. Or, to re-phrase the question, which quantity (voltage or current), when maintained at a constant value, results in which other quantity (current or voltage) steadily ramping either up or down over time? Therefore, the subsequent differentiation stage, perfect or not, has no slope to differentiate, and thus there will be no DC bias on the output. What would the output of this integrator then represent with respect to the automobile, position or acceleration? 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. Speed is the derivative of distance; distance is the integral of speed. How are they similar to one another and how do they differ? 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. Differential Equations and Transforms: Differential Equations, Fourier Series, Laplace Transforms, Euler’s Approximation Numerical Analysis: Root Solving with Bisection Method and Newton’s Method. 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. In these calculations:V = voltage (in volts)I = current (in amps)R = resistance (in ohms)P = power (in watts) A forward-biased PN semiconductor junction does not possess a “resistance” in the same manner as a resistor or a length of wire. Hopefully the opening scenario of a dwindling savings account is something they can relate to! Being air-core devices, they lack the potential for saturation, hysteresis, and other nonlinearities which may corrupt the measured current signal. (ex) 4. A passive integrator circuit would be insufficient for the task if we tried to measure a DC current - only an active integrator would be adequate to measure DC. The latter is an absolute measure, while the former is a rate of change over time. Calculus is a branch of mathematics that originated with scientific questions concerning rates of change. Follow-up question: draw the schematic diagrams for these two circuits (differentiator and integrator). 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). It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve. %%EOF For an integrator circuit, the rate of output voltage change over time is proportional to the input voltage: A more sophisticated way of saying this is, “The time-derivative of output voltage is proportional to the input voltage in an integrator circuit.” However, in calculus there is a special symbol used to express this same relationship in reverse terms: expressing the output voltage as a function of the input. Challenge question: the integrator circuit shown here is an “active” integrator rather than a “passive” integrator. Hence, calculus in … Some students may ask why the differential notation [dS/dt] is used rather than the difference notation [(∆S)/(∆t)] in this example, since the rates of change are always calculated by subtraction of two data points (thus implying a ∆). What practical use do you see for such a circuit? 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). 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). What I’m interested in here is the shape of each current waveform! h�bbd```b``: "k���d^"Y��$�5X��*���4�����9$TK���߿ � 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. However, this does not mean that the task is impossible. %PDF-1.5 %���� The book is in use at Whitman College and is occasionally updated to correct errors and add new material. Hopefully, the challenge question will stir your students’ imaginations, as they realize the usefulness of electrical components as analogues for other types of physical systems. ... AC Motor Control and Electrical Vehicle Applications Seconds Edition by Kwang Hee Nam PDF Free Download. The easiest rates of change for most people to understand are those dealing with time. 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. CALCULUS MADE EASY Calculus Made Easy has long been the most populal' calculus pl'imcl~ In this major revision of the classic math tc.xt, i\'Iartin GardnCl' has rendered calculus comp,'chcnsiblc to readers of alllcvcls. 1 offer from $890.00. Calculus is a branch of mathematics that originated with scientific questions concerning rates of change. 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. This is a radical departure from the time-independent nature of resistors, and of Ohm’s Law! Which electrical quantity (voltage or current) dictates the rate-of-change over time of which other quantity (voltage or current) in an inductance? 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 What relationship is there between the amount of resistance and the nature of the voltage/current function as it appears on the graph? The expression [di/dt] represents the instantaneous rate of change of current over time. Ask your students to come to the front of the class and draw their integrator and differentiator circuits. 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? To this end, computer engineers keep pushing the limits of transistor circuit design to achieve faster and faster switching rates. This much is apparent simply by examining the units (miles per hour indicates a rate of change over time). ... An Engineers Quick Calculus Integrals Reference. connect the output of the first differentiator circuit to the input of a second differentiator circuit)? Calculus is widely (and falsely!) By the way, this DC bias current may be “nulled” simply by re-setting the integrator after the initial DC power-up! One of the variables needed by the on-board flight-control computer is velocity, so it can throttle engine power and achieve maximum fuel efficiency. Usually students find the concept of the integral a bit harder to grasp than the concept of the derivative, even when interpreted in graphical form. I have also uploaded all my Coursera videos to YouTube, and links are placed at You may want to have them phrase their responses in realistic terms, as if they were describing how to set up an illustrative experiment for a classroom demonstration. Take this water tank, for example: One of these variables (either height H or flow F, I’m not saying yet!) $930.35. Differentiation is fundamentally a process of division. Calculate the size of the resistor necessary in the integrator circuit to give the integrator output a 1:1 scaling with the measured current, given a capacitor size of 4.7 nF: That is, size the resistor such that a current through the conductor changing at a rate of 1 amp per second will generate an integrator output voltage changing at a rate of 1 volt per second. In today’s world, if one wants to be a true, creative professional, practically in any field one has to command En- Special Honors. Thankfully, there are more familiar physical systems which also manifest the process of integration, making it easier to comprehend. This Calculus Handbook was developed primarily through work with a number of AP Calculus classes, so it contains what most students need to prepare for the AP Calculus Exam (AB or BC) or a first‐year college Calculus course. Be as specific as you can in your answer. Challenge question: derivatives of power functions are easy to determine if you know the procedure. 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. Regardless of units, the two variables of speed and distance are related to each other over time by the calculus operations of integration and differentiation. Everyone inherently understands the relationship between distance, velocity, and time, because everyone has had to travel somewhere at some point in their lives. Show this both in symbolic (proper mathematical) form as well as in an illustration similar to that shown above. Lower-case variables represent instantaneous values, as opposed to average values. 0 I’ll let you figure out the schematic diagrams on your own! 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. In calculus terms, we would say that the induced voltage across the inductor is the derivative of the current through the inductor: that is, proportional to the current’s rate-of-change with respect to time. It is a universal language throughout engineering sciences, also in computer science. Electrical phenomena such as capacitance and inductance may serve as excellent contexts in which students may explore and comprehend the abstract principles of calculus. Integrator and differentiator circuits are highly useful for motion signal processing, because they allow us to take voltage signals from motion sensors and convert them into signals representing other motion variables. The fundamental definition of resistance comes from Ohm’s Law, and it is expressed in derivative form as such: The fundamental equation relating current and voltage together for a PN junction is Shockley’s diode equation: At room temperature (approximately 21 degrees C, or 294 degrees K), the thermal voltage of a PN junction is about 25 millivolts. Also, what does the expression [di/dt] mean? The derivative of a linear function is a constant, and in each of these three cases that constant equals the resistor resistance in ohms. The fact that we may show them the cancellation of integration with differentiation should be proof enough. This is one of over 2,200 courses on OCW. Of these two variables, speed and distance, which is the derivative of the other, and which is the integral of the other? Follow-up question: manipulate this equation to solve for the other two variables ([de/dt] = … ; C = …). Calculus for Engineering Students: Fundamentals, Real Problems, and Computers insists that mathematics cannot be separated from chemistry, mechanics, electricity, electronics, automation, and other disciplines. I like to use the context of moving objects to teach basic calculus concepts because of its everyday familiarity: anyone who has ever driven a car knows what position, velocity, and acceleration are, and the differences between them. The studies of electricity and electronics are rich in mathematical context, so exploit it whenever possible! 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”). This last statement represents a very common error students commit, and it is based on a fundamental misunderstanding of [di/dt]. 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. Then, ask the whole class to think of some scenarios where these circuits would be used in the same manner suggested by the question: motion signal processing. Students should also be familiar with matrices, and be able to compute a three-by-three determinant. My purpose in using differential notation is to familiarize students with the concept of the derivative in the context of something they can easily relate to, even if the particular details of the application suggest a more correct notation. 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. �]�o�P~��e�'ØY�ͮ�� S�ე��^���}�GBi��. A very important aspect of this question is the discussion it will engender between you and your students regarding the relationship between rates of change in the three equations given in the answer. Whenever you as an instructor can help bridge difficult conceptual leaps by appeal to common experience, do so! Normally transformers are considered AC-only devices, because electromagnetic induction requires a changing magnetic field ([(d φ)/dt]) to induce voltage in a conductor. The d letters represent a calculus concept known as a differential, and a quotient of two d terms is called a derivative. Create one now. Find materials for this course in the pages linked along the left. A computer with an analog input port connected to the same points will be able to measure, record, and (if also connected to the arm’s motor drive circuits) control the arm’s position. Being able to differentiate one signal in terms of another, although equally useful in physics, is not so easy to accomplish with opamps. of Statistics UW-Madison 1. For an integrator circuit, this special symbol is called the integration symbol, and it looks like an elongated letter “S”: Here, we would say that output voltage is proportional to the time-integral of the input voltage, accumulated over a period of time from time=0 to some point in time we call T. “This is all very interesting,” you say, “but what does this have to do with anything in real life?” Well, there are actually a great deal of applications where physical quantities are related to each other by time-derivatives and time-integrals. (ex) 40 thousand dollars L'Hospital's Rule It's good for forms 1. Here, I ask students to relate the instantaneous rate-of-change of the voltage waveform to the instantaneous amplitude of the current waveform. One common application of derivatives is in the relationship between position, velocity, and acceleration of a moving object. Now we send this voltage signal to the input of a differentiator circuit, which performs the time-differentiation function on that signal. calculus in order to come to grips with his or her own scientific questions—as those pioneering students had. Draw a block diagram for a circuit that calculates [dy/dx], given the input voltages x and y. To integrate the [dS/dt] values shown on the Credit Union’s statement so as to arrive at values for S, we must either repeatedly add or subtract the days’ rate-of-change figures, beginning with a starting balance. This is true whether or not the independent variable is time (an important point given that most “intuitive” examples of the derivative are time-based!). calculus stuff is simply a language that we use when we want to formulate or understand a problem. Substituting 1 for the non-ideality coefficient, we may simply the diode equation as such: Differentiate this equation with respect to V, so as to determine [dI/dV], and then reciprocate to find a mathematical definition for dynamic resistance ([dV/dI]) of a PN junction. Follow-up question: manipulate this equation to solve for the other two variables ([di/dt] = … ; L = …). Resistance and the nature of the variables associated with moving objects way to show the importance of calculus ” )... Purpose of this complementarity a dwindling savings account is something they can relate to you is which operation goes way! Dictates the rate-of-change of current principle is important to understand are those dealing time! The task is impossible is changing over time that for simple resistor circuits, the units ( miles per ”. ’ Hôpital ’ s Law so it can throttle engine power and achieve maximum fuel efficiency rest! 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