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Transcript
Mathematical Basis for Electronic Design Wentworth Institute of Technology Electronic Design I Prof. Tim Johnson Designs with a Purpose In order to design a component, you don’t have to rely on inspiration or resort to the Edison solution: methodical testing of all possible variables until the key is found. You can instead ask if the system can be described mathematically? Since a design is only a realization of a solution to a problem, what you are really asking is: can the problem or need be described mathematically? Is there a formula that describes the operation? Then implement the design solution using electronics. Designing by variable When working on the Bell project, you had the benefit of knowledge of the formula describing an electromagnet. In the formula (N*I/L)*μ=B each letter represents a variable or electrical parameter of the design. The results of the calculation (B) was a value we were trying to maximize by means of the inputs. When the speaker worked, we could define B as a parameter we had to reach. We discovered its value by experimentation just as Bell did in his design. Some of the inputs may have limits. Clarifying the purpose To find a formula in other designs, we need to examine what it is that the system or component does and how math plays a part. Assume a system takes a measurement of something and reports the information back to the user. We’ll look first at some various measurement of a design then look at how they are implemented as components. Distance measurements Distance can be measured directly. A ruler can measure distance between two objects (or marks) directly. This is a counting of units. The units can be inches, centimeters, volts, amperes, ohms, and etc. Counting is a summation process. Summation is addition. Time measurements Time can be measured. Time can be measured by an electronic counter. Counters can display the sequence of units. Stopping the counter freezes the display giving the summation. You are measuring from the start (t0=0) until the finish (tf=X). Time, con’t. Summation for time is mathematically the same as summation for distance…it’s addition. It’s also a difference between time zero and time of finish: t0– tf =X Alternatively, two different times can be subtracted from each other giving the difference. Time can be measured by addition or subtraction. Indirect measurements Determining size from a distance. This employs the trigonometry of similar triangles. Knowing the height (H) and distance (D) of one object (#2) and either the height or distance of the other object (#1) determines the unknown. Math used is multiplication and division. H1/D1=H2/D2 is the formula. Solving for the height of unknown object: H1=(D1/D2)*H2 A simplification of math used in electronics Multiplication is the summation of a number, over and over for a specific number of times. Division is the subtraction of a number, over and over for a specific number of times. Microprocessors are really stupid but fast. If you are multiplying x by y, it merely adds y to itself (x-1) times and keeps track of the total in various registers. Amplification A telescope and a microscope are optical amplifiers. The size of the object seen is a function of the focal length of the lenses times the angle the object actually occupies without the lens. Tan =d/(2*f) or 2*tan *f=d Where is the angle that subtends the object, d is the image size we see in the lens and f is the focal length (distance from the lens that the object comes into focus). The focal length is changed by the curvature of the lens. Amplification con’t. Mechanically, force can be multiplied using a lever. The mathematical formula is Fa*Da=Fb*Db If we know Fb and Da&b then we can calculate Fa easily by use of multiplication and division: Fa=Fb*Db/Da By rearrangement of terms: Fa=(Db/Da)*Fb where (Db/Da) is the amplification factor of Fb to get Fa. Other means of amplification mechanically is by using gears or hydraulics. Transportation Transportation is the movement of objects from x1 to x2. This mathematically is the multiplication of x1 by some value. A*x1= x2 This is also known as translation or projection. Translation The value A can be a constant or some function. A is a constant (and linear) if for example you are rearranging furniture in a room. On the other hand if you are moving to another apartment across town then A is a function (and non-linear). Systems Let’s expand our understanding of systems beyond just measurements. We’ll consider the system itself as a mathematical entity. There is an input, some work is performed and there is an output. Control If you can control the output, there is a feedback loop. System, Part II Let’s consider a radio receiver as a system. The input is normally a very low powered electromagnetic signal. The output is a audio wave at a much higher power level. Transfer Function If we know the value of the input to a system that we are measuring at the output, we can establish a relationship. Value out/value in=amplification Vout/Vin is a transfer function (the A coefficient from a few slides back) for a system that is measuring (for example) voltage levels. Communication Communication is the movement of information with fidelity through a medium from point x1 to point x2. Transmission through a medium causes a loss. The loss is a known value and can be expressed in units (the signal is being measured in) per some standard distance (mile, meter, etc.). Communication Amplification Since the loss is a function of distance, if the distance between a receiver and a transmitter is known then I know what my gain has to be. V2=A*V1 where A is the gain which is equal to loss per mile times the number of miles. Since V2/V1=A; then A is also a value for the system’s transfer function. Gain is adjustable and accomplished by amplifiers which is electronic components made up of transistors. The feedback loops can vary or control the gain. Our Design Tools Resistor It’s math is Ohm’s Law: V=R*I Capacitor d It’s math is calculus based: ic C dt V c Inductor It’s math is also calculus based: d V L L dt iL Differential Calculus (in a nutshell) d/dt is an operator Whenever you see it, it means it’s measuring change. d Thus ic C V c dt Means the current (ic) is measured by the change in voltage across the capacitor times the value of C. What does d V L L dt iL mean? Application of Differential Calculus If we had a capacitor in a circuit across a voltage that we wished to measure, And an ammeter to measure current in series with the capacitor, then The value read for the current is actually translatable into the value of the voltage. You’ve all done differential calculus in Circuit Theory 1 when you measured the voltage rise across a capacitor. Application of Differential Calculus If we had an inductor in series with a current that we wished to measure, And a volt meter across the inductor, then The value read for the voltage is directly translatable into the value of the current flowing if we knew the rate the voltage was changing. Since the time constant is R/L, this is known. There is an additional “connection” that can be made. Application of Differential Calculus Because of the time constant, changing the resistor and/or the capacitor/inductor will change the time for the device to charge up, we have control over the summation. Switching resistors into a circuit that are proportionately scaled to reflect the multiplier allows different number to be added. Integral Calculus, part 1 if we integrate both sides of the formula for the Capacitor, we’d get: ic C d d 1 C ic dt V c dt V c C i V c c • Plus some initial current flow (assume zero) The integral sign means: the sum of what it’s applied to. In this case the current going into the capacitor. Integral Calculus, part II As a result of the information on the previous slide… The meaning of: 1 ic V c C Is: if we sum the value of the current into a capacitor, we determine the voltage. If a meaning is assigned to the current then the voltage is an answer. Application in a design Laser light travels at the speed of light. Most microprocessors operate at too slow a speed to allow accurate measurement. How can you tell the distance without parsing the time delay of the return pulse. In a distance measuring device, if a light pulse is sent out and a photo-voltaic cell operates on the reflected levels received, more current flows because the pulses return quicker. Thus a higher voltage across the capacitor indicates a shorter distance being measured. You merely calibrate Vc to the distance. Integral Calculus, part III Integrate both sides of the inductor equation to get: 1 iL V L L Which means that to know the sum of the voltages applied to the inductor, we only have to measure the current going into it (x’s the inverse of the value of L). Integration can be implemented electronically to add. Subtraction is an addition of a negative value. Conclusions Calculus only takes about two weeks to explain…the rest is just practice, familiarity, and some other tricks they throw in. It’s summation and becoming familiar with strange symbols. Since we can add, we can multiply. Since we can subtract, we can divide. Basic arithmetic can be implemented electronically. Your Task Write a memo that explains the basis for the math that we used to control the temperature of the soldering iron project.