## Math for the big four

### General Math

The formulas needed are the linear function, the reciprocal function, the pythagorean theorem, the sine definition, the Cosine definition, and the sine law.

This is a very short list which provides all the ability most scientists and engineers actually use.

#### The linear function

The linear function is the most important and simplest formula. It occurs in several forms: Y=KX (the simplest with a constant and two variables), Y=KX+B (two variables and a multiplied and an added constant), Y=XZ (three variables giving a family of lines), and forms with fractions instead.

The important forms with fractions are:
 Y Y --- = K, and --- = Z X X
The variables X and Z are said to be inversely proportional.

The functions in fraction form are derived from the others by dividing both sides by the same letter.

It is also possible to think of the linear function as a proportion:
 Y W --- = --- X Z
This is often a clearer way to think of a problem.

#### The reciprocal function

A variation of the linear function is the reciprocal or inverse function:
 1 --- = Y X
This isn't a linear function, though it works like one in most instances. Multiplying both sides by X gives 1=XY. Obviously, as X goes up, Y must go down.

#### The Pythagorean Theorem

This ancient discovery, that the square of the long side of a right triangle equals the sum of the squares of the sides. Is basic to so much math that it is included here even though it is not, strictly speaking necessary to understand the big four or do design work with them. The long side of the right triangle, the one opposite the right angle, is called the hypotenuse, and is usually symbolized by the letter H. The other sides are usually called A and B. The formula is:

A2+B2=H2

#### The Sine Definition

The sine of an angle in a right triangle is the fraction formed by dividing the side opposite the angle by the hypotenuse.
 O sin(angle) = --- H
The values of the sine for each angle are calculated by a calculator or computer or stored in a table. Since it is often easy to measure an angle (for instance in surveying with a transit or theodolite), the sine of the angle will give the hypotenuese if the opposite side is known or the opposite side if the hypotenuse is known. In formula form: H*sin(angle)=O. and
 O ----------- = H sin(angle)

The inverse sine or arcsine is the angle when the opposite side and the hypotenuse is known. Just divide the opposite side by the hypoteneuse and use the result with the arcsine or inverse sine function.

Note that the sine definition is a constant for any given angle so the working with sines is only a variation of working with the linear function. The sine itself is a function and much is made of that fact in advanced mathematics, but it is of little interest to someone doing the big four. It's just a special linear function for right triangles.

#### The Cosine Definition

The cosine of an angle in a right triangle is the fraction formed by dividing the side adjacent to the angle by the hypotenuse.
 A cosin(angle) = --- H
None of the equations in the big four need the cosine, but it (and the tangent) are needed for completeness and for certain other problems. From the pythagorean theorem, it is easily possible to show that sin2 + cos2 = 1.

#### The Sine Law

The sine, cosine and pythagorean theorem give the ability to analyze any right triangle (the tangent is a convenience, but not neccessary). Other formulas are needed to do the rest of the triangles. The simplest ( the only one everyone remembers) is the sine law. It is normally expressed in proportion form:
 A B C ------ = ------ = ------ sin(a) sin(b) sin(b)
This works with any triangle and allows measurements where only the angles and a side are known. Here is a picture of how it is used in surveying: ### Optics

#### Lens Formulas

In order to see the lens formulas in the purest form some reciprocals need to be defined. We define the power of a lens as the reciprocal of the focal length (the shorter the focal length, the more powerful), and the curvature of a side of a lens as the reciprocal of the radius. The total curvature is the sum of the curvatures of the sides. If the center of the lens is thicker that the edge, the total curvature is positive. If the center is thinner than the edge, the curvature is negative In symbolic form:

 1 P = --- F

 1 C = --- R

The lensmaker's equation is P = nC, where P is the power, n is the refractive index (air is 1), and C is the total curvature.

If two lenses are in contact, P total=P1+P2.

If I and O are the reciprocals of the image and object distance, P=I+O (though this needs some care in placing the signs for object and image).

It should be noted that these formulas are almost never shown this way. The problem is that focal length and radius values are easy to measure and power and curvature are not. Looked at this way, however, optics is easy and intuitive. In essence, the more a lens bulges, the more powerfully it magnifies.

When a lens system is designed so the color fringes are minimized, the result is achromatic. We can use the above equations to derive a formula for the two lens achromatic system. This is just to show the power of the above formulas.

The criteria for an achromatic lens is equal power at both colors, P red=P blue.

For two lenses in contact,P total=P1+P2.

So, P Red=P1red-P2red and P blue=P1blue-P2blue (the powers are subtracted because one lens is positive and the other negative).

Putting the two together, P1red-P2red=P1blue-P2blue.

Substituting with P = nC, n1redC1-n2redC2 =n1blueC1-n2blueC2.

Getting all similar things to the same side,n1redC1-n1blueC1 =n2redC2-n2blueC2.

Calling the diffrence between the refractive indices d, we rewrite, d1C1=d2C2, or

 d1C2 C1 = ---------- d2

#### Other Formulas

The two most significant formulas in optics are the law of refraction and thelaw of reflection. The law of refraction is used when light goes through something and the law of reflection when light bounces off something.

The law of refraction is : N*sin(incoming angle) = N'*sin(outgoing angle), where N is the refractive index (a measured value of the bending power of the glass) of the incoming surface and N' is the refractive index of the outgoing surface. We are going to spend little time on this. This formula resides at the heart of computer ray tracing programs, but isn't needed at the rough and ready stage of this book. It is the basis of trigonometric or exact ray tracing in optics (which isn't difficult with the right software).

The law of reflection is that the incoming angle and the outgoing angle are equal.

### Machine tools

Machine tool math is simple: Linear functions in the gearing, and an occasional sine or cosine, for the most part. A device called a "sine bar" for example is used to make very accurate angles by measuring the hieght with a micrometer and using the sine of the angle.

### Heat engines

The main useful formula is the PV = nRT (pressure times volume is proportional to temperature), though some other linear relations (power = torque times rpm, for example) are employed as well.

Using the ideal gas law, it is possible to get a reasonable estimate of how an engine will work. First the volume is decided, then the stroke and the top dead center volume. The combustion chamber temperature is settled on and established, if necessary, by exhaust gas recirculation or fuel mixture.

With the above facts, given the composition of the hot gas (usually carbon dioxide and water), and the amount of gas in the cylinder (from the fuel-air ratio), the maximum pressure can be gotten. The volume at bottem dead center is known, so the minimum pressure can be gotten as well.

With the pressures known, and the stroke, the crankarm torque can be gotten, and some idea of power obtained. All this with only a few linear functions.

### Electronics

The main equation is ohm's law, voltage = current times resistance. If we have a circuit with a known voltage across a known resistance load, the current through the resistor can be gotten by dividing by the resistance. If a known current flows through a known resistance, multiplying the two gives the voltage across the resistor. If the resistance in unknown, measure the voltage and current and divide the voltage by the current.

Ohm's law is not hard. Think about it a little. Voltage is always across the two ends of the resistor, current is always flowing through the resistor. A one ohm resistor with one volt across it will have one ampere of current flowing through it.

#### Using a Meter

A multimeter has a dial or buttons for mode selection, two probes, and a display. Most are able to read voltage, current and resistance in direct and alternating circuits. Most meters indicate the alternating current settings with a squiggly line (one cycle of a sine function). The direct current settings use a straight line. You may also see "AC" and "DC".

Electronics measurements are almost always DC (most AC measurements are too dangerous for untrained consumption anyway, and are beyond this treatise). The only three measurements commonly used are of voltage across a resistor or source, resistance, or diode conduction.

To read voltage across a resistor, place the meter on a DC scale (auto scaling meters only have one) and place the two probes across the bare conductors at each end of the resistor. To read a source, identify the two points of supply (like the two ends of a plashlight battery), and place a probe on each end. Often, one probe can be clipped to a "ground"--a piece of metal case or part of the circuit board (a "ground plane", for example) which is held at zero volts. The other probe then gives values with respect ot ground. Often schematics give expected voltage values with respect to ground as it's a convenient way to work.

To read resistance, place the meter on the resistance scale (usually a greek omega) and place one probe on each conductor exiting the resistor.

To read a diode, place the meter on the diode position, place a probe on each conductor, take a reading and reverse the probes to get the reading the other way. One should be extremely high and one low.

That's about all to using a meter. An oscilloscope is a fancy meter which only measures voltage and displays it as a function of time.

#### Magnetic devices

The math for magnetic devices is fairly simple. Magnetic force is measured in ampere-turns (the number of amperes or current times the number of turns of wire in the coil), and Magnetic force equals the field strength times the reluctance--an ohm's law for magnetism. Other equations are similar.

#### Amplifiers

The most important two considerations for an amplifier are the gain and the frequency response. The gain (usually voltage gain, and only occasionally current gain) is simple to measure and design. The frequency response is sometimes more complicated.

The gain of an operational amplifier is determined by the feedback to the negative input. There are two configurations: The inverting amp has a resistor between the input and negative pin (we'll call it Rin) and a resistor between the output and the negative pin (Rout) with the positive input grounded. The non-inverting configuration inputs the signal at the positive pin, puts a resistor between the ground and the negative input(Rground), and puts a resistor between the output and the negative input(Rout).

In the inverting amp, the voltage gain of the amp (G) is
 Rout ---- Rin

In the non-inverting amp, the voltage gain of the amp (G) is
 Rout + Rground -------------------- Rground

The frequency response of an op-amp can be tailored by adding capacitors or inductors. The low frequency response is raised by putting a capacitor in series with Rground. The high frequency response can be lowered by putting a capacitor in parallel with Rout. The formulas for calculating this are the next order of business.

A capacitor is two metal plates separated by a thin insulator. When the two plates are connected to a DC source, electrons build up on the negative plate and are pulled away from the positive. This builds an electric field which holds the charges in place after the source is removed. Current flows while the voltage is building up, and stops when the full voltage is reached. If the electric voltage is changing, current flows with the change in a way which opposes the change in voltage. This opposition produces an effect like resistance, without heating, and, therefore, without energy loss. This effect is called reactance.

The essence of the study of reactance is that the capacitor stores energy in an electic field which opposes the change of voltage, and the inductor (or coil) stores energy in a magnetic field which opposes the change in current. When an alternating current flows through a capacitor, the reactance lowers as the frequency goes up. When alternating current flows through an inductor, the reactance increases as the frequency goes up.

The formula for capacitive reactance XC is
 1 ---- 6.283 F C
F is the frequency in cycles per second (or hertz). C is the capacitance in Farads. 6.283 is the short version of two times PI. This is a linear inverse function which reflects the fact that reactance goes down as frequency goes up.

The formula for inductive reactance XL is 6.283 F L. L is the inductance in Henrys.

There are really only two points where we are interested in the value of the reactance: when the reactance equals the resistance (Rout, Rin, and Rground), and when capacitive reactance equals the inductive reactance. The resistance case represents the half power point--the point where the power out of the circuit is cut in half. The inductor-capacitor case is the point where the reactance totally disappears in the series case, and becomes a maximum in the parallel case. This makes tuned circuits (bandpass and bandstop filters) possible.

(sorry, this is still incomplete)