Mathematical Techniques and Formulae:

A1. Mathematical Techniques and Formulae

A1-1. Differentiation.
A1-2. Exponents and Logarithms.

A1-3. The Lorentzian line-shape function.
A1-4. Dimensional consistency.

A1-1. Differentiation:
A function is an equation or formula with a single variable quantity on the left hand side. An example is:
y = ax + b
Let us say that in this case there are two variable quantities y and x, and the other quantities, a and b, do not vary. Formulae of this type can be represented by the general statement:
y = f(x)
which should be read: "y is a function of x", or "y depends on x".
y is known as the dependent variable, because its value is dictated by x.
x is known as the independent variable, because it can take on arbitrary values.

The full definition for a function is as follows:
"A function is a rule relating one set to another; such that if we take one element of one set (xn say) and apply the rule, we obtain a unique element of the other set (yn say).

In general, y may depend on any number of variables. We can express this situation by means of a comma-separated list:
y = f(p, q, r, s, t)
where p, q, r, s and t are independent variables.

The derivative of a function is its rate of change with respect to one of its variables. It is a number which tels us how much the dependent variable will change for a given change in an independent variable. For the simple case y=f(x), the derivative is written dy/dx and should be understood to mean "the amount by which y changes when x is changed by an infinitesimal (i.e., infinitely small) amount". It is necessary to define derivatives in terms of infinitesimal changes because the relationship between y and x is not necessarily linear, and the numerical value of dy/dx will vary as x varies. Hence, in general, the process of differentiation gives a new function, which gives the numerical derivative for a particular value of x when that value of x is inserted.

For the simple linear function y=ax + b:
dy/dx = a
this tells us that a change in x will give 'a' times as much change in y. The constant 'b' makes no difference to the rate of change and so does not appear in the derivative. Only coefficients associated with (i.e., operated on by) the independant variable make a contribution.

Any quantity raised to the power of zero is 1. If we look at what happened when we differentiated y=ax, we see that differentiation subtracted 1 from its power, i.e.:
if y = ax

, then dy/dx = ax

The general rule for differentiation is:

if y = ax

then dy/dx = anx

Differentiation is a simple additive process. If y can be written as a number of functions of x added together (an arithmetic series), i.e.:
y = f1(x) + f2(x) + f3(x) + . . . . .
then
dy/dx = df1(x)/dx + df2(x)/dx + df3(x)/dx + . . . . .

It is however, not always possible to break a formula down into a series of terms of the form
ax

. In such cases we must apply special rules.

Function of a function:
If y is a function of z and z is a function of x, i.e.:

if y = f(z) and z = f(x)
then
dy/dx = (dy/dz) (dz/dx)

The product rule:

if y = u v
where both u and v are functions of x
then
dy/dx = v(du/dx) + u(dv/dx)

The Quotient rule:
By using the product rule and the function of a function rule we obtain:

If y = N/D
where both N and D are functions of x
then
dy/dx = [D(dN/dx) - N(dD/dx)] / D²

Special functions:

if y = Sin(ax) then dy/dx = a Cos(ax)

if y = Cos(ax) then dy/dx = -a Sin(ax)

Since Tan(ax) = Sin(ax)/Cos(ax), we may use the quotient rule to obtain:

if y = Tan(ax) then dy/dx = a / Cos²(ax)

The exponential is a special function which is equal to its own rate of change.

if y = e

then dy/dx = e

if y = Loge(x) then dy/dx = 1/x

Partial differentiation:
If the value of a function is dependent on more than one independent variable, we may define a partial derivitave as the rate of change of the function with respect to one of its independent variables, all other variables being held constant. A partial derivative is denoted by the special symbol '¶' (known as "partial d" or "curly d"), i.e.:
¶y/¶x is the rate of change of y with respect to x when all other variables treated as constants.

Maxima and Minima:
If a function (e.g., a resonance curve) has a peak or a trough, the gradient of the function at the tip of the peak or trough is 1 (level) and its rate of change at this point is 0. Hence, we can find the peaks and troughs in a function by applying the condition:
¶y/¶x = 0
If a function has more than one peak or trough, there will be more than one solution for ¶y/¶x=0. In general, the degree (highest power of x) of the derivative will be the same as the number of peaks and troughs. E.g., for an electrical system with a single resonant frequency, there will be a positive and a negative frequency solution for f0. Hence the derivative with respect to frequency of the function which describes the resonance curve will be a quadratic equation (see section 1-9).

Analytical solutions exist for all equations up to degree 4 (quartic). Beyond that, no general solutions exist and numerical methods (matrix diagonalisation) may be required.

A1-2. Exponents and Logarithms:
>>> article to be inserted
>>> explain how to use differentiation rules above
>>> conversion between bases

A1-3. The Lorentzian line-shape function:
The various expressions for the electrical resonance curve derived in chapter 1 are closely related to a simple curve known as the Lorentzian (or Cauchy) line-shape function, which has the general form:

y =	h w² w² + (x - x0)²

A1-3.1

where h is the peak height and w is called the half-width. The expression can also be written:

1 + [(x - x0)/w]²

A1-3.2

which is the form most similar to equation (1-26.9).
The Lorentzian is regarded as the characteristic signature of natural electromagnetic resonance processes. In particular, the peaks in molecular and atomic spectra in the microwave, optical, x-ray and gamma-ray regions are all of this form when displayed on a linear amplitude (y-axis) scale. The curve is called a line-shape function because the narrow spikes which occur when dense spectra are drawn by a chart-recorder or otherwise displayed are tradionally known as lines. It is only when the frequency scale is expanded that the individual peaks resolve into Lorentzians.
In comparing the Lorentzian to the electrical resonance curve, we may first note that the Lorentzian is always exactly symmetric about x0, and that x0 can be set to zero. We have noted before (section 1.26a) that the electrical resonance curve is skewed when plotted against linear frequency, but becomes symmetric to a good approximation when the Q is high. We also noted that the resonance curve can be made perfectly symmetric by plotting it on a logarithmic frequency scale; in which case, since the logarithm of unit frequency is zero, the curve can also be symmetric about Log(f)=0. In fact, natural resonance processes have such high Q that they appear symmetric on linear, logarithmic, and even reciprocal (wavelength) scales; but to find the relationship between the Lorentzian and the electrical curve, it is obvious that we must identify the x-axis as corresponding to logarithmic frequency, i.e., x=Loga(f), where the base a can be chosen arbitrarily. Here we will use Naperian logarithms because it will allow us to use the series expansion of e to solve the problem. Hence we choose:
x = Loge(f)
which means that:
f = e

and
f0 = e

Substituting these identities into the electrical resonance curve (1-26.9) we obtain:

1 + { Q0[ (e

/ e

) - (e

/ e

) ] }²

but, from the rules of logarithms discussed in section 1-25:
e

/ e

= e

and
e

/ e

= e

Hence:

1 + [ Q0(e

- e

) ]²

The quantity e

- e

is related to a function known as the hyperbolic sine, which is defined as:
sinh(x) = (e

- e

)/2
Hence:

1 + [ 2 Q0 sinh(x - x0) ]²

A1-3.3

The function e

can be expanded as an infinite series:

e = 1 + x +

x²

x³

+ . . . . . . . . .

where an exclamation mark indicates a factorial number, the factorials being defined as:

Factorial	0!	1!	2!	3!	4!	n!			(n+1)!
Value	1	1	2	3´2	4´3´2	n(n-1)(n-2) . . . . . . 1			(n+1)´n!

It follows that the series for e

is:

e = 1 - x +

x²

x³

+ . . . . . . . . .

and that by subtracting one series from the other we can obtain a series for e

-e

=2sinh(x)

2sinh(x) = 2x +

2x³

+ . . . . . . . . .

sinh(x) = x +

x³

+ . . . . . . . . .

Now notice that when the magnitude of x is somewhat less than 1, the magnitudes of the terms in which x is raised to a high power become very small, and so we can make the approximation:
sinh(x) » x when |x| < 1.
Substituting this into equation (A1-3.3) we get:

1 + [ 2 Q0 (x - x0) ]²

which is a Lorentzian with w=1/(2Q0). Hence the electrical resonance curve is approximately Lorentzian when |x - x0| is less than 1.

A1-4. Dimensional Consistency:
It should be obvious from the circuit analysis techniques demonstrated in these articles, that even relatively simple circuits can require complicated mathematical expressions to describe their behaviour. There is however, a certain reality-check which makes the derivation of equations much easier, and gives an immediate indication of the likely correctness of the result. This is the test of dimensional consistency which, with a certain amount of practice, can be carried out at a glance. The rules are as follows:

A1-4.1) If two quantities are to be added together (or subtracted) it must be possible to express them in the same units.

It would make no sense to add a distance in metres to a temperature in degrees centigrade. It would also make no sense to add a distance in metres to a distance in centimetres, but in that case the distance in centimetres can be divided by 100 to convert it into metres, and then the addition can be performed. It follows, that if a '+' or a '-' symbol appears anywhere in an equation, the dimensions of the quantities on either side of that symbol must be the same.

A1-4.2) An equation, which supposedly represents a certain quantity, must have dimensions appropriate to that quantity.

Take, for example, the expression for the real part of two impedances in parallel derived in section 1-11:

R =

R1R2(R1+R2) +R1X2² +R2X1²
(R1 + R2)² +(X1 + X2)²

The truth of this statement is not immediately obvious, but a check of dimensional consistency can very quickly tell us if it is capable of being true. In this case, the denominator (the bottom part) of the fraction has two brackets each containing quantities having the units of resistance (Ohms). Hence the terms (R1+R2)² and (X1+X2)² have dimensions of [W²], and the overall dimensions of the denominator are [W²]. In the case of the numerator; there are three terms to be added, each having the dimensions of [W³], and the overall dimensions of the numerator are [W³]. Dividing the dimensions of the numerator by the dimensions of the denominator we obtain: [W³]/[W²]=[W]; and so the equation is dimensionally consistent and represents a quantity which can be expressed in Ohms.
It is also possible to test the dimensional consistency of equations involving mixed units. The point here is that units have aliases, which are composites of other units; and so we can check any equation, provided that we know the relationships between the units used. In the context of circuit analysis, these relationships are easily obtained, because they are embedded in the basic formulae from which the mathematical argument is constructed. Ohm's law, V/I=Z, for example, tells us that Ohms are equivalent to volts divided by amperes, and so a quantity having the latter dimensions, i.e., a voltage divided by a current, may legitimately replace a quantity measured in Ohms. Similarly, the reactance laws XL=2pfL and XC=-1/2pfC, tell us that Ohms can also be replaced with [Henrys ´ radians / second], or by [1/(Farads ´ radians / second)]. Thus we should not be confused by structures such as:
Z = R +j(2pfL -1/[2pfC])
The bracket after the j is internally consistent, and represents a quantity measured in Ohms.

Balanced vector equations:
A further implication of the need for dimensional consistency in mathematics relates to the nature of vectors: A mathematical object must have the same number of dimensions as the quantity it represents, and the left hand side of an equation must have the same number of dimensions as the right hand side. In these articles we have kept track of this requirement by noting vectors in bold, a practice which results in the de-facto rule that if there is vector on one side of an equation, there must be either one bold variable or a j on the other (unless some vector operation which produces a scalar, such as taking a magnitude or a dot product, is involved). This rule is not usually followed by other writers on AC theory, and results in both dimensional inconsistency and confusion over which items are meant to be vectors and which are magnitudes.

Dimensions in tables:
One final point on the subject of units, is that in the rest of this work you will find tables with headings such as: "Resistance / KW" and "Diameter / mm". This significance of this notation is that the numbers in the table, if they have no units written next to them, are just numbers, i.e., they are dimensionless. Thus the heading tells you that the quantity shown has been divided by some unit quantity in order to make it dimensionless, e.g., a resistance of 10KW, divided by KW is just the number 10. Thus the humble slash, often omitted in both writing and pronunciation, in this context should be read as "per", or "divided by", or better still as "in units of ".

TX to Ae