How Are Imaginary Numbers Used In Electrical Engineering

Complex Numbers

While the central bespeak used in electric engineering is the sinusoid, it tin can exist expressed mathematically in terms of an even more fundamental signal: the complex exponential. Representing sinusoids in terms of complex exponentials is not a mathematical oddity. Fluency with complex numbers and rational functions of complex variables is a critical skill all engineers master. Agreement information and power arrangement designs and developing new systems all hinge on using complex numbers. In short, they are critical to mod electrical engineering, a realization fabricated over a century agone.

Definitions

The notion of the square root of

originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity

could be defined. Euler start used

for the imaginary unit but that notation did non take hold until roughly Ampère's fourth dimension. Ampère used the symbol

to denote electric current (intensité de current). It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using

for current was entrenched and electric engineers chose

for writing complex numbers.

An imaginary number has the form

. A complex number,

, consists of the ordered pair (

is the real component and

is the imaginary component (the

is suppressed considering the imaginary component of the pair is ever in the second position). The imaginary number

equals (00,

). Note that

and

are existent-valued numbers.

Figure 1 shows that we tin can locate a circuitous number in what we call the complex plane. Here,

, the real office, is the

-coordinate and

, the imaginary function, is the

-coordinate.

The Complex Plane

Figure ane. A complex number is an ordered pair (

) that tin be regarded as coordinates in the plane. Complex numbers can too be expressed in polar coordinates as

From analytic geometry, we know that locations in the plane tin be expressed as the sum of vectors, with the vectors corresponding to the

and

directions. Consequently, a complex number

can be expressed as the (vector) sum

where

indicates the

-coordinate. This representation is known as the Cartesian form of

. An imaginary number can't be numerically added to a real number; rather, this notation for a circuitous number represents vector addition, but it provides a convenient note when nosotros perform arithmetic manipulations.

Some obvious terminology. The real part of the complex number

, written every bit

, equals

. We consider the existent part equally a part that works by selecting that component of a complex number not multiplied by

. The imaginary office of

, equals

: that part of a complex number that is multiplied past

. Again, both the real and imaginary parts of a complex number are real-valued.

The complex conjugate of

, written every bit

, has the same existent part every bit

but an imaginary part of the reverse sign.

Using Cartesian notation, the following properties easily follow.

If we add ii complex numbers, the existent part of the result equals the sum of the real parts and the imaginary role equals the sum of the imaginary parts. This property follows from the laws of vector add-on.

In this mode, the real and imaginary parts remain separate.

Exercise

Apply the definition of addition to show that the existent and imaginary parts can be expressed every bit a sum/difference of a circuitous number and its conjugate.

and

. Similarly,

Circuitous numbers can also be expressed in an alternating form, polar grade, which we volition find quite useful. Polar form arises arises from the geometric estimation of complex numbers. The Cartesian class of a circuitous number can exist re-written as

Past forming a correct triangle having sides

and

, nosotros see that the real and imaginary parts correspond to the cosine and sine of the triangle'due south base bending. We thus obtain the polar form for circuitous numbers.

The quantity

is known as the magnitude of the complex number

, and is frequently written equally

. The quantity

is the circuitous number's angle. In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.

Exercise

Convert

to polar form.

To convert

to polar form, we offset locate the number in the circuitous airplane in the quaternary quadrant. The distance from the origin to the complex number is the magnitude

, which equals

. The angle equals

radians (

degrees). The final answer is

degrees.

Euler's Formula

Surprisingly, the polar form of a complex number

tin can be expressed mathematically as

To show this result, we use Euler'south relations that express exponentials with imaginary arguments in terms of trigonometric functions.

The first of these is easily derived from the Taylor'southward serial for the exponential.

Substituting

for

, we find that

because

, and

. Grouping separately the real-valued terms and the imaginary-valued ones,

The real-valued terms correspond to the Taylor'southward serial for

, the imaginary ones to

, and Euler's first relation results. The remaining relations are easily derived from the first. We see that multiplying the exponential in Equation by a existent abiding corresponds to setting the radius of the complex number to the constant.

Computing with Complex Numbers

Adding and subtracting circuitous numbers expressed in Cartesian form is quite easy: You add together (subtract) the existent parts and imaginary parts separately.

To multiply two complex numbers in Cartesian class is not quite every bit like shooting fish in a barrel, but follows directly from following the usual rules of arithmetics.

Note that nosotros are, in a sense, multiplying two vectors to obtain another vector. Complex arithmetic provides a unique way of defining vector multiplication.

Practice

What is the product of a complex number and its cohabit?

. Thus,

Division requires mathematical manipulation. We convert the division problem into a multiplication problem past multiplying both the numerator and denominator by the conjugate of the denominator.

Because the final result is so complicated, information technology's best to call up how to perform sectionalisation—multiplying numerator and denominator by the circuitous conjugate of the denominator—than trying to recall the final effect.

The properties of the exponential brand computing the product and ratio of ii complex numbers much simpler when the numbers are expressed in polar form

To multiply, the radius equals the production of the radii and the angle the sum of the angles. To carve up, the radius equals the ratio of the radii and the bending the departure of the angles. When the original complex numbers are in Cartesian form, it's usually worth translating into polar form, and so performing the multiplication or division (especially in the case of the latter). Addition and subtraction of polar forms amounts to converting to Cartesian form, performing the arithmetic operation, and converting dorsum to polar form.

When we solve circuit problems, the crucial quantity, known as a transfer function, will ever be expressed equally the ratio of polynomials in the variable

. What nosotros'll demand to understand the circuit'southward effect is the transfer function in polar form. For case, suppose the transfer function equals

Performing the required division is nigh easily accomplished past first expressing the numerator and denominator each in polar class, and so calculating the ratio. Thus,