# Complex Numbers and Phasers

In Electrical Engineering, the mathematical system used to combine resistors, currents or DC voltages uses the set of "real numbers" used as integers or fractions.

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But actual numbers are not the only type of number that we should use, especially when dealing with frequency-dependent sinusoidal sources and vectors.In addition to using regular or actual numbers, Complex Numbers are used to solve complex equations with numbers that are square root of √ -1 negative **numbers.**

In electrical engineering, such a number is called "imaginary number", and in electrical engineering the letter "j", commonly known as **the j-operator,** is used to distinguish a virtual number from a real number.Thus, the letter "j" is placed in front of a real number to indicate its imaginary number operation.

**Complex Numbers**represent points in a two-dimensional complex or s-plane that references two different axes.The horizontal axis is called the "real axis", and the vertical axis is called the "imaginary axis".The real and virtual parts of a complex number are shortened to Re(z) and Im(z), respectively.

Complex numbers consisting of real (active component) and imaginary (reactive component) numbers can be added, removed and used in exactly the same way as the basic algebra used to analyze DC Circuits.

The rules and laws used in mathematics for the collection or subtracting of imaginary numbers are the same as the actual numbers, j2 + j4 = j6, etc. The only difference is in multiplication, because when two imaginary numbers are multiplied together, there is a negative real number.Actual numbers can also be considered a complex number, but there is a zero imaginary part labeled j0.

**The J operator** has a value exactly equal to √-1, so the consecutive product " j ", ( j x j ) causes j to have the following values -1, -j, and +1. Operator J is widely used to indicate the counterclockwise rotation of a vector. Each consecutive multiplication or power will force the vector to rotate at a fixed angle of 90^{o} counterclockwise. Similarly, if the multiplication of the vector results in the a-j operator, phase shift-90^{o}, that is, there will be a clockwise rotation.

### j-Operator's Vector Rotation

Thus, multiplying an imaginary number by j^{2} returns the vector counterclockwise to 180^{o,} j^{3} multiplers 270^{o,} and j^{4 }multiplys 360 o or returns^{it} back to its original position. Multiplying by j^{10} or j^{30 }causes the vector to rotate counterclockwise in the appropriate amount.With each successive rotation, the size of the vector always remains the same. Therefore, no matter how large the j's strength, if you take the large force in mode(4) when calculating, your operations will be easier.

In Electrical Engineering, there are different ways to represent a complex number graphically or mathematically.Using the cosine and sinus rule, we can make **Cartesian** or **Rectangular Form** representations.

## Complex Numbers Using a Rectangular Form

In the final tutorial on phasers, we found that a complex number is represented by a real part and a virtual part that takes the generalized form of:

- Here:
- Z – Complex Number representing vector
- x – The real part or the Active ingredient
- y – Imaginary part or Reactive component
- j – defined by √-1

In rectangular format, a complex number can be represented as a point in a two-dimensional plane, called a **complex** or **s-plane.**For example, Z = 6 + j4 represents a single point whose coordinates represent 6 on the horizontal real axis and 4 on the vertical imaginary axis, as shown.

### Complex Numbers That Use a Complex or s-plane

However, both the real and virtual parts of a complex number in the rectangular format can be a positive or negative number, so both the real and virtual axes must extend in both positive and negative directions.This then produces a complex four-dial plane called **the Argand Diagram,** as shown below.

### Four Quarter Argand Diagram

In the Argand diagram, the horizontal axis represents all positive actual numbers to the right of the vertical virtual axis and all negative actual numbers to the left of the vertical virtual axis.All positive virtual numbers are shown on the horizontal axis, while all negative virtual numbers are below the horizontal real axis.This then produces a two-dimensional complex plane with four different dials labeled QI, QII, QIII, and QIV.

The Argand diagram above can also be used to represent a phaser that rotates the radius as a point in the complex plane given by the size of the phaser, drawing a full circle around it every 2π/ω seconds.

Next, we can further expand this idea to show the definition of a complex number, both polar and rectangular, for turns of 90^{o.}

**Complex Numbers** can also have "zero" real or virtual parts, such as Z = 6 + j0 or Z = 0 + j4.In this case, the dots are drawn directly on the real or imaginary axis.In addition, the angle of a complex number can be calculated using simple trigonometry to calculate the angles of right-angled triangles, or counterclockwise around the Argand diagram, starting with the positive actual axis.

Then 0^{angles} between^{o} and 90 o in the first quarter ( I ), 90^{o} with 180^{the} angles between it (ε) will be in the second quarter (II). Third dial ( III ) 180^{o} to 270^{the} fourth and final dial (IV) 270, which completes the full circle, while containing the angles between it^{o} to 360^{o.} The corresponding angles on all four dials can be found as follows:

tan ^{-1} (imaginary component ÷ real component)

## Aggregation and Subtraction in Complex Numbers

The collection or subtracting of complex numbers can be done mathematically or graphically in rectangular form.For aggregation, the real parts are first put together to form the real part of the total, then the virtual parts are collected to form the virtual part of the total, which can be easily processed by naming two complex numbers, A and B:

### Complex Addition and Subtraction

### Complex Numbers Question Example 1

The two vectors are defined as A = 4 + j1 and B = 2 + j3, respectively.Determine the sum and difference of the two vectors both in rectangular (a + jb ) format and graphically as an Argand Diagram.

### Mathematical Addition and Subtraction

Collection

Subtraction

### Collecting and Subtract Graphics

## Multiplication and Division in Complex Numbers

The rectangular product of complex numbers, along with some additional rules for the successive multiplication of the j-operator, follows more or less the same rules as normal algebra, where: j ^{2} = -1.For example, if we multiply our vectors A = 4 + j1 and B = 2 + j3 from above, it gives us the following result.

Mathematically, dividing complex numbers into rectangular shapes is slightly more difficult to perform because it requires the use of the denominator map function to convert the denominator of the equation to a real number.It's called rationalization.The division of complex numbers is best carried out using the "Polar Format", which we will examine later.However, as an example in a rectangular format, let's find the value of vector A by dividing it by vector B .

## Complex Conjugate

**A complex conjugate**, or simply **conjugat,** is used to hold the same real number of pre-signs of a complex number and detect someone's complex map only by reversing the imaginary number of complex numbers.For example, the conjugate z = 6 + j4 is Z = 6 + J4, which is z = 6 – j4, which is the conjugate z = 6 + J4.

For a complex map, the dots in the Argand diagram have the same horizontal position as the original complex number on the actual axis, but contrast to vertical positions.Therefore, complex conjugates can be considered as a reflection of a complex number.The following example shows a complex number, 6 + j4, and its map on a complex plane.

### Conjugate Complex Numbers

The sum of a complex number and its complex map will always be a real number, as we see above.Then, adding a complex number and its map gives the result only as an actual number or active component, while removing them returns only a virtual number or only a reactive component.The mapping of a complex number is an important element used in Electrical Engineering to determine the apparent power of an AC circuit using its rectangular form.

## Complex Numbers That Use a Polar Form

Unlike the rectangular format that draws points in a complex plane, the **Polar Format** of a complex number is written with its size and angle. In this case, a polar was presented as a vector in some way , Z = 1 ∠ ± ε , where: Z is a complex number in a polar form, the size of a vector or module, and the angle or argument of ε can be A.The size and angle of the point remain the same as in the rectangular form above, this time in the polar form the position of the point is represented in the "triangular form", as shown below.

### Polar-Shaped Representation of a Complex Number

Since the polar representation of a point is based on the form of a triangle, we can use the simple geometry of the triangle, especially trigonometry and pythagorean theorem, on triangles to find both the size and angle of the complex number.As we remember from school, trigonometry deals with the relationship between the edges and angles of triangles, so that we can define the relations between the parties as follows:

Again, using trigonometry, angle ε between A is given as follows.

Then, in polar form, the length and angle of A represents a complex number instead of a period. Also in polar form, the conjugate of the complex number has the same size or module, it is a sign of the changing angle, so for example, the conjugate of 6 ≤ 30^{o}will be 6 ≤ – 30^{o.}

## Converting Between Rectangular Form and Polar Form

In rectangular form, we can express a vector in rectangular coordinates; horizontal axis is the actual axis and vertical axis is the imaginary axis or j component.In the polar form, these real and virtual axes are simply represented by " A ∠ε ".Then, using our example above, it can be defined as the relationship between the rectangular form and the polar form.

### Converting Polar Form to Rectangular Form, ( P→R )

We can also return from the rectangular form to the polar form as follows.

### Converting a Rectangular Form to a Polar Form, ( R→P )

## Polar Form Multiplication and Division

The rectangular format is best for collecting and subtracting complex numbers, as we see above, but the polar form is usually better for multiplication and division.To hit the two vectors in a polar way, we first have to multiply the two modules or the magnitude together, and then collect their angles.

### Polar Multiplication

Multiplying together is 6 ∠30 ^{o} and 8 ∠-45 in ^{that} polar form.

### Split in Pole Format

Similarly, in order to divide the two vectors polarly, we must divide the two modules and then remove their angles, as shown.

Fortunately, today's modern scientific calculators have mathematical functions (check your book) that allow the rectangle to be easily transformed into a polar form (R → P) and a rectangular form (R → P) from the pole.

## Complex Numbers That Use Exponential Form

So far, we have considered complex numbers **in Rectangular Form** ( a+jb ) and **Polar Form** ( A ∠±ε ).However, there is also a third method to represent a complex number similar to the polar form that corresponds to the length (size) and phase angle of the sinusoid, but uses the base of the natural logarithm *e* = 2.718 281… to find the value of the complex number.This third method is called **Exponential Form.**

Exponential Form uses the trigonometric functions of both sinus ( sin ) and cos ( cos ) values of a right-angled triangle to define it as a rotating point in a complex plane with a complex exponent. The exponential form to locate the point is based on the Id of Euler, named after the Swiss mathematician Leonhard Euler, and is given as follows:

Euler's identity can then be represented by the following rotating phaser diagram on the complex plane.

We can see that Euler's identity is very similar to the polar form above, and shows us that a number such as Ae^{jε,} which has a magnitude of 1, is also a complex number. Not only does it easily convert complex numbers in exponential form to polar form such as 2e ^{j30} = 2 √ ^{30,} 10e ^{j120} = 10 √ 120 or -6e j90 = -6 √ 90, but Euler's identity also gives us a way to convert a complex number from exponential form to rectangular form. Then, in defining a complex number, the relationship between exponential, polar, and rectangular form is given as follows.

### Complex Number Forms

## Phaser Display

So far, we've explored different ways to represent a rotating vector or a stationary vector using complex numbers to identify a point in a complex plane.Phaser representation is a single complex number generation process with the amplitude and phase angle of the given sinusoidal waveform.

Then the phaser representation, or phaser transformation as sometimes called, transfers the actual part of the sinusoidal function: A(t) = Am cos(wt ± Φ) from the time zone to the complex number field, also called the frequency field. Eg:

Please note that √ 2 converts the maximum amplitude to *an effective* or RMS value with the given phase angle (ω) in radians.

## Summary of Complex Numbers

Then, to summarize this tutorial on **Complex Numbers** and the use of complex numbers in electrical engineering.

- Complex Numbers consist of two different numbers: one real number and one virtual number.
- Virtual numbers are distinguished from the actual number using the j-operator.
- A number with the letter " j " in front of it defines it as an imaginary number on a complex plane.
- By definition, j-operator j ≡ √ -1
- Imaginary numbers can be aggregated, subtracted, multiplied, and divided in the same way as actual numbers.
- Returns j 2 =
^{-1}multiplied by " j " j "^{} - A complex number in a rectangular form is represented by a point in space on a complex plane.
- In Polar Form, a complex number is represented by a line with amplitude in length and a phase angle.
- In Exponential Form, a complex number is represented by a line that uses the base of the natural logarithm and the corresponding angle.
- A complex number can be represented in one of three ways:
- Z = x + jy » Rectangular Form
- Z = A ∠Φ » Polar Form
- Z = A
*e*^{jΦ}» Exponential Form

- Euler's IDENTITY can be used to convert Complex Numbers from exponential to rectangular format.

In previous courses, including this, we have found that we can use phasers to represent sinusoidal waveforms, and their amplitude and phase angles can be written in the form of a complex number. We also found that Complex Numbers can be presented in rectangular, polar or exponential form with the transformation between each complex form of number algebra, including addition, subtraction, multiplication, and division.

In the next few tutorials on the phaser relationship in AC series circuits, we will look at the impedance of some common passive circuit components and draw phaser diagrams for both the current flowing from the component and the voltage applied.