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Calculating Phasors and Complex Numbers

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Phasors are complex numbers which are used to represent the magnitude and phase angle of sinusoidal waveforms in AC circuits.

When you are working with frequency-dependent sinusoidal sources and vectors in electrical engineering, real numbers alone are insufficient.

The introduction of Complex Numbers which include imaginary numbers denoted by the “j” operator, addresses the need for solutions involving the square roots of negative numbers (e.g., √-1).

In electrical engineering an imaginary number is referred to as such, and we use the “j” operator to distinguish it from real numbers.

Consequently we can create a complex number by combining a “Real Number” and an “Imaginary Number.” Examples of imaginary numbers include j3, j12, and j100.

We can show Complex Numbers graphically in a two-dimensional complex or s-plane, with the “real axis” as the horizontal axis and the “imaginary axis” as the vertical axis. The real and imaginary components of a complex number are abbreviated as Re(z) and Im(z) respectively.

When we are dealing with complex numbers in DC circuit analysis, we work with both real (active) and imaginary (reactive) components, applying the same rules as elementary algebra.

Operations like adding or subtracting imaginary numbers follow the principles used for real numbers, such as j2 + j4 = j6.

But, multiplication differs, because the product of two imaginary numbers results in a negative real number. Real numbers can essentially be seen as complex numbers with a zero imaginary part, represented as j0.

The j-operator, which is equivalent to √-1, yields specific values (e.g., -1, -j, +1) when we successively multiply (j x j).

We commonly use it to indicate anticlockwise vector rotation. Each multiplication or power of “j” (j2, j3, etc.) causes a fixed 90o anticlockwise rotation of the vector. On the other hand if the multiplication results in a -j operator, the phase shift is -90o, signifying a clockwise rotation.

Rotation of the vector by the j-operator

Vector Rotation of the j operator diagram

Multiplying an imaginary number by j2 causes the vector to turn 180 degrees counterclockwise; multiplying by j3 causes it to rotate 270o; multiplying by j4 causes it to rotate 360o, or return to its initial form.

The vector will spin anticlockwise by the proper amount when multiplied by j10 or j30. The vector’s magnitude never changes throughout a subsequent rotation.

In the field of Electrical Engineering there are several methods for representing complex numbers, both technically and aesthetically. Using the cosine and sine rules, one method is the Cartesian or Rectangular Form.

Representing complex numbers in the Rectangular Form

Our recent Phasors lesson demonstrated that a complex number can be described by an imaginary part and a real part, which may be broadly expressed as follows:

Z = x + jy

In this context:

When representing a complex number in its rectangular form, you visualize it as a point on a two-dimensional plane, commonly referred to as the complex or s-plane.

Take, for instance, Z = 6 + j4; this corresponds to a specific point where your coordinates indicate 6 on the horizontal real axis and 4 on the vertical imaginary axis, as depicted.

Utilizing the complex or s-plane to express complex numbers

As both the real and imaginary parts of a complex number in rectangular form can be either positive or negative, you need to extend both the real and imaginary axes in both positive and negative directions.

This as a result creates a complex plane with four quadrants, referred to as an Argand Diagram, as shown below.

Argand Diagram divided into four quadrants

On the Argand diagram, you observe that the horizontal axis represents positive real numbers to the right of the vertical imaginary axis and negative real numbers to the left.

Above the horizontal axis, you find positive imaginary numbers, and below the horizontal real axis, negative imaginary numbers are located.

This arrangement forms a two-dimensional complex plane with four specific quadrants labeled as QI, QII, QIII, and QIV.

In this Argand diagram, you can utilize it to depict a rotating phasor, essentially portraying it as a point in the complex plane. The magnitude of the phasor sets the radius, guiding it to complete a full circle every 2π/ω seconds.

Taking it a step further, you can extend this concept to illustrate the definition of a complex number in both polar and rectangular forms, specifically focusing on rotations of 90 degrees.

Representation of Complex Numbers

Complex numbers may have zero real or imaginary parts, denoted as Z = 6 + j0 or Z = 0 + j4, and their points are plotted on the corresponding axis.

Calculation of Complex Number Angles

The angle of a complex number is determined using trigonometry, involving right-angled triangles or measuring anti-clockwise around the Argand diagram from the positive real axis.

Quadrants and Angles

Angles between 0 and 90 degrees fall in the first quadrant (I), while those between 90 and 180 degrees are in the second quadrant (II).

The third quadrant (III) encompasses angles from 180 to 270 degrees, and the fourth quadrant (IV) covers angles from 270 to 360 degrees.

Angle Calculation Formula

Angles in all quadrants can be found using the formula:

tan-1(imaginary component ÷ real component)

Addition and Subtraction of Complex Numbers

Mathematically or graphically, addition involves adding real and imaginary parts separately, illustrated with two complex numbers A and B.

Addition and Subtraction of Complex Numbers

Example 1: Complex Number Operations

Consider two vectors, A = 4 + j1 and B = 2 + j3. Find the sum and difference of these vectors in rectangular form (a + jb) and represent them graphically on an Argand Diagram.

Mathematical Operations: Addition and Subtraction

How to Add

How to Minus

Representing Addition, Subtraction through Graph

The process of multiplying complex numbers, in rectangular form, adheres largely to the rules of standard algebra, with additional considerations for the consecutive multiplication of the j-operator, where j2 = -1.

To illustrate the multiplication of the two given vectors A = 4 + j1 and B = 2 + j3 yields the following outcome.

It can be somewhat trickier to divide complex numbers statistically in rectangular form as you have to apply the denominators conjugate function to change the denominator of the equation into a real integer.

We refer to this as “rationalizing.” Subdividing complicated numbers is optimally accomplished by the utilization of “Polar Form,” which will be examined subsequently. Let’s calculate the value of vector A divided by vector B, though, using a rectangle example

Understanding Complex Conjugate

The complex conjugate, or in simple words the conjugate, of a complex number is determined by negating the algebraic sign of its imaginary component while maintaining the sign of the real component.

The complex conjugate of a given complex number “z” is denoted by the symbol “z.”

For instance, if z = 6 + j4, then its conjugate is z = 6 – j4, and vice versa. On the Argand diagram, the points that indicate the complex conjugate, share the same horizontal position along the real axis as the original complex number, but occupy opposite vertical positions.

Consequently, now the complex conjugates can be conceptualized as reflections of complex numbers. The subsequent example illustrates a complex number, 6 + j4, and its conjugate in the complex plane.

Conjugate Complex Numbers

As we have seen previously, the combined value of a complex number and its complex conjugate continues to be a real number.

When a complex number and its conjugate are added, the outcome is a real number or simply the active component, and when they are subtracted, the result is an imaginary number or only the reactive component.

In the field of electrical engineering, the conjugate of a complex number is a crucial component that is utilized to calculate the apparent power of an AC circuit employing rectangular form.

Complex Numbers using Polar Form

A complex number’s Polar Form is expressed by means of its quantity and angle rather than its points being shown on the complex plane like a rectangle form does.

A polar form vector may therefore be expressed as follows: Z = A ∠±θ, where Z is the complex number expressed in polar form, A is the vector’s magnitude or modulo, and θ is its angle, or argument, that may be either positive or negative.

When the point is depicted in polar form, its position is displayed as a “triangular form,” as seen below, but its magnitude and angle are still the same as they were in the rectangular form before.

Representing Complex Number in Polar Form

Since the triangular form serves as the foundation for the polar depiction of a point, we may determine the complex number’s magnitude and angle by applying basic trigonometric and Pythagoras’s triangle theorem.

We may characterize the connections between the sides of triangles as follows because, as we recall from school, trigonometry examines the relationship between a triangle’s sides and angles.

Following this, rather than a point, the complex number is represented in Polar form by the length of A and its angle.

In polar form as well, the conjugate of a complex number has identical value or modulus; the sign of the angle varies. For instance, the conjugate of 6 ∠30o would be 6 ∠– 30o.

How to Implement Conversions across Polar Form and Rectangular Form

A vector could be expressed by means of its rectangular coordinates in the rectangular form, where the vertical axis represents the j-component or imaginary axis and the horizontal axis represents the real axis.

These real and imaginary axes are effectively indicated by “A ∠θ” in polar form. The link between rectangular form and polar form may then be stated using the example we used previously.

Calculations Showing Polar Form into Rectangular Form Conversion (P→R)

Calculations Showing Rectangular Form to Polar Form Conversion (R→P)

(5.2+ j3) = A∠θ
where: A = √5.22 + 32 = 6 and 0 = tan-1 (3/5.2) = 30° Therefore, (5.2 + j3) = 6∠30°

How to Multiply and Divide using Polar Form

When multiplying or dividing complex numbers, it’s advantageous to use polar form rather than rectangular form.

To multiply two vectors in polar form, start by multiplying their magnitudes and then combine their angles, as shown below.

As demonstrated earlier, rectangular form is more suitable for adding and subtracting complex numbers.

Z1 x Z2 = A1 x A2 = ∠θ1 + θ2

If we multiply 6 ∠30o and 8 ∠– 45o with each other in polar form we get the following results.

Z1 x Z2 = 6 x 8∠30° + (-45°) = 48∠-15°

Calculations Showing How to Divide in Polar Form

Z1/Z2 = (A1/A2) ∠θ1 + θ2

Z1/Z2 = (6/8) ∠30o – (-45°) = 0.75 ∠75°

Modern scientific calculators come equipped with built-in mathematical functions, facilitating seamless conversion between rectangular (R → P) and polar (P → R) forms.

Expressing Complex Numbers with Exponential Form

When dealing with complex numbers, we’ve explored Rectangular Form (a + jb) and Polar Form (A ∠±θ).

However, there’s a third representation, Exponential Form, akin to Polar Form but utilizing the natural logarithm base, e = 2.718 281.

This method relies on trigonometric functions (sin and cos) to define a complex exponential, treating it as a rotating point in the complex plane.

Euler’s Identity, named after mathematician Leonhard Euler, serves as the foundation for this Exponential Form. This can be expressed using the formula:

Therefore, Euler’s identity can be demonstrated through the depicted rotating phasor diagram in the complex plane below.

Euler’s identity closely resembles the polar form, highlighting that a number like Aejθ, with a magnitude of 1, is a complex number.

This identity not only simplifies the conversion of exponential to polar forms (e.g., 2ej30 = 2∠30, 10ej120 = 10∠120, -6ej90 = -6∠90) but also provides a method for converting exponential to rectangular forms.

This establishes a comprehensive connection among Exponential, Polar, and Rectangular forms in defining complex numbers, as indicated below.

Z = x + jy = A∠θ = A(cosΦ + sinΦ)

Wrapping up

In electrical engineering, Complex Numbers encompass both real and imaginary components, the latter identified by the j-operator.

Imagination numbers, marked with “j,” are defined by j ≡ √-1 and undergo arithmetic operations akin to real numbers.

Represented in Rectangular Form as a point on the complex plane, Complex Numbers are visualized in Polar Form by amplitude and phase angle and in Exponential Form using the natural logarithm base.

Three representations exist: Rectangular (Z = x + jy), Polar (Z = A ∠Φ), and Exponential (Z = A e jΦ). Euler’s identity facilitates the conversion from exponential to rectangular forms.

Previous tutorials, including this one, introduced the use of phasors to represent sinusoidal waveforms, expressing amplitude and phase angle as complex numbers.

Complex Numbers can be portrayed in Rectangular, Polar, or Exponential Forms, with seamless algebraic conversion.

Upcoming tutorials on AC series circuits will explore the impedance of common passive components, starting with AC Resistance, illustrated through phasor diagrams depicting current and voltage relationships.

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