Complex Numbers – An Alternative View

Preamble

A typical way of introducing complex numbers is as below:

                                                       Z  =  (a + ib) 

where ‘Z’  is a “complex” number and i = √(-1)  is the “imaginary” number.   In addition, ‘a’ is called the “real part” and ‘ib’ is called the “imaginary part”. 

The terms “real”, “imaginary” and “complex” do not convey the underlying concepts and lead to eternal confusion.  Novice students are at a loss, and succumb to rote learning to get through exams.  The basic concepts remain a mystery.

This blog provides an alternative way to introduce complex numbers and the concepts involved.  It concludes by demonstrating that complex numbers are the most general form of a number!

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Three Phase Systems – Part 1 – Why Three Phase?

Preamble

The blogs on Electric Power provided the concepts in AC power flow including the concept of Reactive (Oscillatory) power.  We also derived the generalised equation for AC power in complex form, namely S  =  (P + jQ) = E I* .

The above concepts and equations were developed for AC single phase circuits.  However, in practice, except for the low power domestic and commercial systems, the bulk of the power system uses a three phase system.  Hence, the power system engineer must have a clear understanding of the concepts in three phase power systems.

 The main objective this series of blogs is to present the basic concepts in three phase systems.  The three phase system is assumed to have no mutual coupling between phases.  However, a brief introduction to the solution of three phase systems with mutual coupling is included in this series.  A detailed presentation on mutual coupling and  ‘Symmetrical Components’ will be done in a later series. 

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Electric Power – Part 4 – AC Power Equation

 

Preamble

From the last two blogs (Electric Power – Parts 2 & 3), we know that the AC power flow consists of two components. 

 The first is the ‘Average’ power flow over one cycle, which is associated with the energy conversion or the ‘Useful’ power flow. 

 The second is the ‘Oscillatory’ power flow due to energy storage elements, namely inductances and capacitances.  The ‘Oscillatory’ power flow is not relevant for the transfer of ‘Useful’ power.  However, ‘Oscillatory’ power uses up the available capacity of the power system.  Hence, it is an  important factor in the operation of a power system. 

 Due to some quirk of history, the ‘Oscillatory’ power is called the ‘Reactive’ power!  Consequently, the ‘Average’ or ‘Useful’ power flow is called the ‘Active’ power.  The term ‘Active’ power is more commonly used in practice, and the term ‘Average’ power is rarely used! 

The above alternative terms for power have caused a lot of confusion in the power system community.  We have no choice but to accept them and move on.

We derived the equation for ‘Average’ power (P) in Part 2 of this series as given below:

                        P = E I cos(φ)  watts       ( where E = Em / √2  and I = Im / √2 )

We defined the equation for ‘Reactive’ power (Q) in Part 3 of this series as given below:

                        Q = E I sin(φ)  vars

Note that the ‘root mean square (rms)’ values of voltage (E) and current (I) are used for the calculation of AC power.

In previous blogs, we used the above equations for power calculations.  This resulted in some confusion regarding the calculation of phase angle difference (φ).  In addition, the above equations do not adequately specify the direction of power flow.

 In this blog, we shall derive a more versatile equation for power in complex form, namely S = V I*, which resolves the above issues. 

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