Application of complex numbers in AC circuits

Application of complex numbers in AC circuits

Complex numbers are generally written in the form a+bi. From this equation, a and b are called real numbers while i is called an imaginary number. In Argand diagrams, it is in the form (a,b). Complex numbers allow computations of absolute equations whose solution in real numbers format is not possible to get. In polynomial equations having real and complex coefficients in a single variable, the use of complex number computations efficiently arrives at the solution. In today, complex numbers can be defined as an algebraic extension of an ordinary real name by an imaginary number i. Complex numbers can, however, be applied in multiplication, subtraction, addition, and divisions in polynomial equations (Castro, 2019). In the geometrical analysis of two-dimensional complex planes, the horizontal axis represents the real part of the complex number, while the vertical axis represents the imaginary section.

In ac circuit analysis, analyzing voltages, currents, resistance, capacitance, and inductance in a circuit containing resistors, capacitors, and inductors is impossible by the use of one-dimensional scalar quantities. This is made possible by the use of multidimensional amounts. It is in this study of electricity where now the complex numbers play an essential role in easing any computations involving linear and non-linear electrical devices.  Since the theoretical value I in electrical studies represent the current, it cannot, therefore, be used as an imaginary number in equations involving electricity. Consequently, it is replaced by an imaginary number j when it comes to computations involving power; hence the general equation now becomes a+jb where a and b are real numbers, and j is the imaginary number.

Therefore, complex numbers play an essential role in ac electrical circuit as it makes it possible to compute mathematical analysis that cannot be done by the use of a one-dimensional format. In this context, we will exploit more on how complex numbers are applied in the following electrical aspects:

• Impedance and phase angle
• The impedance of a resistor
• The impedance of a capacitor
• The impedance of an inductor
• The reactance of capacitors and inductors

In impedance and phase angle, impedance can be defined as the measure of the opposition of the circuit to the flow of the current when a voltage is applied. It extends the knowledge of resistance in ac electrical circuit, and therefore, in mathematical analysis, it is easily computable if it is in complex number format. Unlike strength, impedance has both magnitude and phase angle. Although both have the same units, resistance has a zero phase angle, and that why it has no theoretical value in a complex number format. Impedance is denoted by letter Z. In AC electrical circuits, the mathematical representation of an impedance can be written as:

Z=R+jX……………… (I)

This is now the complex format of impedance. From equation (I) above, we can see that there are both real numbers and an imaginary value; hence the equation is in complex representation. Z from the equation stands for impedance, R stands for the actual name of the equation, and j stands for the notional value of the complex equation (Norouzi, 2019 ).

In polar form, Z can be written as |Z|<ϴ. |Z| represents the complex number of the impedance and ϴ the phase angle.

Capacitors in electrical circuits are described as non-linear devices. It lags the voltage by 90ᵒ compared to the current in the time domain. Therefore the impedance of a capacitor can be represented as:

Zc=-j/ωC.

From the above equation, we can see that the impedance of the capacitor is expressed in a complex number format where C is the capacitance of the capacitor, and ω is the angular frequency. This expression, therefore, make it possible for calculations involving negative number which in one-dimensional format cannot be computed. In our equation above, the real value is zero, while the notional amount is -j/ωC.

Inductors are also non-linear devices that play an essential role in electrical circuits, depending on where it is applied. They introduce inductances into a course. Inductor stores electrical energy temporarily in a loop in the form of magnetic fields. When used in the circuit, current lag by 90ᵒ compared to the voltage in the time domain.

When expressed in a multiplex format, the impedance of an inductor can be expressed as  ZL=jωL

From the equation above, the real part of the equation is not shown because it is zero, and the imaginary part is jωL. L denotes the inductance of the inductor, while ω is the angular frequency.

Reactance is an ac electrical circuit that can be defined as the opposite of an electrical course to the flow of alternating current due to circuit capacitances and inductances. When an AC flows through a component that has reactance, energy is alternately stored in and leased from the magnetic field. Increasing inductance also inductive reactance increases, and increasing capacitance, capacitive reactance decreases. From this knowledge, inductive reactance and capacitive reactance can be expressed in equations form (Norouzi, 2019). For these equations to be computable in a mathematical application, complex numbers representation will be the rule out. In a circuit containing a capacitor, the current flowing through it will be proportional to quickly we are changing the voltage at any instant.  Its reactance is therefore denoted by Xc.

Xc=1/j2ΠωfC

From the equation above, the real value is zero, and the notional value is 1/j2ΠωfC.  For the inductor, reactance will be the rate of change of current, which is proportional to the applied voltage. Xl denotes its reactance.  It can, therefore, be written as Xl=j2ΠωfL. This equation has the real number as zero.

In this knowledge of electrical circuits involving resistors, capacitors, and inductors, we can, therefore, conclude that through the use of complex numbers in mathematical analyses comprising the same, all computations that cannot be solved in one-dimensional criteria are now made possible in this application. This allows the circuit analysis solutions to be executed using simple algebra involving complex numbers instead of dealing with integral differential equations.  By also expressing currents and voltages in an electrical circuit in either Cartesian form or polar form, it will make it easier to compute other parameters in a loop like active power denoted by P, reactive power denoted by Q and apparent power denoted by S (Akagi, 2017). All these three main types of energy can, however, be used to compute other variables like capacitive losses, resistance losses, and inductive losses in an electrical circuit. Depending also on the personal choice, complex numbers in ac electrical circuits provide three primary forms. That is Cartesian form, extreme form, and exponential form; hence it does not limit the electrical engineering scholars and engineers in problem computations.

References

Castro-Núñez, M., Londoño-Monsalve, D., & Castro-Puche, R. (2019). Theorems of compensation and Tellegen in non-sinusoidal circuits via geometric algebra. The Journal of Engineering, 2019(4), 3409-3417.

Norouzi, A. (2019). Fault Currents, Circuit Breakers, and a New Method for X/R Calculations in Parallel Circuits.

Akagi, H., Watanabe, E. H., & Aredes, M. (2017). Instantaneous power theory and applications to power conditioning. John Wiley & Sons.