What is a sinusoid?

Sinusoids are defined as the mathematical waveforms that are used to describe the nature of periodic oscillations.

What is phasor?

Phasor is defined as the complex number that is used to define the amplitude and phase domain of the sinusoids.

Functions of Sinusoids

Sinusoidal currents are typically referred to as alternating currents (AC). Currents of this kind are opposite at every intervals and changes with positive and negative values. Circuits that work on the sinusoidal voltage or current are referred to as AC circuits.

The sinusoidal wave have both transient response and steady-state response, similar to the step feature. The transient response dies out with time such that the most effective response that remains is the steady state response. The transient reaction has negligibly small time period in comparison with the steady-state response, and it can be considered that the circuit operates in sinusoidal steady-state.

Sinusoids are mathematical curves. They are in the form of cosine and sine functions. Sinusoids are used to represent the waveform of alternating current.

What is an alternating current?

Alternating current (AC) is the flow of charges that reverses its flow at a specific time period. AC waveform starts from zero and it increases to the average value and then decreases till zero after a specific period waveform continues in a reverse direction following the same manner and starts from zero then increases to maximum RMS value then decays to zero. The time required to complete one cycle including positive and negative waveform is known as the period. The same frequency is defined as the number of cycles per second.

Sinusoidal circuit

Sinusoidal currents are referred to as AC. Circuit which has sinusoidal voltage and current as its input and output is referred to as AC circuit. There are two types of the response generated by the sinusoidal circuit. One is a transient response and the other is a steady-state response.

Transient and steady-state response

Consider the example of starting off the ceiling fan. When we switch on the fan, the fan attains its maximum speed after a few seconds. The time required by the fan to reach its maximum speed is termed as transient time. After reaching the maximum speed fan continues to run at the same speed. Therefore, the time at which a fan runs at a steady speed is termed as steady time.

The transient response dies out after a certain amount of time and only steady-state remains. The time at which only steady-state remains is called sinusoidal steady state.

Benefits of sinusoids

• All the natural phenomenon we see around us the example: vibration of string, movement of a pendulum, etc. have sinusoidal characteristics.
• Any practical periodic signal can be represented in the form of a sinusoidal function by the use of the Fourier series.
• Sinusoidal functions are easier to calculate and operate.
• Sinusoids can be easily transformed into phasors.
• It saves time.

Function of Phasors

Phasors are used to define the amplitude and phase difference of sinusoidal function. Phasors are represented by vectors in a diagram. Phasors are generally a set of complex numbers in mathematical form.

Complex numbers

The complex numbers conjugate that contains both real and imaginary parts are termed as complex numbers. Complex numbers can be represented in two forms, one is a rectangular form which is defined as (a + bi) where the real axis part is represented by “a” and the imaginary part is represented by “I”. The second form is phasor form which is written as r<Φ where r represents the amplitude of sinusoidal function and Φ represents the phase of a sinusoidal function.

Basics of a complex numbers

Consider a complex number written in rectangular form.

$Z=x+iy$

Here, x represents the real part, y represents the imaginary part.

Z can also be represented in polar form as shown below.

$Z=r\left(\cos \varphi +i\sin \varphi \right)$

Here, r represents amplitude which can be calculated as:

$r=\sqrt{x^2+y^2}$

The ϕ represents the phase domain of sinusoidal that can be calculated as shown below:

$\varphi ={\tan }^{-1}\frac{y}{x}$

Phasor representation

Euler’s identity is given by:

$e^{\pm j\varphi }=\cos \varphi \pm i\sin \varphi$

Here, cos ϕ and sin ϕ are the real and imaginary parts of e.

Sinusoidal voltage can be represented in rectangular as well as phasor form. The phasor form representation is shown below:

$$\begin{gathered} v\left(t\right)=V_m\cos \left(\omega t+\varphi \right) \\ v\left(t\right)=Re\left(V_m{e}^{j\left(\omega t+\varphi \right)}\right) \\ v\left(t\right)=Re\left(V_m{e}^{j\omega t}{e}^{j\varphi }\right) \end{gathered}$$

Here, $V=V_m{e}^{j\varphi }$

Therefore, the sinusoidal voltage will result in $v\left(t\right)=V{e}^{j\omega t}$.

Hence, in this way the phasor representation can be found out for any signal including current and voltage.

Explain Phasor Diagram?

A pictorial representation of phasor representation including magnitude and phase domain is termed a phasor diagram. The phasor diagram contains voltage vector, current vector, and phase angle. With the help of the nature of phase angle, the behavior of the circuit can be determined that is whether it is lagging or leading.

Benefits of phasor diagram

With the help of a phasor diagram, it is easier to represent waveform and carry out calculations involving ac waves. It can be used to determine the nature of the circuit whether it is lagging or leading. It can be used to determine the root mean square value.

When the circuit consists of only a resistor, the current and voltage will be in the same phase difference. When the circuit contains the inductor, the current will lag the voltage and the circuit will have lagging nature whereas when the circuit contains a capacitor, the current will lead the voltage, therefore, the circuit will have a leading nature.

Application of sinusoids and phasors

The application of sinusoids and phasors to AC circuits are:

• The motivation behind the use of sinusoids in the evaluation of AC circuits is that almost all the natural phenomena have sinusoidal characteristics.
• Signals in the sinusoidal shape are simple to generate and transmit.
• The usage of Fourier evaluation, any realistic periodic signal may be represented as a sum of sinusoids.
• Mathematically, a sinusoid is easy for mathematical calculations.

 Common Mistakes

Students may get confused between instantaneous value and the phasor representation. Following are the points to avoid those confusions.

• v(t) represents the instantaneous value of sinusoidal voltage whereas V represents the phasor representation of sinusoidal voltage.

• The value of instantaneous voltage is time-dependent whereas a value of V is time-independent.

• Instantaneous voltage is always real whereas phasor representation is the complex plane.

• It may also be possible to get confused between the time domain and phasor representation. Note the following points to avoid mistakes.

• The time-domain function will be represented using small alphabets whereas phasor representation will be given by capital alphabet.

• Time-domain represents sinusoidal voltage as a function of time whereas phasor is independent of time.

Context and Applications

Some applications of sinusoids and phasors are listed below.

• Solving real-life practical problems.
• Signal analysis.
• Network theory.
• Control theory
• Fluid dynamics.
• Determining the response of the system.
• Analysis of bulk power system reliability.
• Solving RLC circuits.
• Trigonometry
• Linear algebra.

This subject matter is tremendous inside the expert exam for each undergraduate and graduate publication, mainly for:

• Bachelor of Technology in the electrical and electronic department
• Bachelor of Science Physics
• Master of Science Physics

Related Concepts

Sinusoids and phasor involve the following concepts.

• Euler’s identity
• Trigonometry identities

Practice Problems

Question 1- Transform $i=5 \cos \left(40t-40\right)$ sinusoids into phasors.

1. $5\left(\angle 30\right)$
2. $5\left(\angle 80\right)$
3. $5\left(\angle -140\right)$
4. $5\left(\angle 40\right)$

Correct answer: (c)

Explanation: To find the phasor of the sinusoidal time domain using the magnitude and phase angle.

$\begin{array}{rcl}5 \cos \left(40t+40\right)& =& 5 \cos \left(40t+40-180\right)\\ & =& 5\angle \left(-140\right)\end{array}$

Question 2- Transform $v=3\sin \left(30t-50\right)$ into phasor.

1. $3\left(\angle 140\right)$
2. $3\left(\angle -140\right)$
3. $3\left(\angle 30\right)$
4. None of the above

Correct answer: (b)

Explanation: Use identity $\sin A=\cos \left(A-90\right)$

To find phasor use magnitude and phase angle.

$\begin{array}{rcl}v& =& 3\sin \left(30t-50\right)\\ & =& 3\cos \left(30t-50-90\right)\\ & =& 3\angle \left(-140\right)\end{array}$

Question 3- Transform phasor $I=-4+i7$ into sinusoids.

1. $7\left(\angle 25\right)$
2. $8.06\left(\angle 60.25\right)$
3. $5\left(\angle 90\right)$
4. $8\left(\angle 54\right)$

Explanation: Find magnitude and phase using the formula discussed earlier.

$\begin{array}{rcl}i& =& \sqrt{{\left(-4\right)}^2+{\left(7\right)}^2}\angle \left({\tan }^{-1}\left(\frac{-7}{4}\right)\right)\\ & =& 8.06\angle 60.25\end{array}$

Question 4- When current leads voltage the nature of the circuit is _____.

1. Leading
2. Lagging
3. Unity
4. All of the above

Correct answer: (a)

Explanation: When current leads voltage, the circuit is leading in nature.

Question 5- When voltage leads to the current the nature of the circuit is _____.

A. Lagging

B. Leading

C. Unity

D. None of these

Correct answer: (a)

Explanation: When current lags voltage, the circuit is lagging in nature.