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# Steady State Sinusoidal Analysis Using Phasors

By BYJU'S Exam Prep

Updated on: September 25th, 2023

steady state sinusoidal analysis using phasors is a powerful technique used in electrical engineering to analyze the behavior of circuits driven by sinusoidal voltage or current sources. This method is particularly useful when dealing with circuits that contain multiple frequency components, as it simplifies the analysis process and allows for quick and accurate results. Phasors are mathematical tools used in electrical engineering to represent sinusoidal signals in a compact and convenient way. A phasor is a vector that represents the amplitude and phase of a sinusoidal signal at a particular frequency.

By representing sinusoidal signals as phasors, we can perform algebraic operations such as addition, subtraction, multiplication, and division, which are much simpler than working with complex sinusoidal functions. Phasors are particularly useful when dealing with AC circuits, where multiple sinusoidal sources with different frequencies and phases may be present. In this article, we will explore the principles of steady state sinusoidal analysis using phasors, including how to represent sinusoidal signals as phasors, how to perform phasor arithmetic, and how to use phasors to analyze AC circuits.

Table of content

- 1. Steady State Sinusoidal Analysis Using Phasors
- 2. What is Power in AC Circuits?
- 3. Instantaneous Power
- 4. Average Power
- 5. Apparent Power
- 6. Complex Power
- 7. Steady-State Sinusoidal Analysis
- 8. Impedance AC Circuits
- 9. Phase Diagrams in Passive Elements Pure resistance R
- 10. Phasor Voltage/Current Relationships

## Steady State Sinusoidal Analysis Using Phasors

In the steady state sinusoidal analysis using phasors, we assume that the circuit has reached a steady state, where all the voltages and currents have settled to their steady-state sinusoidal values. This means that we can represent all the sinusoidal signals in the circuit as phasors, which simplifies the analysis process. By using phasors, we can transform the circuit analysis problem from a time-domain problem to a phasor-domain problem, which is much simpler to solve.

The principles of steady-state sinusoidal analysis using phasors are based on the concept of impedance, which is the AC equivalent of resistance. Impedance is a complex quantity that represents the opposite of a circuit element to the flow of AC current. By representing circuit elements as impedances, we can perform phasor analysis using Ohm’s law and Kirchhoff’s laws, just as we would in DC circuits. By combining phasors using algebraic operations, we can calculate the phasor voltage and current values at any point in the circuit, which allows us to analyze the behavior of the circuit under different conditions.

## What is Power in AC Circuits?

## Instantaneous Power

- It is used for the special case of steady-state sinusoidal signals.
- Instantaneous power supplied to Impedance: p(t) = v(t) i(t) = i
^{2}(t) R

- v(t) = V
_{m}sin(ωt), where the radian frequency in radians per second ω = 2πf and the frequency in Hertz f = 1 T and T is the period of the sine wave.

## Average Power

If voltage and current are in phase:

_{v}+ θ

_{i}⇒ P = (1/2) V

_{M}I

_{M}(Purely Resistive)

If voltage and current are in quadrature:

θ_{v} – θ_{i }= ± 90° ⇒ P = 0 (Purely Inductive or Capacitive)

## Apparent Power

Apparent power is a fundamental concept in electrical engineering that refers to the total power consumed by an AC circuit, including both active power and reactive power. Active power represents the power dissipated by the circuit and is measured in watts, while reactive power represents the power that is stored and released by the circuit’s reactive elements such as capacitors and inductors, and is measured in volt-amperes reactive (VAR). The total power consumed by an AC circuit is known as apparent power, which is measured in volt-amperes (VA). Understanding apparent power is crucial in the design and analysis of AC circuits, as it determines the size of the components required to handle the total power consumed by the circuit.

The product of rms voltage (V) and rms current (I) is called apparent power and is denoted by S. It is measured in volt-ampere (VA).

S=VI

Reactive power

Reactive power is a concept used in electrical engineering to describe the portion of AC power that is not used to perform useful work but is required to maintain the electric and magnetic fields in AC circuits. Reactive power is measured in units of volt-amperes reactive (VAR) and is an important parameter in the design and operation of power systems. Understanding reactive power is essential for ensuring the efficient and reliable operation of power systems and for minimizing losses and disruptions in the distribution of electric power.

Q=VI sin φ

Active power

Active power, also known as real power or true power, is the actual power consumed by an electrical device or system. It represents the energy that is converted into a useful form, such as mechanical work or heat and is measured in watts (W). Active power is the primary component of the total power consumed by a device or system, and it is essential for determining the efficiency of electrical systems. The power is called active power, true power or real power. The unit of Active or Real power is Watt where 1W = 1VA. Active power is denoted by P.

P = VI Cosφ

## Complex Power

Complex power is a concept used in electrical engineering to describe the power consumed or supplied by a circuit element in AC circuits. Unlike the simple concept of real power, which represents the rate at which energy is being transferred, complex power takes into account both the real and imaginary components of the power, which represent the active and reactive power, respectively. Understanding complex power is essential in the design and analysis of AC circuits, particularly those with reactive elements such as capacitors and inductors.

Power Factor (cos φ): It is defined as the ratio of true power to apparent power. It is a measure of the angle between current and voltage phasors.

Power Factor = (True Power/ Apparent Power) = V_{i} cos∅ / V_{i }= cos∅ Apparent Power is measured in VA, Reactive Power is measured in VAR, and True Power is measured in Watts.These three types of power are trigonometrically related to one another. In the above right triangle, P = adjacent length, Q = opposite length, and S = hypotenuse length. The opposite angle is equal to the circuit’s impedance (Z) phase angle.

The numerical value of the cosine of the phase angle between the applied voltage and the current drawn from the supply voltage gives the power factor.

cos ∅ = R/Z

- The nature of the power factor is always determined by the position of the current with respect to the voltage.
- For the pure inductor, the power factor is cos 90o(zero lagging)
- For the pure capacitors, the power factor is cos 90o (zero but leading).
- For a purely resistive circuit voltage and current are in phase i.e., φ = 0 [cos 0 = 1]. A such circuit is called a unity power factor circuit.

Example-1:What is the resistance of a light bulb that uses an average power of 75.0 W when connected to a 60 Hz power source with a peak voltage of 170 V?

Solution:

P_{Avg} = (V_{RMS}) 2 / R = (V_{Peak}) 2 / (2R)

R = (V_{Peak}) 2 / (2 P_{Avg}) = (170)2 / (2 * 75) = 193 Ohms

Example-2:An inductor has a 54.0 Ohm reactance at 60 Hz. What will be the maximum current if this inductor is connected to a 50 Hz source that produces 100 V rms.

Solution:

XL = ω L

XL 50 Hz = 2 π (50) L

XL 60 Hz = 2 π (60) L

XL 50 Hz / XL 60 Hz = 2 π (50) L / ( 2 π (60) L ) = 5/6

XL 50 Hz= 5/6 (XL 60 Hz) = 5/6 (54) = 45 Ohms

I Peak 50 Hz = VPeak / XL 50 Hz = (1.41) (100) /45 =3.13 A

## Steady-State Sinusoidal Analysis

The steady-state sinusoidal analysis is a fundamental technique used in electrical engineering to analyze circuits that are driven by a sinusoidal voltage or current source. In steady-state sinusoidal analysis, we assume that the circuit has reached a steady state, where all the voltages and currents have settled to their steady-state sinusoidal values. This simplifies the analysis process and allows for quick and accurate results. By using the principles of steady-state sinusoidal analysis, we can determine the behavior of circuits under different conditions, such as the effect of changing the frequency or amplitude of the input signal. This technique is essential in the design and analysis of AC circuits, and it provides the foundation for more advanced techniques such as phasor analysis and Fourier analysis.

- Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal.
- Solvesteady-state ac circuits using phasors and complex impedances.
- Compute power for steady-state ac circuits
- FindThevenin and Norton equivalent circuits. Lightly.
- Determineload impedances for maximum power transfer.

## Impedance AC Circuits

Impedance is a fundamental concept in electrical engineering, particularly in AC circuits. It is a complex quantity that represents the opposite of a circuit element to the flow of AC current. Unlike resistance, which is a real quantity, impedance has both a magnitude and a phase angle. In AC circuits, impedance plays a crucial role in determining the behavior of the circuit, including the voltage and current waveforms, power distribution, and frequency response. Understanding the principles of impedance in AC circuits is essential for designing, analyzing, and troubleshooting electrical systems that operate on AC power.

- The inductances are represented by inductive reactances in AC circuits

XL= ωL = 2πfLΩ

- The capacitances are represented by capacitive reactances in AC circuits

X_{c} = (1/ω_{C}) = (1/ 2πfC)Ω

- The combination of R, XL and XC present in the circuit is called an impedance of the circuit.

## Phase Diagrams in Passive Elements Pure resistance R

Phase diagrams are graphical representations used in electrical engineering to visualize the relationship between two or more sinusoidal signals. In passive elements such as pure resistance R, phase diagrams can be used to illustrate the relationship between the voltage and current in a circuit. By plotting the voltage and current phasors on a polar graph, we can determine the phase difference between the two signals, which is a critical parameter in AC circuit analysis.

- Characteristics: V and I are in phase
- Impedance is rectangular from Z = R + j0
- Impedance polar from = RL0°
- Phasor diagram:

**Pure inductance L:**

- Characteristics: I lag V by 90°
- Impedance rectangular from: Z = 0 + jXL
- Impedance polar from: Z = XLL + 90°
- Phasor diagram:

**Pure capacitance C:**

- Characteristics:i lead V by 90°
- Impedance rectangular from: Z = 0 – jXC
- Impedance polar from: Z = XCL – 90°
- Phasor diagram:

Impedance R.L and R.C Series Circuit with Phasor Diagram

- For R– L series circuit, the impedance is represented as,

- For R – C series circuit, the impedance is represented as,

## Phasor Voltage/Current Relationships

Phasor voltage/current relationships refer to the mathematical representation of voltage and current as phasors in the frequency domain. A phasor is a complex number that represents the amplitude and phase of a sinusoidal waveform. By using phasor analysis, we can simplify the calculation of AC circuit behaviour, especially in steady-state conditions. Phasor analysis allows us to transform complex waveforms into simpler phasors and analyze the magnitude and phase relationships between voltages and currents in AC circuits. This technique is widely used in electrical engineering to design and analyze circuits and systems that operate at AC frequencies.

(i) VI Relationship for R, L and C in Time Domain:

(ii) VI Relationship for R, L and C in the frequency domain:

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