Steady State Sinusoidal Analysis Using Phasors

By Himanshu Mittal|Updated : July 19th, 2018
The complete response of a linear electric circuit is composed of two parts:
  • Natural response: The natural response is the short-lived transient response of a circuit to a sudden change in its condition.
  • Forced response: The forced response is the long-term steady-state response of a circuit to any independent sources present.
Another very common forcing function is the sinusoidal waveform. This function describes the voltage available at household electrical sockets as well as the voltage of power lines connected to residential and industrial areas.
A sinusoidal source has three attributes namely amplitude, frequency, and phase angle.
  • Phasors are vector representation of sinusoidal signals.
  • Phasors suppress the element of time.
  • The length of the phasor or its magnitude is the amplitude or maximum value of the cosine function.
  • Phasors can be added or subtracted using vector addition.  
  • Phasors are used to represent the relationship between two or more waveforms with the same frequency.
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) = i2(t) R
  • v(t) = Vm 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: Power absorbed or supplied during one cycle. For sinusoidal (and other periodic signals) we can compute averages over one period. image3
If voltage and current are in phase:

If voltage and current are in quadrature:

image5 Apparent Power: The product of rms voltage (V) and rms current (I) is called apparent power and denoted by S. It is measured in volt-ampere (VA).


Reactive power: The product of applied voltage and the reactive component of current (I sin φ) is called reactive power and is denoted by Q. It is measured in reactive volt-ampere.

Q=VI sin φ

Active power: While 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 Cos φ

Complex Power: It helps in measure of power using phasors. image2 image6 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. 04-Power-and-Power (1) 1 Apparent Power measured in VA, Reactive Power measured in VAR, and True Power 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 cosine of the phase angle between the applied voltage and the current drawn from the supply voltage gives the power factor.

04-Power-and-Power (2)

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


PAvg = (VRMS) 2 / R = (VPeak) 2 / (2R)

R = (VPeak) 2 / (2 PAvg) = (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 a 100 V rms.


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:

  1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal.
  2. Solve steady-state ac circuits using phasors and complex impedances.
  3. Compute power for steady-state ac circuits
  4. Find Thevenin and Norton equivalent circuits. Lightly.
  5. Determine load impedances for maximum power transfer.

Impedance in AC Circuits

  • The inductances are represented by inductive reactance’s in AC circuits

X= ωL = 2πfLΩ

  • The capacitances are represented by capacitive reactance’s in AC circuits


  • 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 :

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


Pure inductance L:

  • Characteristics: i lags V by 90°
  • Impedance in rectangular from: Z = 0 + jXL
  • Impedance in polar from: Z = XLL + 90°
  • Phasor diagram:


Pure capacitance C:

  • Characteristics: i leads V by 90°
  • Impedance in rectangular from: Z = 0 – jXC
  • Impedance in polar from: Z = XCL – 90°
  • Phasor diagram:


Impedance in 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

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

Timedomain (ii) VI Relationship for R, L and C in Frequency Domain:




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