1. SINUSOIDAL - PHASOR

SINUSOIDAL STEADY-STATE ANALYSIS  

Introduction:

Sinusoidal current is usually referred as ALTERNATING CURRENT (ac).Such a current reverses at regular time intervals and has alternately positive and negative values. Circuits driven by sinusoidal current or voltage sources are called ac circuits.

SINUSOIDAL  -  it is a signal that has the form of sine/cosine function.

Steady-state sinusoidal time-varying voltage and current waveforms can be given by:

where v and i are the time-varying voltage and current, and V_m and I_m are the peak values (magnitudes or amplitudes) of the voltage and current waveforms. In equation 3.2, θis known as the phase angle, which is normally defined with reference to the voltage waveform.The term cos θ is called a power factor. Remember that we assumed a voltage having a zero phase. In general, the phase of the voltage may have a value other than zero. Then θ should be taken as the phase of the voltage minus the phase of the current.

PHASORS 

In most ac circuit studies, the frequency is fixed, so this feature is used to simplify the analysis. Sinusoidal steady-state analysis is greatly facilitated if the currents and voltages are represented as vectors in the complex number plane known as phasors. The basic purpose of phasors is to show the rms value (or the magnitude in some cases) and phase angle between two or multiple quantities, such as voltage and current.

The phasors can be defined in many forms, such as rectangular, polar, exponential, or trigonometric:

where R is the real part and X is the imaginary part of the complex number, and |Z| is the absolute value of Z.

Complex Number z can be also e written in polar or exponential forms as:

where r is the magnitude of z, and φ is the phase of z. We notice that z can be represented in three ways:

• z = x + jy    Rectangular form
• z = r φ       Polar form
• z = rejφ      Exponential form

Phase Shifts - Often e does not pass through zero at t = 0 sec, and we account for this by a phase shift 
(θ) . If e is shifted left (Leading), then 
 e = Em sin (ω t + θ)

If e is shifted right (Lagging), then 
 e = Em sin (ω t - θ




The angle by which the wave LEADS or LAGS the zero point can be calculated based upon the Δ