Z-TransformThe Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Final Value Theorem of Z-TransformThe final value theorem of Z-transform enables us ... Read More

Autocorrelation FunctionThe autocorrelation function defines the measure of similarity or coherence between a signal and its time delayed version. The autocorrelation function of a real energy signal $\mathit{x}\mathrm{(\mathit{t})}$ is given by, $$\mathit{R}\mathrm{(\mathit{\tau})} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x\mathrm(\mathit{t})}\:\mathit{x}\mathrm{(\mathit{t-\tau})}\:\mathit{dt}$$Energy Spectral Density (ESD) FunctionThe distribution of the energy of a signal in the frequency domain is ... Read More

For a continuous-time function $\mathit{x(t)}$, the Fourier transform of $\mathit{x(t)}$ can be defined as$$\mathrm{\mathit{X\left ( \omega \right )\mathrm{\mathrm{=}}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt}}$$And the inverse Fourier transform is defined as, $$\mathrm{\mathit{F^{\mathrm{-1}}\left [ X\left ( \omega \right ) \right ]\mathrm{\mathrm{=}}x\left ( t \right )\mathrm{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega ... Read More

The infinite series of sine and cosine terms of frequencies $0, \omega_{0}, 2\omega_{0}, 3\omega_{0}, ....k\omega_{0}$is known as trigonometric Fourier series and can written as, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t… (1)}$$Here, the constant $a_{0}, a_{n}$ and $b_{n}$ are called trigonometric Fourier series coefficients.Evaluation of a0To evaluate the coefficient $a_{0}$, we shall integrate the ... Read More

What is Fourier Series?In the domain of engineering, most of the phenomena are periodic in nature such as the alternating current and voltage. These periodic functions could be analysed by resolving into their constituent components by a process called the Fourier series.Therefore, the Fourier series can be defined as under ... Read More

Trigonometric Fourier SeriesA periodic function can be represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are the trigonometric functions, then it is known as trigonometric Fourier series.Mathematically, the standard trigonometric Fourier series expansion of a periodic signal is, ... Read More

Units of Synchronizing Power Coefficient (𝑷𝐬𝐲𝐧)Generally, the synchronizing power coefficient is expressed in Watts per electrical radian, i.e., $$\mathrm{𝑃_{syn} =\frac{𝑉 𝐸_{𝑓}}{𝑋_{𝑠}}cos\:𝛿 \:\:Watts/electrical\:radian …(1)}$$$$\mathrm{∵ \:𝜋\:radians = 180\:degrees}$$$$\mathrm{\Rightarrow\:1\:radian =\frac{180}{𝜋}\:degrees}$$$$\mathrm{∵ \:𝑃_{syn}=\frac{𝑑𝑃}{𝑑𝛿}\:\:Watts/ \left(\frac{180}{𝜋}\:degrees \right)}$$$$\mathrm{\Rightarrow\:𝑃_{syn}=\left( \frac{𝑑𝑃}{𝑑𝛿}\right)\left(\frac{𝜋}{180}\right)\:\:Watt/electrical\:degree …(2)}$$If p is the total number of pole pairs in the machine, then$$\mathrm{𝜃_{electrical} = 𝑝 \cdot 𝜃_{mechanical}}$$Therefore, the synchronizing ... Read More

When the flux density distribution in the alternator is non-sinusoidal, the induced voltage in the winding will also be non-sinusoidal. Thus, the pitch factor or coil span factor, distribution factor and winding factor will be different for each harmonic voltage.Pitch Factor for nth HarmonicAs the electrical angle is directly proportional ... Read More

A 1-phase supply produces a pulsating magnetic field which does not rotate in the space. Therefore, a 1-phase supply cannot produce rotation in a stationary rotor. Although, like a 3-phase supply, the 2-phase supply can also produce a rotating magnetic field of constant magnitude. Therefore, all the single-phase induction motors, ... Read More

The differences between a single-phase and a three-phase induction motor are given below −A 1-phase induction motor requires single-phase AC supply, whereas a 3-phase induction motor need a source of 3-phase AC supply for its operation.1-phase induction motors produce low starting torque, whereas 3phase induction motors produce high starting torque.The ... Read More