Digital Electronics Articles - Page 38 of 57

Water TurbineIn a hydroelectric power plant, a device which is used to convert the energy of falling water into mechanical energy is known as water turbine. There are two types of water turbines used in a hydroelectric power plant viz. −Impulse TurbineReaction TurbineImpulse TurbineIn an impulse turbine, the entire pressure of water is converted into kinetic energy in a nozzle and the velocity of the water jet drives the turbine wheel. Impulse turbines are used for high heads. The pelton wheel is an example of the impulse turbine. It consists of a wheel fitted with elliptical buckets along its periphery. ... Read More

An electric power generating station in which the chemical energy of diesel is converted into electrical energy is known as diesel power plant. In other words, the diesel power plant is a power generating plant in which diesel engine is used as the prime mover for the generation of electrical energy.Schematic Diagram and Working of Diesel Power PlantThe schematic arrangement of a typical diesel power plant is shown in the figure given below.In a diesel power plant, the diesel engine is used as the prime mover to drive an alternator. The diesel (fuel oil) burns inside the engine and the ... Read More

Thermal Power PlantA generating station which converts the heat energy of combustion of coal into electrical energy is called a thermal power plant or steam power station.Schematic Arrangement of Thermal Power PlantThe thermal power plant has many arrangements for proper and efficient working. The schematic of a modern thermal power plant is shown in the figure. This whole schematic arrangement can be divided into the following segments −Coal and Ash Handling PlantThe coal is transported to the site of power plant from the coal mines by rail or road and it is stored in the coal storage plant. From the ... Read More

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 to calculate the steady state value of a sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{x}\mathrm{\left(\mathit{\infty}\right)}$ directly from its Z-transform, without the need for finding its inverse Z-transform.Statement - If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a causal sequence, then the final value theorem of Z-transform states that if, $$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$And if the Z-transform X(z) has no poles outside ... 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 called the energy spectral density.The ESD function of a signal is given by, $$\mathit{\psi}\mathrm{(\mathit{\omega})}\: \mathrm{=}\: \mathrm{|\mathit{X}\mathrm{(\mathit{\omega})}|}^\mathrm{2} \:\mathrm{=}\: \mathit{X}\mathrm{(\mathit{\omega})} \mathit{X}\mathrm{(\mathit{-\omega})}$$Autocorrelation TheoremStatement − The autocorrelation theorem states that the autocorrelation function $\mathit{R}\mathrm{(\mathrm{\tau})}$ and the ESD (Energy Spectral Density) function $\mathit{\psi}\mathrm{(\mathit{\omega})}$ of an energy signal $\mathit{x}\mathrm{(\mathit{t})}$ form a Fourier transform pair, i.e., $$\mathit{R}\mathrm{(\mathit{\tau})} ... 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, $$\mathrm{\mathit{F^{\mathrm{-1}}\left [ X\left ( \omega \right ) \right ]\mathrm{\mathrm{=}}x\left ( t \right )\mathrm{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega \right )e^{j\omega t}d\omega }}$$Multiplication Property of Fourier TransformStatement – The multiplication property of continuous-time Fourier transform (CTFT) states that the multiplication of two functions in time domain is equivalent to the convolution of their spectra in the frequency domain. The multiplication property is also called frequency convolution theorem of Fourier ... Read More

ConvolutionConvolution is a mathematical tool for combining two signals to produce a third signal. In other words, the convolution can be defined as a mathematical operation that is used to express the relation between input and output an LTI system.Consider two signals $\mathit{x_{\mathrm{1}}\left( t\right )}$ and $\mathit{x_{\mathrm{2}}\left( t\right )}$. Then, the convolution of these two signals is defined as$$\mathrm{ \mathit{\mathit{y\left(t\right)=x_{\mathrm{1}}\left({t}\right)*x_{\mathrm{2}}\left({t}\right)\mathrm{=}\int_{-\infty }^{\infty }x_{\mathrm{1}}\left(\tau\right)x_{\mathrm{2}}\left(t-\tau\right)\:d\tau=\int_{-\infty }^{\infty }x_{\mathrm{2}}\left(\tau \right)x_{\mathrm{1}}\left(t-\tau\right)\:d\tau }}}$$Properties of ConvolutionContinuous-time convolution has basic and important properties, which are as follows −Commutative Property of Convolution − The commutative property of convolution states that the order in which we convolve two signals does not ... 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 equation (1) on both sides over one period, i.e., $$\mathrm{\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt=a_{0}\int_{t_{0}}^{(t_{0}+T)}dt+\int_{t_{0}}^{(t_{0}+T)}\left(\sum_{n=1}^{\infty}a_{n}\:cos\:n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t\right)dt}$$$$\mathrm{\Rightarrow\:\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt=a_{0}T+\sum_{n=1}^{\infty}a_{n}\int_{t_{0}}^{(t_{0}+T)}cos\:n\omega_{0} t\:dt+\sum_{n=1}^{\infty}b_{n}\int_{t_{0}}^{(t_{0}+T)}sin\:n\omega_{0} t\:dt… (2)}$$As we know that the net areas of sinusoids over complete periods are zero for any non-zero integer n and any time $t_{0}$. Therefore, $$\mathrm{\int_{t_{0}}^{(t_{0}+T)}cos\:n\omega_{0} t\:dt=0\:\:and\:\:\int_{t_{0}}^{(t_{0}+T)}sin\:n\omega_{0} t\:dt=0}$$Hence, from equation (2), we get, $$\mathrm{\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt=a_{0}T}$$$$\mathrm{\therefore\:a_{0}=\frac{1}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)\:dt… (3)}$$Using equation (3), ... 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 −“The representation of periodic signals over a certain interval of time in terms of linear combination of orthogonal functions (i.e., sine and cosine functions) is known as Fourier series.”The Fourier series is applicable only to the periodic signals i.e. the signals which repeat itself periodically over an interval from $(-\infty\:to\:\infty)$and ... 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, $$\mathrm{x(t)=a_{0}+ \sum_{n=1}^{\infty}a_{n}\:cos\:\omega_{0}nt+b_{n}\:sin\:\omega_{0}nt\:\:… (1)}$$Exponential 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 exponential functions, then it is known as exponential Fourier series.Mathematically, the standard exponential Fourier series expansion for a periodic ... Read More