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#### Application of The Laplace Transform to the Network Synthesis

Network synthesis may be regarded as the process of obtaining an appropriate network to represent a given transfer function. Network synthesis is easier in the $s$ domain than in the time domain. In network analysis, we find the transfer function of a given network. In network synthesis, we reverse the approach: given a transfer function, we are required to find a suitable network. Network synthesis is finding a network that represents a given transfer function. Keep in mind that in synthesis, there may be many different answers-or possibly no answers - because there are many circuits that can be used

#### Application of The Laplace Transform to the Network Stability

So far we have considered three applications of Laplace's transform: circuit analysis in general, obtaining transfer functions, and solving linear integrodifferential equations. The Laplace transform also finds application in other areas in circuit analysis, signal processing, and control systems. Here we will consider two more important applications: network stability and network synthesis. Network Stability A circuit is stable if its impulse response $h(t)$ is bounded (i.e., $h(t)$ converges to a finite value) as $t \rightarrow \infty$; it is unstable if $h(t)$ grows without bound as $t \rightarrow \infty$. In mathematical terms, a

#### Application to Integrodifferential Equations

The Laplace transform is useful in solving linear integrodifferential equations. Using the differentiation and integration properties of Laplace transforms, each term in the integrodifferential equation is transformed. Initial conditions are automatically taken into account. We solve the resulting algebraic equation in the $s$ domain. We then convert the solution back to the time domain by using the inverse transform. The following examples illustrate the process.
Example 1: Use the Laplace transform to solve the differential equation $$\frac{d^{2} v(t)}{d t^{2}}+6 \frac{d v(t)}{d t}+8 v(t)=2 u(t)$$ subject to $v(0)=1, v^{\prime}(0)=-2$.
Solution: We take the Laplace

#### The Convolution Integral

The term convolution means "folding." Convolution is an invaluable tool to the engineer because it provides a means of viewing and characterizing physical systems. For example, it is used in finding the response $y(t)$ of a system to an excitation $x(t)$, knowing the system impulse response $h(t)$. This is achieved through the convolution integral, defined as $$y(t)=\int_{-\infty}^{\infty} x(\lambda) h(t-\lambda) d \lambda \tag{1}$$ or simply $$y(t)=x(t) * h(t) \tag{2}$$ where $\lambda$ is a dummy variable and the asterisk denotes convolution. Equation (1) or (2) states that the output is equal to the input convolved with the unit

#### Transfer Function of The Laplace Transform

The transfer function is a key concept in signal processing because it indicates how a signal is processed as it passes through a network. It is a fitting tool for finding the network response, determining (or designing for) network stability, and network synthesis. The transfer function of a network describes how the output behaves in respect to the input. It specifies the transfer from the input to the output in the $s$ domain, assuming no initial energy. The transfer function H(s) is the ratio of the output response Y(s) to the input excitation X(s), assuming all initial conditions
TRENDING PAGES

#### Transformer

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Chapter 11 discussed the self-inductance of a coil. We shall now examine the mutual inductance that exists between coils of the same or different dimensions. Mutual inductance is a phenomenon basic to the operation of the transformer, an electrical device used today in almost every field of electrical engineering. This device plays an integral part in power distribution systems and can be found in many electronic circuits and measuring instruments. In this chapter, we will discuss three of the basic applications of a transformer: to build up or step down the voltage or current, to act as an impedance matching device, and to isolate

#### Branches of Electrical Engineering

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Electrical Engineering and Telecommunications is arguably the origin of most high technology as we know it today. Based on fundamental principles from mathematics and physics, electrical engineering covers but not limited to the following fields:

#### Mutual Inductance

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A transformer is constructed of two coils placed so that the changing flux developed by one will link the other, as shown in Fig. 1. This will result in an induced voltage across each coil. To distinguish between the coils, we will apply the transformer convention that the coil to which the source is applied is called the primary, and the coil to which the load is applied is called the secondary. For the primary of the transformer of Fig. 1, an application of Faraday's law will result in $$\bbox[10px,border:1px solid grey]{e_p = N_p { d \phi \over dt}} \,\, \text{(volts, V)} \tag{1}$$ revealing

#### Current Sources in Parallel

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We found that voltage sources of different terminal voltages cannot be placed in parallel because of a violation of Kirchhoff's voltage law. Similarly, Current sources of different values cannot be placed in series due to a violation of Kirchhoff's current law. However, current sources can be placed in parallel just as voltage sources can be placed in series. In general, Two or more current sources in parallel can be replaced by a single current source having a magnitude determined by the difference of the sum of the currents in one direction and the sum in the opposite direction. The new parallel internal resistance is the total resistance of the resulting parallel

#### Ohms Law for Magnetic Circuits

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Recalling the equation introduced for Ohm's law for electric circuits. $$\text{Effect} = {\text{cause} \over \text{opposition}}$$ The same equation can be applied for magnetic circuits. For magnetic circuits, the effect desired is the flux $\Phi$. The cause is the magnetomotive force (mmf) , which is the external force (or "pressure") required to set up the magnetic flux lines within the magnetic material. The opposition to the setting up of the flux $\Phi$ is the reluctance $S$. Substituting, we have $$\bbox[10px,border:1px solid grey]{\Phi = {m.m.f \over S}} \tag{1}$$ The magnetomotive force is proportional to the product of the number of turns around the core
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