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analogous-systems

# analogous-systems - ECE 382 NOTES ON ANALOGOUS SYSTEMS...

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ECE 382 – NOTES ON ANALOGOUS SYSTEMS Systems that are governed by the same type of differential equations are called analogous systems . If the response of one physical system to a given excitation is found, then the response of all other systems that are described by the same set of equations are known for the same excitation function. We study analogous systems for the following reasons: Compact size of electric components ( R , L , C ); Circuit theory is well understood; Ease of setting up experimental verification. We shall approach this study of analogous systems using the impedance approach . 1 Series Connection of Electrical Components Consider a series connection of three electrical components (an inductor, a resistor, and a capacitor) as shown below (see Figure 1). ~ L R C i + _ e Figure 1: Series connection of RLC electrical components. Using the impedance approach, the equation describ- ing this system is E ( s ) = ( sL + R + 1 sC ) I ( s ) 4 = Z eg I ( s ) . (1) Or it can be written as E ( s ) = ( s 2 L + sR + 1 C ) I ( s ) s . (2) Thus, for serially-connected electrical elements, the total (or equivalent) impedance is the sum of the individ- ual impedance of each element. 2 Parallel Connection of Electrical Components Next consider the three basic electrical components connected in parallel as shown below (see Figure 2). i i L i L i R R C i C e Figure 2: Parallel connection of RLC electri- cal components. Using the impedance approach again, the equation describing the system in the Laplace transform domain is I ( s ) = ( 1 R + 1 sL + 1 1 sC ) E ( s ) = ( 1 R + 1 sL + sC ) E ( s ) . (3) Or it can be rewritten as I ( s ) = ( s 2 C + s R + 1 L ) E ( s ) s . (4) Thus, for parallelly-connected electrical elements, the total impedance is equal to one divided by the sum of the reciprocal of the individual impedance of each element. 1

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3 Series Connection of Mechanical Components Similar to the electrical components ( R , L , C ) that can be connected either in series or parallel, mechanical elements (mass, M ; dashpot, f ; and coil spring, K ) can also be connected in either series or parallel arrangement. x F M f K 1 K 2 (a) x F f K 1 K 2 M (b) Figure 3: Series connection of mechanical components. We shall use the concept of equivalent “grounded-chair” representation. Figure 3(a) is represented by its grounded-chair representation in Fig. 3(b). Figure 3(b) shows that the mass is in series with the other elements. The inertia force ( M ¨ x ( t )) is always taken with respect to the ground. For serially-connected mechanical ele- ments, the force F is equal to the summation of the forces acting on each individual component and each component undergoes the same displacement F ( t ) = M ¨ x ( t )+ f ˙ x ( t )+( K 1 + K 2 ) x ( t ) . (5) Its Laplace transform equivalence is F ( s ) = ( Ms 2 + fs + K 1 + K 2 ) X ( s ) 4 = Z eg X ( s ) . (6) The equivalent impedance is Z eq = Ms 2 + fs + K 1 + K 2 . Thus, for serially-connected mechanical elements, the total (or equivalent) impedance is the sum of the individual impedance of each element. (Note that the mechanical elements look like are connected in “parallel” when viewed as electrical connection, but they are actually connected in series) 4
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analogous-systems - ECE 382 NOTES ON ANALOGOUS SYSTEMS...

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