barcode reader code in asp.net Impedance of a resistor in Software

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Impedance of a resistor
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(455)
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Equation 455 corresponds to Ohm s law in phasor form, and the result should be intuitively appealing: Ohm s law applies to a resistor independent of the particular form of the voltages and currents (whether AC or DC, for instance) The ratio of phasor voltage to phasor current has a very simple form in the case of the resistor In general, however, the impedance of an element is a complex function of frequency, as it must be, since it is the ratio of two phasor quantities, which are frequency-dependent This property will become apparent when the impedances of the inductor and capacitor are de ned The Inductor Recall the de ning relationships for the ideal inductor (equations 49 and 412), repeated here for convenience: vL (t) = L 1 iL (t) = L diL (t) dt vL (t )
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(456)
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Let vL (t) = vS (t) and iL (t) = i(t) in the circuit of Figure 433 Then the following expression may be derived for the inductor current: iL (t) = i(t) = iL (t) = = 1 L 1 L vS (t ) dt (457)
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A cos t dt
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A sin t L
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Note how a dependence on the radian frequency of the source is clearly present in the expression for the inductor current Further, the inductor current is shifted in phase (by 90 ) with respect to the voltage This fact can be seen by writing the inductor voltage and current in time-domain form: vS (t) = vL (t) = A cos t i(t) = iL (t) = A cos t L 2 (458)
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It is evident that the current is not just a scaled version of the source voltage, as it was for the resistor Its magnitude depends on the frequency, , and it is shifted (delayed) in phase by /2 radians, or 90 Using phasor notation, equation 458 becomes VS (j ) = A 0 I(j ) = A /2 L (459)
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Circuits
Thus, the impedance of the inductor is de ned as follows: VS (j ) = L /2 = j L I(j )
ZL (j ) =
Impedance of an inductor
(460)
Note that the inductor now appears to behave like a complex frequency-dependent resistor, and that the magnitude of this complex resistor, L, is proportional to the signal frequency, Thus, an inductor will impede current ow in proportion to the sinusoidal frequency of the source signal This means that at low signal frequencies, an inductor acts somewhat like a short circuit, while at high frequencies it tends to behave more as an open circuit The Capacitor An analogous procedure may be followed to derive the equivalent result for a capacitor Beginning with the de ning relationships for the ideal capacitor, iC (t) = C 1 vC (t) = C dvC (t) dt iC (t ) dt
(461)
with iC = i and vC = vS in Figure 433, the capacitor current may be expressed as: iC (t) = C =C dvC (t) dt d (A cos t) dt (462)
= C(A sin t) = CA cos( t + /2) so that, in phasor form, VS (j ) = A 0 I(j ) = CA /2 (463)
The impedance of the ideal capacitor, ZC (j ), is therefore de ned as follows: VS (j ) 1 = /2 I(j ) C Impedance of a capacitor
ZC (j ) =
j 1 = = C j C
(464)
where we have used the fact that 1/j = e j /2 = j Thus, the impedance of a capacitor is also a frequency-dependent complex quantity, with the impedance of the capacitor varying as an inverse function of frequency; and so a capacitor acts
4
AC Network Analysis
ZR = R
L ZL
Z R 2
ZL = j L R
ZC 1 C
ZC = 1 j C
like a short circuit at high frequencies, whereas it behaves more like an open circuit at low frequencies Figure 434 depicts ZC (j ) in the complex plane, alongside ZR (j ) and ZL (j ) The impedance parameter de ned in this section is extremely useful in solving AC circuit analysis problems, because it will make it possible to take advantage of most of the network theorems developed for DC circuits by replacing resistances with complex-valued impedances The examples that follow illustrate how branches containing series and parallel elements may be reduced to a single equivalent impedance, much in the same way resistive circuits were reduced to equivalent forms It is important to emphasize that although the impedance of simple circuit elements is either purely real (for resistors) or purely imaginary (for capacitors and inductors), the general de nition of impedance for an arbitrary circuit must allow for the possibility of having both a real and an imaginary part, since practical circuits are made up of more or less complex interconnections of different circuit elements In its most general form, the impedance of a circuit element is de ned as the sum of a real part and an imaginary part: Z(j ) = R(j ) + j X(j ) (465)
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