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CHAPTER 9 Impedance Transformation
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to changes in power levels). The unit most often used to measure power ratios is called the decibel (abbreviated dB ), and is de ned by the equation dB 10 log P where log P is the logarithm of P to the base 10.* Problem 164 (a) If the input power to an ampli er system is 8 watts and the output power is 16 watts, nd the power gain in decibels. (b) If the input power to a network is 1.65 watts and the output power is 0.26 watt, express the loss in decibels. (c) If the input power to a system is 0.75 watt, and the system is known to have a power gain of 18 decibels, nd the output power. To continue, let s next consider the condition of FREQUENCY DISTORTION or frequency discrimination. To begin, frequency distortion (or discrimination ) is produced by the unequal treatment of the di erent frequency components (harmonics) in a given signal. Frequency distortion is thus produced by the presence of L or C, or both, in a circuit; this is because XL and XC are both functions of frequency (!L and 1=!C). Thus, suppose a certain non-sinusoidal signal wave is applied to the input terminals of a network containing inductance L. Since the value of XL is di erent for each harmonic component, it follows that the relative values of the di erent harmonic frequencies appearing at the output side of the network will be di erent from their relative values at the input side of the network, and thus the output signal will not have exactly the same waveshape as the input signal. Thus the output wave will be distorted to some extent, relative to the given input signal wave. It should be noted that frequency discrimination is sometimes an undesirable condition and sometimes a necessary condition. Thus, while such discrimination is undesirable in, say, an audio ampli er system, it is a necessary condition in the operation of frequency lter networks. To complete the above discussion, it should be noted that there are really two factors to be considered when evaluating the e ect of frequency distortion. The rst concerns the amount of AMPLITUDE distortion produced, and the second concerns what is called TIME-DELAY or PHASE distortion. Thus, the various harmonic frequencies at the input side of a system have certain amplitudes and time relationships with one another, and if either the relative amplitudes or the relative time relationships change, the resultant waveshape at the output side of the system will not be exactly the same as the input waveshape.{ The presence of frequency distortion can be found experimentally by applying a constant-amplitude sine wave to the input side of a system and measuring the resulting amplitude and phase angle at the output side as the frequency of the input sine wave is slowly varied over any desired frequency range (keeping the amplitude of the input sine wave constant). The output amplitudes and phase angles, thus found, are then plotted on graph paper. Actually, in most practical work, only the variations in output amplitudes are recorded and plotted, since if a system has acceptable amplitude response it will generally
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* See note 19 in Appendix. { See note 20 in Appendix.
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CHAPTER 9 Impedance Transformation
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Fig. 189
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also have acceptable phase response. A typical frequency response curve, showing decibel gain versus frequency, drawn on semi-log paper,* is shown in Fig. 189. With the foregoing in mind, let s now apply the algebra of sinusoidal steady-state circuit analysis to some basic circuits, as follows.
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CASE I: BASIC LOW-PASS RC NETWORK
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This is a series RC circuit in which the output voltage appears across the capacitor, as shown in Fig. 190.
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Fig. 190
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In the gure, since R and C remain constant in value, and since the reactance of C decreases with increasing frequency, it follows that as the FREQUENCY INCREASES the OUTPUT VOLTAGE ACROSS C DECREASES; in other words, the higher the frequency, the lower is the output voltage (we are, of course, assuming the amplitude of " Vi remains constant as the frequency increases). Thus the circuit of Fig. 190 discriminates against the higher frequencies, and is therefore called a low-pass type of network. " In Fig. 190 let the input voltage be the reference vector; thus Vi Vi =08 Vi . Then, in " " " " the gure, let us denote the ratio of Vo to Vi by G (G Vo =Vi ), which we ll call the gain of the circuit. " We now wish to nd the manner in which G varies in magnitude and phase as the frequency of Vi changes (the amplitude of Vi always remaining constant). To do this we rst note that, by Ohm s law, in Fig. 190, " I