Critique of Derivation of Modulator Equations (Young, Page 330)
Critique of “Derivation of Modulator Equations” (Young, page 330)
The author’s derivation is fraught with conceptual errors and symbolic inconsistencies, therefore virtually impossible to follow. If you try to follow it and get confused, that’s a good thing.
In the paragraph above equation (9-34), the author indicates that the phase shift imparted by the circuit of Figure 9-9 “ . . . follows a tangent curve that is the phase of the impedance Z; that is, (f) = -tan-1(Q) . . .” This equation (as modified below) should appear as equation (9-34), which is referenced just above equation (9-43) on the following page. The remainder of the paragraph following this equation, as well as equation (9-34) as it appears in the text, is total hooey. The intent of the hooey is expressed in equation (9-46).
He previously defines the symbol using equation (9-33a):
(9-33a)
Where f0 is the resonant frequency of the tuned circuit (per eq 9-31), and fcis the frequency of the input carrier oscillator signalin figure 9-9. The author blithely uses the symbol f, and doesn’t bother to tell us what it represents. In this derivation, the frequency of the carrier oscillator is fixed, and the resonant frequency of the tuned circuit is being varied by changing the capacitanceCd; thus the phase function should not be stated as a function of f, or fc but as a function of fo :(f0) = -tan-1(Q) , which is the proper form for equation (9-34) referred to above. (Recall that we have previously substituted Shape Factor, denoted by the symbol S , for Q .) This sets the stage for total confusion in the subsequent discussion, wherein the author refers to “frequency” and “frequency deviation” in a way that implies that he is referring to mystery symbol“ f ” , presumably meaning the frequency of the carrier oscillatorfc, which exhibits no frequency deviation. The chain rule relation, equation (9-35) should appear as:
(9-34)
Note that the author incorrectly uses the symbol Vm (uppercase “V” indicates an AC amplitude) for the modulation voltage, shown in the figure as vm, correctly indicating an AC time function. This correction has been incorporated in equation (9-34) above, and subsequently.
In evaluating the first term of equation (9-34), equation (9-40) correctly identifies the correct derivative with respect to reverse voltage, VR, but neglects to account for the fact that VR = VB–vm in figure 9-9; and we really want the derivative of Cd with respect to vm. Incorporating this oversight, equation (9-40) should appear as:
(9-40)
{term in braces included for clarity only}
Note the absence of the negative sign when compared to the form in the text, due to the second term in the braces being equal to minus one.
In evaluating the second term of equation (9-34), the author equates his improper term given in the text with the correct form given above. The leftmost term of equation (9-41) should be deleted (it merely contributes to the confusion), although the result is correct.
In equation (9-42) it is important to note that the symbolf, although it looks like a frequency deviation in the oscillator frequency, really represents the difference between the fixed oscillator frequency and the variable resonant frequency of the tuned circuit. It would have more useful to keep the difference termin equation (9-42a), and forget the f altogether. Now equation 9-42b looks like this:
(9-42b)
Equation (9-43) takes the derivative of with respect to f, (the oscillator frequency, fc ), whereas it should be taken with respect to f0, the varying resonant frequency of the tuned circuit. When equation (9-42b) is expressed as above, one can see by inspection that the derivative of with respect to f0 will have the opposite sign as the author’s result, having taken the derivative with respect to fc ; thus (9-43) should appear as:
(9-43)
They say two wrongs don’t make a right, but the two errors in the derivation each contributed a negative sign, which cancel out in the result. Equation (9-44) should appear as:
(9-44a)
We want to evaluate the nominal sensitivity when vm = 0 (or very small compared to VB), so under these conditions we have:
(9-44b)