(i) Sketch qualitatively, as a function of $x = w/w_0,$ the magnitude of the current drawn from the source by the two circuits shown in the figure; here $w_0= (LC)^{-1/2}.$

(ii) Using three or more of the components shown in figure ( $a$ ), construct five new circuits, each of which shows current resonance (maximum current drawn from the source at some frequency), but all at different frequencies.

Question

(i) Sketch qualitatively, as a function of [latex] x = w/w_0, [/latex] the magnitude of the current drawn from the source by the two circuits shown in the figure; here [latex] w_0= (LC)^{-1/2}. [/latex]

(ii) Using three or more of the components shown in figure ([latex] a [/latex] ), construct five new circuits, each of which shows current resonance (maximum current drawn from the source at some frequency), but all at different frequencies.

Accepted Answer

(i) These circuits are tricky to analyse using differential equations, but become straightforward if complex impedances are employed. The impedance of an inductor of inductance [latex] L [/latex] at angular frequency [latex] w [/latex] is [latex] iwL,[/latex] whilst that of a capacitor of capacitance [latex] C [/latex] is [latex] 1/iwC. [/latex] Here [latex] i [/latex] is the square root of [latex] -1,[/latex] i.e. [latex] i^2 = -1.[/latex] Combining impedances is governed by the same rules as those which apply to resistances [latex]R_1 [/latex] and [latex] R_2 [/latex] in series [latex] [R = R_1 + R_2] [/latex] and in parallel [latex] [R = R_1R_2/(R_1 + R_2)]. [/latex]

Circuit [latex] (a) [/latex] thus has total complex impedance

[latex] Z=\frac{i w L imes i w L}{i w L + i w L} +\frac{(1/i w C) imes (1/ i w C)}{(1/i w C) + (1/ i w C)} = \frac{i w L}{2}+\frac{1}{2 i w C}=\frac{1- w^2 L C}{2 i w C} . [/latex]

The magnitude of the current drawn from the source is therefore

[latex] |I| = \frac{|V_0|}{|Z|} = \frac{V_0 |2 i w C |}{|1 - w^2LC|} =C V_0 w_0 \frac{2x}{|1-x^2|} , [/latex]

when [latex] LC [/latex] is written as [latex] w_0^{-2} [/latex] and [latex] w/ w_0 [/latex] is written as [latex] x. [/latex] This is plotted in Fig. S170.1(a). It will be seen that, theoretically, the current increases without limit as [latex] w [/latex] approaches [latex] w_0; [/latex] in practice the source will be unable to supply such a current. In any case, real inductors and connectors have non-zero resistance, and the calculated peak in Fig. S170.1(a) is then of finite amplitude.

In a similar way the impedance of circuit (b) is

[latex] Z=\frac{i w L imes (1/ i w C)}{i w L + (1/ i w C)} +\frac{i w L imes (1/ i w C)}{i w L + (1/ i w C)} , [/latex]

leading to

[latex] |Z|=\frac{2w L}{|1- w^2 L C |} [/latex] and [latex] |I|=\frac{C V_0 w_0 |1- x ^2|}{2x} . [/latex]

This is plotted in Fig. S170.1(b). It will be noticed that at [latex] w = w_0 [/latex] the circuit has an infinite impedance and no current is drawn. It is also of interest that the [latex] x [/latex]-dependences in Fig. S170.1(a) and (b) are reciprocal functions.

(ii) Inductances of [latex] L, 2L, L/2, [/latex] and capacitances of [latex] C, 2C, C /2,[/latex] constructed from identical components in series or parallel as needed, are used to make circuits resonant at [latex] w_0, w_0/2, 2w_0, w_0/ \sqrt{2}, \sqrt{2}w_0.[/latex] As shown in Fig. S170.2, for two of these there are two alternatives.

Question P.170: (i) Sketch qualitatively, as a function of x = w/w0, the mag...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions