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13.7: The RLC Series Acceptor Circuit

A resistance, inductance and a capacitance in series is called an "acceptor" circuit, presumably because, for some combination of the parameters, the magnitude of the inductance is a minimum, and so current is accepted most readily.  We see in figure XIII.5 an alternating voltage File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gifapplied across such an R, L and C.

 

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image003.wmz

FIGURE XIII.5

C

L

R

 
  File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif

 

 

 

 

 

 

 

 

 

 

 The impedance is

 

                                                            File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif                                          13.7.1

 

We can see that the voltage leads on the current if the reactance is positive;  that is, if the inductive reactance is greater than the capacitive reactance;  that is, if File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif  (Recall that the frequency, n, is w/(2p)).  If File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif the voltage lags behind the current.  And if File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif the circuit is purely resistive, and voltage and current are in phase.

 

The magnitude of the impedance (which is equal to File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif) is

 

                                                       File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif                                     13.7.2

 

and this is least (and hence the current is greatest) when File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif the resonant frequency, which I shall denote by w0.

 

It is of interest to draw a graph of how the magnitude of the impedance varies with frequency for various values of the circuit parameters.  I can reduce the number of parameters by defining the dimensionless quantities

 

                                                            File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image019.gif                                                                13.7.3

 

                                                            File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image021.gif                                                              13.7.4

 

and                                                      File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image023.gif                                                                     13.7.4

 

You should verify that Q is indeed dimensionless.   We shall see that the sharpness of the resonance depends on Q, which is known as the quality factor (hence the symbol Q).  In terms of the dimensionless parameters, equation 13.7.2 becomes

 

                                                File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image025.gif                                                 13.7.5

 

This is shown in figure XIII.6, in which it can be seen that the higher the quality factor, the sharper the resonance. 

                        File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image027.gif

 

 

In particular, it is easy to show that the frequencies at which the impedance is twice its minimum value are given by the positive solutions of

 

                                                File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image029.gif                                             13.7.6

 

If I denote the smaller and larger of these solutions by W- and W+, then W+ - W- will serve as a useful description of the width of the resonance, and this is shown as a function of quality factor in figure XIII.7.

                        File:C:/Users/DELMAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image031.gif