646 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



PULLING EFFECT 



In a perfect FM system the carrier wave can be written 



E,{t) = A sin [pt + <^(0] (1) 



where .4 is a constant and the signal is ^S(0 = dxp/dt = (p'(t), measured 

 in radians/second. As mentioned in the Introduction, we assume that 

 when the FM oscillator is connected directly to a transmission line with a 

 slightly mismatched antenna at the far end, its frequency is changed. 

 The reactive component of the input impedance of the line "pulls" the 

 frequency of the oscillator to its new value. When the antenna is perfectly 

 matched, there is no change in oscillator frequency. 



If the characteristic impedance of the line is Zr and the impedance of 

 the antenna is Zr , the impedance Z looking into the line is 



y ^ J ^R + ^K ^-^"h P 



Zk + Zr tanh P 

 _ 7 I + pe 



— Zjr 



1 - pc--^ 

 where p is the reflection coefficient 



Zr + Zk 



and P is the propagation constant of the line. If the loss of the line is 

 negligible and the reflection coefficient is small, the input impedance is 

 approximately 



Z = Zk[\ + 2p(cos o:T - i sin aT)] ohms 



where w is the oscillator frequency in radians per second and T is twice 

 the delay of the line. 



It will be observed that the magnitude of the reactive component of Z 

 oscillates as the phase angle coT increases. 



The dependence of the frequency of an oscillator upon the load reac- 

 tance has been expressed by earlier workers as a "pulling figure." This 

 figure is customarily defined as the difference between the maximum and 

 minimum frequencies observed when the load reactance is varied over 

 one cycle of its oscillation (the variation being accomplished, say, by 

 increasing T). The load is taken to be such that it causes a voltage stand- 

 ing wave ratio of 1.5. This corresponds to a reflection coefficient of 0.20 

 and 14 db return loss. 



In our work, we assume that the change in frequency is flirectly pro- 



