550 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



From the above equations it can be seen that gain does not increase 

 rapidly with a when a is sufficiently near unity. In the case of our numerical 

 example, increasing a from 0.9785 to unity increases the gain by only 4.1 db. 

 The gain of the stage is approximately proportional to Tc and inversely 

 proportional to r^ and can therefore be increased by operating at higher 

 emitter currents or by fabricating the transistor in such a way as to obtain 

 higher values of tc . In the latter case it would be desirable also to increase 

 a in order to keep the output impedance from becoming unreasonably high. 



It has been seen that, for the case of our numerical example, the matched 

 output impedance is large compared to the input impedance (671,000 ohms 

 compared to 619 ohms). This means that if the maximum available gain is 

 to be obtained in cascaded stages, step-down interstage transformers must 

 be used. But an appreciable amount of gain can be obtained without inter- 

 stage impedance matching. This is because of the short-circuit current 

 amplification previously mentioned which amounts approximately to 



1 - a 



or 45.5 times in the case of our numerical example. For this transistor, then, 

 the iterative gain per stage without impedance transformation would be 

 33.2 db. This gain increases very rapidly as a approaches unity, not only 

 because the short-circuit current amplification increases, but also because 

 the output impedance decreases and the input impedance increases so that 

 a better interstage impedance match is obtained. From equations (41) and 

 (42), it is seen that the matched input and output impedances are equal 

 when 



fe — rm — fb 

 or when 



1-. 



re + n 



In this case the gain per stage would be approximately rc/fg . This says 

 that if fm could be increased in the numerical example until 



a = 0.999821 



the gain per stage (without impedance transformation) would be increased 

 to 57.1 db. Values of a this near to unity have not even been approached in 

 transistors made to date. This unrealistic example is included only to indi- 

 cate one of the reasons for seeking to make a very near unity. 



Consider next one possible way of supplying biases to a grounded emitter 



