7.1.2 Simplifed Expressions for Widely Separated Poles

Chapter 7.1.2 Simplifed Expressions for Widely Separated Poles

Radio Frequency Integrated Circuit Design Second Edition Book
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Radio Frequency Integrated Circuit Design Second Edition Book

  • 7.1 Introduction and Basic Amplifiers 175Example 7.3: Calculation of Gain of Single Pole AmplifierFor the above example, for Avo = 20 with fP1 = 2.76 GHz, calculate the gain at 5.6 GHz.Solution:With fP1 = 2.76 GHz, at 5.6 GHz, the gain can be calculated to be 8.84, or 18.9 dB. This is down by about 7 dB from the low-frequency gain.The exact expression for vo/vs (after about one page of algebra) is -=æöéù éù¢++++êú êúç÷¢¢èøëû ëû2s111mosmSLSLgscvvgssZZC RC RCCC R C (7.12)where vs is the source voltage, RS¢ = RS + rb, RSp = RS¢||rp , Cmp = Cm in series with Cp , and ZL¢ = ZL||ro. As expected, this equation features a zero in the right-half plane, and real, well-separated poles, similar to that of a pole-splitting opamp [1].Example 7.4: Calculation of Exact Poles and ZerosCalculate poles and zeros for the transistor amplifier as in the previous example.Solution:Results for the previous example: 15´ npn, 5 mA, ZL =100W, Cm = 23.2 fF, Cp = 700 fF, RS = 50W, rb = 5W. Using (7.12), the results are that the poles are at 2.66 GHz and 118.3 GHz; the zero is at 1,372 GHz. Thus, the exact equation has been used to verify the original assumptions that the two poles are well separated, that the dominant pole is approximately at the frequency given by the previous equations, and that the second pole and feed-forward zero in this expression are well above the frequency of interest.7.1.2  Simplified Expressions for Widely Separated PolesIf a system can be described by a second-order transfer function given by: 2()oivA s zvssb c-=++ (7.13)then the poles of this system are given by: 1,22412 2b bcPb= - ±- (7.14)If the poles are well separated, then 4c/b2 << 1, therefore: 1,22212 2b bcPbæö» - ±-ç÷èø (7.15)