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Unformatted text preview: Harmonic Distortion in Electronic Amplifiers Hayden McCabe : 0750372 March 19, 2010 1 Small signal analysis using Taylor series In the analysis of amplifiers made from nonlinear devices, such as transistors, the assumption that a device behaves linearly for small disturbances away from its bias condition. This small signal modeling will predict the most significant portion of the systems output  the amplitude of the fundamen tal harmonic. The assumption of linearity, however, ignores the sometimes significant effect the actual response of the device will have on the output signal. A more accurate model of the small signal output of an amplifier can be achieved using Taylor series form of the output voltage equation. If a formula V o ( V i ) can be defined that is accurate for a range of allowable input voltage including the biasing voltage V b , the Taylor series of the output voltage is given by V o ( V i ) = X n =0 V ( n ) o  V b n ! ( V i V b ) n Where V i is the instantaneous voltage at the input terminal of the am plifier, and the small signal input, the difference between the instantaneous voltage and the biasing condition v i is given by v i = V i V b The first, constant, term of this series will give the output voltage at the amplifiers quiescent operating point, and the subsequent terms give the smallsignal output voltage. This can be shown by rewriting the equation in the form V o ( V i ) = V o ( V b ,v i ) = V o ( V b ) + X n =1 V ( n ) o  V b n ! ( V i V b ) n The less accurate linear transconductance model can be found by analyz ing the series through n = 1. i.e., V o ( V b ,v i ) V o ( V b ) + V i V o  V b ! ( V i V b ) = V o ( V b ) + gv i and the additional terms of the series, if analyzed, increase the accuracy of the output voltage equation. To analyze the harmonic distortion of the amplifier, the output of the amplifier given a sinusoidal, small signal input can be found. For an input signal sin ( t ), 2 V i = V b + v i = V b + sin ( t ) V o ( V i ) = V o ( V b ,,,t ) = V o ( V b ) + X n =1 V ( n ) o  V b n ! n sin n ( t ) This equation includes terms of the form sin n ( t ), the various powers of the sine. In order to analyze the harmonic distortion, we must first describe the powers of the sine function as a harmonic series. Powers of sine as a finite harmonic series A model of the powers of sin ( t ) can be found using Eulers formula. i.e. sin ( t ) = 1 2 j e jt e jt sin n ( t ) = 1 2 j ! n e jt e jt n For example, for sin 2 ( t ), sin 2 ( t ) = 1 2 j ! 2 e jt e jt 2 = 1 4 e j 2 t 2 + e 2 jt = 1 2 [cos (2 t ) 1] For sin 3 ( t ), sin 3 ( t ) = 1 2 j !...
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