HO21_214W09_HF_disto_2pp

HO21_214W09_HF_disto_2pp - Handout#21 EE 214 Winter 2009...

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Handout #21 EE 214 Winter 2009 Introduction to High-Frequency Distortion Analysis B. Murmann and B. A. Wooley Stanford University Analysis of Weakly Nonlinear Circuits ± The power series approach studied previously in this course ignores any frequency dependence introduced by reactive elements Sufficient for 90% of typical circuits, including some operating at RF ± Assuming weakly nonlinear behavior, the frequency dependence can be included using a Volterra Series model – Vito Volterra, 1887 ± The purpose of this handout is to provide a few basic examples that will allow you to understand the general framework ± Examples – Memoryless nonlinearity followed by a filter Memoryless nonlinearity preceded and followed by a filter – RC circuit with nonlinear capacitance – RC circuit with nonlinear resistance B. A. Wooley, B. Murmann EE214 Winter 2008-09 2
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Example 1 R 23 12 3 cii i i a v a v a v ... = +++ C v o i c 3 11 26 CQ CQ m TT II ag a a VV == = v i V I 1 o c v R K( j ) ij R C ω= = ± Ignoring device capacitances and finite output resistance for simplicity B. A. Wooley, B. Murmann EE214 Winter 2008-09 3 Single-Tone Input ( ) ii ˆ vv c o s t ± Ignoring DC offset and gain expansion, we have () 3 24 ci i i ˆˆ ˆ i a v cos t a v cos t a v cos t ... + ω + ω + ± The output voltage consists of the same tones, with their magnitude and phase altered by the linear filter K(j ω ) ( ) ( ) 1 2 22 1 oi i ˆ vK j a v c o s t ˆ Kj a v c o s t ω ω ω + φ ω + φ ( ) m Km j ω φ =∠ ⋅ ω 3 33 2 1 4 i ˆ a v c o . . . ω + ω⋅ ω+φ + B. A. Wooley, B. Murmann EE214 Winter 2008-09 4
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Two-Tone Input ( )( ) 11 2 2 i ˆˆ vv c o stv c o s t Substituting this input into the power series and using the identities ± Substituting this input into the power series, and using the identities shown below, the complete expression for the collector current is most elegantly expressed as shown on the next slide () () () 1 2 1 cos cos cos cos ⎡⎤ αβ = α + β + α β ⎣⎦ () ( ) ( ) 4 cos cos cos cos cos cos cos α β γ = α+β+γ + α+β−γ β + γ β γ B. A. Wooley, B. Murmann EE214 Winter 2008-09 5 ia v c o st v c o ω+ ω ω ω Frequency Components ( ) ( ) [] () [] 1 2 2 22 2 1 2 2 c a vc o s t vc o s t = + ω ±ω + 1 , 2 0, 2 ω 1 , 2 ω 2 12 1 2 33 3 2 vv cos t a ± ω ω 1 - ω 2, ω 1 + ω 2 [ ] ( ) [ ] ( ) 1 111 2 2 2 2 2 1 2 2 4 3 o s o s t t ± ω ± ω ± ω ± ω + ω ±ω ±ω ω 1 , ω 2 , 3 ω 1 , 3 ω 2 ω 1 ,2 ω 1 - ω 2 ω 1 + ω 2 2 1 1 2 3 v v cos t ... + + ω 2 ω 2 - ω 1 ω 2 + ω 1 B. A. Wooley, B. Murmann EE214 Winter 2008-09 6
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Filtered Output ( ) ( ) ( ) ( ) 12 11 1 1 2 2 2 o ˆˆ va v K jc o st v K j c o ωω ⎡⎤ ω + φ + ω ω + φ ⎣⎦ () 1 22 2 1 2 1 0 2 a Kj v c o s t vK ω ω + φ + ( ) [] ( ) ( ) 2 2 2 2 1 2 0 2 Kj v c o vvc o s t ω ω−ω ω + φ + + ω −ω 1 2 2 o s t a ω+ω + ω +ω [ ] 3 4 ... + B. A. Wooley, B. Murmann EE214 Winter 2008-09 7 Short Hand Notation ( ) ( ) 3 23 3 aa b abc oa i a b i a b c i H(j ) H (j j ) Hj j j v aKj j j j v .
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HO21_214W09_HF_disto_2pp - Handout#21 EE 214 Winter 2009...

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