Lecture 7: Distortion Analysis

Transcription

1 EECS 142 Lecture 7: Distortion Analysis Prof. Ali M. Niknejad University of California, Berkeley Copyright c 2005 by Ali M. Niknejad A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 1/18

2 Introduction to Distortion s i s o s o Linear Region s i Up to now we have treated amplifiers as small-signal linear circuits. Since transistors are non-linear, this assumption is only valid for extremeley small signals. Consider a class of memoryless non-linear amplifiers. In other words, let s neglect energy storage elements. This is the same as saying the output is an instantaneous function of the input. Thus the amplifier has no memory. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 2/18

3 Distortion Analaysis Assumptions We also assume the input/output description is sufficiently smooth and continuous as to be accurately described by a power series s o = a 1 s i + a 2 s 2 i + a 3 s 3 i +... For instance, for a BJT (Si, SiGe, GaAs) operated in forward-active region, the collector current is a smooth function of the voltage V BE I Q I C I C = I S e qv BE kt V BE A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 3/18

4 BJT Distortion I C I Q ideal response V i V BE,Q We shift the origin by eliminating the DC signals, i o = I C I Q. The input signal is then applied around the DC level V BE,Q. Note that an ideal amplifier has a perfectly linear line. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 4/18

5 A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 5/18 The gate current (junction leakage) is typically very JFET Distortion I DS V P 0 V GS JFETs are more common devices in RF circuits. The I-V relation is also approximately square law I D = I DSS (1 V GS V P ) 2

6 MOSFET Distortion I DS I DS VT V GS V GS forward active regime The long-channel device also follows the square law relation (neglecting bulk charge effects) I D = 1 2 µc ox W L (V GS V t ) 2 (1 + λv DS ) This is assuming the device does not leave the forward active (saturation) regime. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 6/18

7 MOSFET Model Note that the device operation near threshold is not captured by our simple square-law equation The I-V curve of a MOSFET in moderate and weak inversion is easy to describe in a piece-meal fashion, but difficult to capture with a single equation. Short-channel devices are even more difficult due to velocity saturation and drain induced barrier lowering. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 7/18

8 Differential Pair I C1 I C2 + V i 2 I EE V i 2 + The differential pair is an important analog and RF building block. For a BJT diff pair, we have V i = V BE1 V BE2 I C1,2 = I S e qv BE1,2 kt The sum of the collector currents are equal to the current source I C1 + I C2 = I EE A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 8/18

9 BJT Diff Pair I EE I o I EE V i The ideal BJT diff pair I-V relationship (neglecting base and emitter resistance) is give by I o = I C1 I C2 = αi EE tanh qv i 2kT Notice that the output current saturates for large input voltages A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 9/18

10 Power Series Relation For a general circuit, let s represent this behavior with a power series a 1 is the small signal gain s o = a 1 s i + a 2 s 2 i + a 3 s 3 i +... The coefficients a 1,a 2,a 3,... are independent of the input signal s i but they depend on bias, temperature, and other factors. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 10/18

11 Harmonic Distortion Assume we drive the amplifier with a time harmonic signal at frequency ω 1 s i = S 1 cos ω 1 t A linear amplifier would output s o = a 1 S 1 cos ω 1 t whereas our amplifier generates or s o = a 1 S 1 cos ω 1 t + a 2 S 2 1 cos 2 ω 1 t + a 3 S 3 1 cos 3 ω 1 t +... s o = a 1 S 1 cos ω 1 t + a 2S1 2 2 (1 + cos 2ω 1 t) + a 3 S (cos 3ω 1 t + 3 cosω 1 t) +... A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 11/18

12 Harmonic Distortion (cont) The term a 1 s 1 cos ω 1 t is the wanted signal. Higher harmonics are also generated. These are unwanted and thus called distortion terms. We already see that the second-harmonic cos 2ω 1 t and third harmonic cos 3ω 1 t are generated. Also the second order non-linearity produces a DC shift of 1 2 a 2S 2 1. The third order generates both third order distortion and more fundamental. The sign of a 1 and a 3 determine whether the distortion product a 3 S cos ω 1t adds or subtracts from the fundamental. If the signal adds, we say there is gain expansion. If it subtracts, we say there is gain compression. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 12/18

13 Second Harmonic Distortion Waveforms second harmonic fundamental The figure above demonstrates the waveform distortion due to second harmonic only. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 13/18

14 Third Harmonic Distortion Waveform third harmonic fundamental The above figure shows the effects of the third harmonic, where we assume the third harmonic is in phase with the fundamental. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 14/18

15 Third Harmonic Waveform (cont) third harmonic fundamental The above figure shows the effects of the third harmonic, where we assume the third harmonic is out of phase with the fundamental. A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 15/18

16 General Distortion Term Consider the term cos n θ = 1 2 n ( e jθ + e jθ) n. Using the Binomial formula, we can expand to For n = 3 = 1 8 = 1 8 (( ) 3 e j3θ + 0 = 1 2 n n k=0 ( ) n e jkθ e j(n k)θ k ( ) 3 e jθ e j2θ + 1 ( ) 3 e j2θ e jθ + 2 ( ) 3 )e j3θ 3 (e j3θ + e j3θ) ( e jθ + e jθ) = 1 4 cos 3θ cos θ A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 16/18

17 General Distortion Term (cont) We can already see that for an odd power, we will see a nice pairing up of positive and negative powers of exponentials For the even case, the middle term is the unpaired DC term ( ) ( ) 2k 2k e jkθ e jkθ = k k So only even powers in the transfer function can shift the DC operation point. The general term in the binomial expansion of (x + x 1 ) n is given by ( ) ( ) n n x n k x k = x n 2k k k A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 17/18

18 General Distortion Term (cont) The term ( n k) x n 2k generates every other harmonic. If n is even, then only even harmonics are generated. If n is odd, likewise, only odd harmonics are generated. Recall that an odd function f( x) = f(x) (anti-symmetric) has an odd power series expansion f(x) = a 1 x + a 3 x 3 + a 5 x Whereas an even function, g( x) = g(x), has an even power series expansion g(x) = a 0 + a 2 x 2 + a 4 x A. M. Niknejad University of California, Berkeley EECS 142 Lecture 7 p. 18/18