Nonlinearity

Short Talk -
Nonlinearity
Jay Chang
July 9 2015

Agenda
Harmonic Distortion
Gain Compression
Cross Modulation
Intermodulation
AM/PM Conversion

 Even-order harmonics result from αj with even j.
 nth harmonic grows in proportion to An.
DC Fundamental Second
Harmonic
Third
Harmonic
Effects of Nonlinearity: Harmonic Distortion
Harmonic Distortion
2 3
1 2 3( ) ( ) ( ) ( ).
where ( ) cos .
y t x t x t x t
x t A t
  

  

2 3
0 1 2 3( ) ( ) ( ) ( )y t x t x t x t      
If only α0 ≠ 0, rectifier convert AC input to DC.
If only α1 ≠ 0, linear attenuator (α1 < 1) and linear amplifier (α1 > 1).
If only α2 ≠ 0, mixer.
And in most PA α3 < 0.
x(t) y(t)

Example of Harmonic Distortion in Mixer
Solution:
An analog multiplier “mixes” its two inputs below, ideally producing y(t) = kx1(t)x2(t), where k
is a constant. Assume x1(t) = A1 cos ω1t and x2(t) = A2 cos ω2t.
(a) If the mixer is ideal, determine the output frequency components.
(b) If the input port sensing x2(t) suffers from third-order nonlinearity, determine the output
frequency components.
(a)
(b)
Ex: B5, B8 and B9 fighting
if ω1 = 2π x (850 MHz) and ω2 = 2π x (900 MHz),
then |ω1 - 3ω2| = 2π x (1850 MHz), an “undesired” component that
is difficult to filter because it lies close to the desired component at
ω1 + ω2 = 2π x (1750 MHz).

Effects of Nonlinearity: Gain Compression
Gain Compression
Expansive Compressive
 Most RF circuit of interest are compressive, we focus on this type.
 Output falls below its ideal value by 1 dB at the 1-dB compression point.
 Peak value instead of peak-to-peak value.
 P1dB is typically in the range of -20 to -25 dBm (63.2 to 35.6 mVpp in 50 Ohm system)
at the input of RF receivers.
3
1 3
2
1 3
3
cos
34
gain compression.
cos 4
v
A A t
G A
A t
  
 

 
 
    

Gain Compression: Effect on FM and AM Waveforms
 FM signal carries no information in its amplitude and hence tolerates compression
(i.e., amplitude limiting).
 AM contain information in the amplitude, hence distorted by compression.

 Desensitization: the receiver gain is reduced by the large excursions produced by the interferer
even though the desired signal itself is small.
 Rx must be able to withstand blockers that are 60 to 70 dB greater than the desired signal.
For A1 << A2
Gain Compression: Desensitization
2 3
1 2 3
1 1 2 2
( ) ( ) ( ) ( ).
where ( ) cos cos .
y t x t x t x t
x t A t A t
  
 
  
 
2
1 3 1 2 1 3 2
3
if 0, when the gain drop to zero.
2
A A A    = In math 

Effects of Nonlinearity: Cross Modulation
Cross Modulation
 Desired signal at output suffers from amplitude modulation at ωm and 2ωm.
Suppose that the interferer is an amplitude-modulated signal
Thus
when a weak signal and a strong interferer pass through
a nonlinear system is the transfer of modulation from the interferer to the signal.
 2 21 cos cosmA m t t 

 Wanted signal(fRX) corruption due to the cross-modulation of an amplitude-modulated blocker with a CW
jammer close to the RX channel.
 Wanted signal(fRX) corruption due to the intermodulation of an amplitude-modulated blocker (the aggressor
TX) with a CW jammer.

 LNA輸入端, 會同時存在三種訊號: Blocker、TX、RX, 兩種訊號便可產生IMD (Intermodulation), 亦即這三種訊號中,
隨意任兩種便可產生IMD.

Harmonic distortion
Desensitization
Intermodulation
So far we have considered the case of:
 Single Signal
 Signal + one large interferer
 Signal + two large interferers
Effects of Nonlinearity: Intermodulation-
Recall Previous Discussion

Effects of Nonlinearity: Intermodulation
Intermodulation
assume
Thus
Intermodulation products:
Fundamental components:

Intermodulation Product Falling on Desired Channel
desired
Interferer
 A received small desired signal along with two large interferers.
 Intermodulation product falls onto the desired channel, corrupts signal.

 IP3 is not a directly measureable quantity, but a point obtained by extrapolation.
Intermodulation: Third Intercept Point

 PA: power range that is limited at the low end by
noise and at the high end by the compression
point. This is essentially the linear operating range
for the amplifier, and is called the linear dynamic
range (DRl).
 LNA or mixer: operation may be limited by noise
at the low end and the maximum power level for
which intermodulation distortion becomes
unacceptable. This is effectively the operating
range for which spurious responses are minimal,
and is called the spurious-free dynamic range (DRf).
Intermodulation: Dynamic Range

Effects of Nonlinearity: Cascaded Nonlinear Stages
Considering only the first- and third-order terms, we have:
Thus,

 To “refer” the IP3 of the second stage to the input of the cascade, we must divide it by α1.
Thus, the higher the gain of the first stage, the more nonlinearity is contributed by the second
stage.
Cascaded Nonlinear Stages: Intuitive results
1 3total IP  

1 2( ) cos cosx t A t A t  
IM Spectra in a Cascade (I)
identify the IM products in a cascade

IM Spectra in a Cascade (II)
Adding the amplitudes of the IM products, we have
 Add in phase as worst-case scenario.
 Heavily attenuated in narrow-band circuits.
 Thus, if each stage in a cascade has a gain greater than unity, the nonlinearity of the latter
stages becomes increasingly more critical because the IP3 of each stage is equivalently scaled
down by the total gain preceding that stage.
For more stages:

Experiment Measurement IP3
Amplifier Two-Tone Test
Mixer Two-Tone Test
1
1 2 1
0
2 33 2
in
IP
P P G
P P P

  
 
 
1 2 12 3
3 0
3 2
3
IP
IP
P P P
IIP P G
    
 

Effects of Nonlinearity: AM/PM Conversion
 AM/PM Conversion arises in systems both dynamic and nonlinear.
Phase shift of
fundamental, Const.
Higher harmonic
If R1C1(t)ω1 << 1 rad
Assume that
obtaining
 Undesired phase deviation (PM) that is caused by amplitude variations (AM) of the system.
AM/PM Conversion (APC)
1 1( ) cosinV t V t
Sinusoid frequency ω1 through a
first-order low-pass RC section

 No AM/PM conversion because of the first-order dependence of C1 on Vout.
First order voltage dependence:
AM/PM Conversion: Time-Variation of Capacitor

Example of AM/PM Conversion: Second
Order Voltage Dependence
Suppose C1 in above RC section is expressed as C1 = C0(1 + α1Vout + α2Vout
2). Study the AM/PM
conversion in this case if Vin(t) = V1 cos ω1t.
 The phase shift of the fundamental now contains an
input-dependent term, -(α2R1C0ω1V1
2 )/2.
 This figure also suggests that AM/PM conversion
does not occur if the capacitor voltage dependence
is odd-symmetric.
Figure below plots C1(t) for small and large input swings, revealing that Cavg indeed depends on
the amplitude.
Phase shift of
fundamental, input-depet.

Nonlinearity

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