Output Waveform

In general, then, the output waveform is a Fourier series

vo = Vo1 cos1t + Vo2 cos 21t + Vo3 cos 31t + . . .

Vo

Vi

Vo2

Vo3

100mV

10mV1mV100V10V1V

10mV

1mV

100V

10V

1V

Gain Compression

Higher Order Distortion Products

Niknejad Distortion

Harmonic Distortion Metrics

Niknejad Distortion

Fractional Harmonic Distortion

The fractional second-harmonic distortion is a commonly citedmetric

HD2 =ampl of second harmonic

ampl of fund

If we assume that the square power dominates thesecond-harmonic

HD2 =a2

S212

a1S1or

HD2 =12

a2a1S1

Niknejad Distortion

Third Harmonic Distortion

The fractional third harmonic distortion is given by

HD3 =ampl of third harmonic

ampl of fund

If we assume that the cubic power dominates the thirdharmonic

HD3 =a3

S214

a1S1or

HD3 =1

4

a3a1S21

Niknejad Distortion

Output Referred Harmonic Distortion

In terms of the output signal Som, if we again neglect gainexpansion/compression, we have Som = a1S1

HD2 =1

2

a2a21Som

HD3 =1

4

a3a31S2om

On a dB scale, the second harmonic increases linearly with aslope of one in terms of the output power whereas the thridharmonic increases with a slope of 2.

Niknejad Distortion

Signal Power

Recall that a general memoryless non-linear system willproduce an output that can be written in the following form

vo(t) = Vo1 cos1t + Vo2 cos 21t + Vo3 cos 31t + . . .

By Parsevals theorem, we know the total power in the signalis related to the power in the harmonics

Tv2(t)dt =

T

j

Voj cos(j1t)k

Vok cos(k1t)dt

=j

k

TVoj cos(j1t)Vok cos(k1t)dt

Niknejad Distortion

Power in Distortion

By the orthogonality of the harmonics, we obtain ParsevalsThem

Tv2(t)dt =

j

k

12jk Voj Vok =

12

j

V 2oj

The power in the distortion relative to the fundamental poweris therefore given by

Power in Distortion

Power in Fundamental=

V 2o2V 2o1

+V 2o3V 2o1

+

= HD22 + HD23 + HD

24 +

Niknejad Distortion

Total Harmonic Distortion

We define the Total Harmonic Distortion (THD) by thefollowing expression

THD =HD22 + HD

23 +

Based on the particular application, we specify the maximumtolerable THD

Telephone audio can be pretty distorted (THD < 10%)

High quality audio is very sensitive (THD < 1% toTHD < .001%)

Video is also pretty forgiving, THD < 5% for mostapplications

Analog Repeaters < .001%. RF Amplifiers < 0.1%

Niknejad Distortion

Intermodulation and Crossmodulation Distortion

Niknejad Distortion

Intermodulation Distortion

So far we have characterized a non-linear system for a singletone. What if we apply two tones

Si = S1 cos1t + S2 cos2t

So = a1Si + a2S2i + a3S

3i +

= a1S1 cos1t + a1S2 cos2t + a3(Si )3 +

The second power term gives

a2S21 cos

2 1t + a2S22 cos

2 2t + 2a2S1S2 cos1t cos2t

= a2S212

(cos 21t + 1) + a2S222

(cos 22t + 1) +

a2S1S2 (cos(1 + 2)t cos(1 2)t)

Niknejad Distortion

Second Order Intermodulation

The last term cos(1 2)t is the second-orderintermodulation term

The intermodulation distortion IM2 is defined when the twoinput signals have equal amplitude Si = S1 = S2

IM2 =Amp of Intermod

Amp of Fund=

a2a1Si

Note the relation between IM2 and HD2

IM2 = 2HD2 = HD2 + 6dB

Niknejad Distortion

Practical Effects of IM2

This term produces distortion at a lower frequency 1 2and at a higher frequency 1 + 2

Example: Say the receiver bandwidth is from800MHz 2.4GHz and two unwanted interfering signalsappear at 800MHz and 900MHz.Then we see that the second-order distortion will producedistortion at 100MHz and 1.7GHz. Since 1.7GHz is in thereceiver band, signals at this frequency will be corrupted bythe distortion.

A weak signal in this band can be swamped by thedistortion.

Apparently, a narrowband system does not suffer from IM2?Or does it ?

Niknejad Distortion

Low-IF Receiver

In a low-IF or direct conversion receiver, the signal isdown-converted to a low intermediate frequency fIF

Since 1 2 can potentially produce distortion at lowfrequency, IM2 is very important in such systems

Example: A narrowband system has a receiver bandwidth of1.9GHz - 2.0GHz. A sharp input filter eliminates anyinterference outside of this band. The IF frequency is 1MHzImagine two interfering signals appear at f1 = 1.910GHz andf2 = 1.911GHz. Notice that f2 f1 = fIFThus the output of the amplifier/mixer generate distortion atthe IF frequency, potentially disrupting the communication.

Niknejad Distortion

Cubic IM

Now lets consider the output of the cubic term

a3s3i = a3(S1 cos1t + S2 cos2t)

3

Lets first notice that the first and last term in the expansionare the same as the cubic distortion with a single input

a3S31,2

4(cos 31,2t + 3 cos1,2t)

The cross terms look like(3

2

)a3S1S

22 cos1t cos

2 2t

Niknejad Distortion

Third Order IM

Which can be simplified to

3 cos1t cos2 2t =

3

2cos1t(1 + cos 22t) =

=3

2cos1t +

3

4cos(22 1)

The interesting term is the intermodulation at 22 1By symmetry, then, we also generate a term like

a3S21S2

3

4cos(21 2)

Notice that if 1 2, then the intermodulation22 1 1

Niknejad Distortion

Inband IM3 Distortion

1 23

22 121 2

S()

Interfering Signalswanted

distortion product

Now we see that even if the system is narrowband, the outputof an amplifier can contain in band intermodulation due toIM3.

This is in contrast to IM2 where the frequency of theintermodulation was at a lower and higher frequency. The IM3frequency can fall in-band for two in-band interferer

Niknejad Distortion

Definition of IM3

We define IM3 in a similar manner for Si = S1 = S2

IM3 =Amp of Third Intermod

Amp of Fund=

3

4

a3a1S2i

Note the relation between IM3 and HD3

IM3 = 3HD3 = HD3 + 10dB

Niknejad Distortion

Complete Two-Tone Response

1 2

22 121 2

S()

32 2131 22

21 22

1 + 22 1

32 131 23132

21 + 2 22 + 1

22 21

2 1

We have so far identified the harmonics and IM2 and IM3products

A more detailed analysis shows that an order n non-linearitycan produce intermodulation at frequencies j1 k2 wherej + k = n

All tones are spaced by the difference 2 1

Niknejad Distortion

Distortion of AM Signals

Consider a simple AM signal (modulated by a single tone)

s(t) = S2(1 + m cosmt) cos2t

where the modulation index m 1. This can be written as

s(t) = S2 cos2t +m

2cos(2 m)t + m

2cos(2 + m)t

The first term is the RF carrier and the last terms are themodulation sidebands

Niknejad Distortion

Cross Modulation

Cross modulation occurs in AM systems (e.g. video cabletuners)

The modulation of a large AM signal transfers to anothercarrier going thru the same amp

Si = S1 cos1t wanted

+S2(1 + m cosmt) cos2t interferer

CM occurs when the output contains a term like

K (1 + cosmt) cos1t

Where is called the transferred modulation index

Niknejad Distortion

Cross Modulation (cont)

For So = a1Si + a2S2i + a3S

3i + , the term a2S2i does not

produce any CM

The terma3S

3i = + 3a3S1 cos1t (S2(1 + m cosmt) cos2t)2 is

expanded to

= + 3a3S1S22 cos1t(1 + 2m cosmt + m2 cos2 mt)12(1 + cos 22t)

Grouping terms we have in the output

So = + a1S1(1 + 3a3a1S22m cosmt) cos1t

Niknejad Distortion

CM Definition

unmodulated waveform (input) modulated waveform due to CM

CM =Transferred Modulation Index

Incoming Modulation Index

CM = 3a3a1S22 = 4IM3

= IM3(dB) + 12dB

= 12HD3 = HD3(dB) + 22dB

Niknejad Distortion

Examples

Niknejad Distortion

Distortion of BJT Amplifiers

+vs

RL

vo

VCC

Consider the CE BJTamplifier shown. Thebiasing is omitted forclarity.

The output voltage is simply

Vo = VCC ICRCTherefore the distortion is generated by IC alone. Recall that

IC = ISeqVBE/kT

Niknejad Distortion

BJT CE Distortion (cont)

Now assume the input VBE = vi + VQ , where VQ is the biaspoint. The current is therefore given by

IC = ISeVQVT

IQ

eviVT

Using a Taylor expansion for the exponential

ex = 1 + x +1

2!x2 +

1

3!x3 +

IC = IQ(1 +viVT

+1

2

(viVT

)2+

1

6

(viVT

)3+ )

Niknejad Distortion

BJT CE Distortion (cont)

Define the output signal ic = IC IQ

ic =IQVT

vi +1

2

( qkT

)2IQv

2i +

1

6

( qkT

)3IQv

3i +

Compare to So = a1Si + a2S2i + a3S

3i +

a1 =qIQkT

= gm

a2 =1

2

( qkT

)2IQ

a3 =1

6

( qkT

)3IQ

Niknejad Distortion

Example: BJT HD2

For any BJT (Si, SiGe, Ge, GaAs), we have the followingresult

HD2 =1

4

qvikT

where vi is the peak value of the input sine voltage

For vi = 10mV, HD2 = 0.1 = 10%We can also express the distortion as a function of the outputcurrent swing ic

HD2 =1

2

a2a21Som =

1

4

icIQ

For icIQ = 0.4, HD2 = 10%

Niknejad Distortion

Example: BJT IM3

Lets see the maximum allowed signal for IM3 1%

IM3 =3

4

a3a1S21 =

1

8

(qvikT

)2Solve vi = 7.3mV. Thats a pretty small voltage. For practicalapplications wed like to improve the linearity of this amplifier.

Niknejad Distortion

Example: Disto in Long-Ch. MOS

vi

VQ

ID = IQ + ioID =

12Cox

W

L(VGSVT )2

io+IQ =12Cox

W

L(VQ+viVT )2

Ignoring the output impedance we have

= 12CoxW

L

{(VQ VT )2 + v2i + 2vi (VQ VT )

}= IQ

dc

+CoxW

Lvi (VQ VT ) linear

+ 12CoxW

Lv2i

quadratic

Niknejad Distortion

Ideal Square Law Device

An ideal square law device only generates 2nd order distortion

io = gmvi +12Cox

W

Lv2i

a1 = gm

a2 =12Cox

W

L= 12

gmVQ VT

a3 0The harmonic distortion is given by

HD2 =1

2

a2a1vi =

1

4

gmVQ VT

1

gmvi =

1

4

viVQ VT

HD3 = 0

Niknejad Distortion

Real MOSFET Device

0

200

400

600

Effective Field

Mob

ility

Triode CLM DIBL SCBE

Routk

Vds (V)

2

4

6

8

10

12

14

0 1 2 3 4

The real MOSFET device generates higher order distortion

The output impedance is non-linear. The mobility is not aconstant but a function of the vertical and horizontal electricfield

We may also bias the device at moderate or weak inversion,where the device behavior is more exponential

There is also internal feedback

Niknejad Distortion