Electro-Optical

Kluges and Hacks

A Lab Rat's Guide to Good Measurements

Phil Hobbs,
IBM T. J. Watson Research Center
Yorktown Heights NY
 Hacks Of The Day

Quantum detection
A little noise theory
Low noise front ends
Design tricks and circuit hacks
Detailed example: bootstrapped cascode TIA
Noise Cancellers & Their Relatives
Motivation
Details
Other linear combinations (locking a laser to an etalon)
High-Performance Pyroelectrics
Low speed wins!
Higher speed
Impedance transformation: transformers, reactive networks,
constant-resistance T-coils
 Quantum Detection
(Optical View)

One photon gets you one electron (η~1)

Shot noise is the intrinsic limit (pace squeezers)
N photons/s gives 0 dB SNR in N/2 Hz, max
Signal and spurious junk are inseparable after detection
2
Etendue (n AΩ ) management for speed and low noise:
2
Achievable BW goes as average radiance (W/cm /sr)
Leakage, background, and capacitance go as the area
Reduce area, increase NA, consider immersion lens
2
High current density (>10 mA/cm ) causes nonlinearity

(And, just between you and me: small detectors are really hard to align)
 Analytic Signals
Circuits people use one-sided BW
Analytic signal convention
Measurable quantities are
real-valued
Analysis is easier in complex
exponentials
Analytic signal definition
Double signal at f >0
Leave DC alone
Chop off all f < 0
A bit problematic at DC
Causes mysterious factors of 2:
Mean square AC power doubled
1-s boxcar has 0.5 Hz noise BW
1/2 1/2
N in 1s is (2N ) in 1 Hz!
 Noise Physics
Johnson Noise:
Classical equipartition & fluctuation-dissipation theorem
Johnson noise PSD pNJ = kT J/s/Hz when matched
1/2 1/2
vN =(4kTR) , iN =(4kT/R) in 1 Hz (unmatched)
Noise temperature TN=Tamb (resistor), TN<< Tamb (LNA)
Shot Noise:
Photodetection is a Poisson process: variance = mean
1/2 1/2
Shot noise limit: iNshot=(2eIdc) > (4kTN/R) when:
Signal drops 50 mV across RL (300K)
Signal power >7 µW in 50Ω (very quiet amp [35K])
NB: It's easy to make currents with no shot noise (metal resistor)
Pauli principle forces electrons to be highly correlated: noise
power suppression is ~ (mean free path)/(length of resistor)

Technical noise (stay tuned)

 Noise Definitions

Noise statistics are ensemble averages or short-time averages

They can be time-varying
Signals at different frequencies add in power since beat term
averages to zero
Noise best specified as power spectral density (PSD): for
reasonable bandwidths, think of this as noise in 1 Hz BW
pN is PSD, PN is total noise power
Noise Bandwidth:
BWN = (total noise power)/(peak noise PSD)
Equivalent width of power spectrum
BWN=1/(autocorrelation width of impulse response)
Generally wider than 3 dB BW (π/2 times for RC rolloff)
 Quantum Detection
(Circuit View)
Output Current:
consists of N Poissonian pulses/s regardless of QE and Idark
Gain can't fix this (PMTs just give bigger pulses)

All fundamental noise sources are white

Circuit Model: current source shunted by Cd

2
Cd ~ 100 pF/cm for a good PIN device, fully depleted

Square law device:

2
Popt = hnN, Pel = (eN) RL
Electrical power theoretically unlimited as RL => infinity
Johnson noise is always kT/s/Hz: weak signals are easily
swamped
 Detection Regimes (Quiet Source)

Photon counting:
8
N < 10 photons/s (40 pW @ 500 nm)
Use PMT or Geiger-mode APD (< 1 MHz)
Useful BW (20 dB SNR) ~ N /200

Shot-noise limited:
Id RL > 50 mV @300K
Can always get there with bigger RL(Si, InGaAs) but BW suffers

Otherwise Johnson-limited:
Nice quiet photoelectrons are immersed in circuit noise
Circuit constants are the problem
Circuit hacks can be the solution
 Escaping Johnson Noise

Additive circuit noise swamps photoelectrons

Very wasteful--we've paid a lot for those photons!

3 dB SNR improvement can save:

Half the laser power needed
Half the measurement time required
Half the cost and 2/3 the weight of the optical system

To escape Johnson
Smaller detectors, higher bias (reduces C)
Low noise amplifiers (reduces noise)
Electron multiplying detectors or cooled CCDs (increases signal)
Impedance transformation networks (increases signal)
Other circuit hacks
 Example:
Low-Level PIN Photodiode Front End

Design Parameters:
Bandwidth: B >= 1 MHz
2
Obese 1 cm Si PIN Photodiode, Cd =100 pF (fully depleted)
Photocurrent: i phot = 2 µA
Photon arrival rate N = iphot/e = 12.4 THz

SNR: Within 2 dB of shot noise limit

Maximum SNR = N /2B = 68 dB in 1 MHz
 Front End Choices

Load resistor
Transimpedance amplifier
Bootstrap + load resistor
Cascode transimpedance amp
Bootstrapped cascode TIA
 Load Resistor

First Try
RL =1 MΩ : BW = 1600 Hz (ick)
Everything is wired in parallel:
Signal and noise roll off together
SNR constant even though
signal rolls off by 55 dB
Subsequent amplifier limits SNR
Optimization:
Lower R increases BW, but SNR
drops due to Johnson noise
Shot = Johnson when IR = 2kT/e
(~50 mV@300K)
Optimum R drops ~ 200 mV
Ropt = 100k, BW = 16 kHz
 Transimpedance Amp
0.5 pF

100K
Connect PD to virtual ground
Op amp wiggles output end
of RF to keep input end still
Improves BW but not SNR
1/2 LF356A
3 dB BW ~ 0.5(fRC*GBW)
Unity gain stability unnecessary
Transimpedance (Ω) CNR (dB
Big improvement but don't push
it too much:
Noise and instability problem
due to capacitive load on
summing junction
Fast amplifiers are worst
0.5 pF Cf helps instability
but can't fix SNR problem
 Transimpedance Amp

Transimpedance BW
Less than closed-loop BW
Depends on values not ratios
Actual BW obtained depends on
frequency compensation

Low noise
Amplifier noise dominates at
large Rf
Active devices can have
TN << 300K (TN = eNiN / 4k)
~ 10K for good bipolar op amps
Even lower for FETs but needs
inaccessible impedance levels
 DIY Op Amps

Current noise of op amp

appears in parallel with Iphot
Treated just like signal:
no high freq SNR penalty

Voltage noise of op amp sees

full noninverting gain
Big noise spike at high freq,
due to Cd (differentiator)

Reducing eNamp means running

the input stage at higher bias
add a BJT stage to the front
Increases iNamp ,but that's OK
 Cascode TIA

Isolate Cd from summing

junction with cascode Q1
BW limited by emitter
impedance rE =1/gm
BW(Hz) = 6.2 IC /Cd

Biasing cascode with

sub-Poissonian I bias reduces rE
--improves BW
Noise now limited by Rb' and
shot noise of Ib
Noise multiplication much
reduced compared to TIA
 Bootstrapping

Bootstrap transistor
Follower forces cold end of D1 to
follow hot end
No voltage swing
->no capacitive current
Speed set by rE Cd not RL Cd
50x faster than RC at Idc=300 µA,
RL=100 kΩ
Superbeta transistor
β ~ 1000: Very low base current noise
Noise Voltage
1/2
Limited by Rb' and rE(2eIC)
Noise multiplication similar to TIA
Can be applied with other techniques
 Bootstrapped
Cascode TIA

Can't use enough Q1 bias to

get 1 MHz BW without being
limited by Ib shot noise and
Rb' Johnson noise
Bootstrap runs at higher
current: lower voltage noise
Reduces effective Cd
Superbeta transistor Q2 has
much lower base current
shot noise, so can run at
higher current than Q1
without ruining the SNR
Bootstrap can be applied
along with cascode
 Bootstrapped Cascode TIA

Final performance:
Within 1 dB of shot
noise, DC-1.3 MHz
600x bandwidth
improvement over naive
approach
Three turns of the crank to
get 1 MHz BW with 100 pF
& 2 µA
Not much more juice
available here:
optical fix needed next
Bottom: Dark noise
time
Top: 2 µA photocurrent
 Detectors With Gain

Electron Multiplication: used in PMTs, APDs, & LLLCCDs

Gain applied to electrons before front end amplifier
Front end noise contribution reduced by M
Allows low load resistances => increased BW

HOWEVER,...

Gain inherently noisy (at least 3 dB noisier than PIN)

Other tradeoffs depend on device (e.g. GBW of APD)

Shot noise doesn't improve:

N photons per second gives 0 dB SNR in N/2 Hz, max
Gain amplifies noise along with signal
 Noise Physics Again

Technical Noise
Usually dominant in laser measurements, especially bright field
2
Dominates in large-signal limit (pN ~ Popt )
Laser RIN, demodulated FM noise, wiggle noise,
below-threshold side modes, mode partition noise, coherence
fluctuations microphonics, 1/f noise, noisy background, phase of
the moon, pink elephants,.....
Many strategies for getting round it, such as:
Reduce background: Dark field and dim field
Move to high frequency: Heterodyne interferometers
Move at least a little away from DC: Chopping
Compare beam before and after sample: Differential detection

NB: Lots of possibilities, because there's no 100% solution

 Shot Noise
Rule of One
One coherently added photon per second gives an ac
measurement with One sigma confidence in a One hertz
bandwidth.

True for bright field or dark field:

Bright field == dark field, except for technical noise
BF: Source instability (RIN)
DF: Johnson noise
DC is actually 3 dB better for a given temporal response, except
for the usual baseband suspects
 Differential Detection Ought To Be
Perfect

Apart from shot noise, Isig and Icomp are perfectly correlated
Optical systems are extremely linear and wideband
Photodiodes can also be extremely linear and pretty wideband:

=> isig/icomp == Isig/Icomp (differential gain == average gain)

If the DC cancels, the noise cancels at all frequencies

Problem: only works with beams of identical strength:
Need to ship a grad student with each system to keep it adjusted
 BJT Differential Pair

With fixed ∆Vbe, the ratio of IC2/IC1

is constant over several decades
of Ie.
Linear splitting => fluctuations
and DC treated alike
(Q1 is in normal bias as
shown--the collector can go 200
mV below the base before
saturation starts)
Transistors can be fast
Adjusting ∆Vbe to null out the
photocurrent doesn't disturb the
subtraction
 Basic Noise Canceller

Add a diff pair to a

current-differencing
amplifier
Use feedback control of
∆Vbe to null the DC
=> Noise cancels identically
at all frequencies
Cancellation BW
independent of FB BW
Linear highpass O/P, log
ratio LP output (∆Vbe)
1k::26Ω divider gets rid of
kT/e factor in ∆Vbe
[2V <==> exp(1)]
 Performance: 3N3904 discrete BJTs
0.75 mW Psig, 1.5 mW Pcomp

Cancellation
He-Ne showing a strong mode
beat (oscilloscope traces)

Upper: TIA mode showing

beat waveforms
due to 4-wave mixing
(comparison beam blocked)

Lower: Cancellation to
0.5 dB above shot noise
(comparison beam unblocked)
 Performance:
Cancellation
He-Ne in quiescent period
Upper: TIA mode, showing
noise and 22 kHz ripple
Lower: Cancellation to
0.5 dB above shot noise

Envelopes of 100 scans,

showing mode beats sweeping
Upper: TIA mode
Lower: >50 dB cancellation,
even with multiple modes

3N3904 discrete BJTs

0.75 mW Psig, 1.5 mW Pcomp
 Performance: 2N3904 discrete BJTs

Cancellation
50-70 dB RIN reduction at low
frequency, ~40 dB to 10 MHz
No critical adjustments
Cancellation at high currents
limited by differential heating

MAT-04 monolithic supermatch quad

 RE Degeneration

Discretes run at different T

=> Less cancellation at high Ic
Use monolithic matching
Main remaining limit is failure of
BJTs to be exponential at high
currents
RE produces negative
feedback on emitters, tending
to even out the current split

Apply positive FB to the bases,

keeping intrinsic VBE constant
 RE Compensator

Requires a current mirror plus a

few extra resistors
Flattens out rejection curve,
10-25 dB improvement
 Differential
Version

Add second signal beam

Run slightly unbalanced
(Isig1 > Isig2)
Differential pair sees only the
slight imbalance
Icomp > (Isig1-Isig2 ) << Isig1
Isig1 =1.48 mA
Limitations of BJTs Isig2 =1.26 mA
circumvented
3 dB noise improvement
(both signal beams contain
information)
Using log output requires
more thought
160 dB SNR (1 Hz)
 Shot Noise
False Alarm Rate

Differential noise canceller,

diode laser, ~0.5 mW/beam
BW = 1.1 MHz
Beam scanning around inside a
chamber with a sandblasted
aluminum back wall (some
mode hopping)
Noise canceller leaves only shot
noise
Very gaussian over >10 orders
(300 kHz - 8 µHz)
Imputed error ~0.1 dB over full
range (1-parameter fit to exact
noise BW)
Multiplicative Noise
Signal beam: 50 kHz AM
Comparison beam vs flashlight
Laser: Distorted 30% AM at 5
kHz
Noise intermod suppression:
>= 70 dB
Power returned to signal
Peak heights are independent
of power level
Intermod suppression depends
on loop gain, but:
The signal being ratioed has
had its additive noise cancelled
at all frequencies
Noise performance greatly
improved--no additive noise!
Log-Ratio Only
Version
Eliminate A1, swap diff pair
inputs to keep FB negative
Gives widest log BW
(> 1 MHz)
BW depends on signal levels
Possible parametric effects
Much less serious than with
analogue dividers
Noise floor 40-60 dB lower
than dividers'
Noise limited by base
resistance Johnson noise
at high currents
RE compensation
applicable
 Performance: Log Noise Floor
Shot noise of Isig and Icomp add in power => noise floor at least 3 dB
above shot noise (but stay tuned)
Noise floor is very flat and stable, generally within 0.5 dB of SNL
except at high currents (and parallelling transistors can improve that)

MAT-04 monolithic supermatch quad

 Log Ratio
Spectroscopy
Sensitivity ~ 1 ppm absorption
Shot noise limited even with huge
dP/dω (∆P~30% over scan range)
Etalon fringes eliminated by
subtracting pressure-broadened scan
 Noise Cancellers and You

The Good News:

A noise canceller will cancel all correlated modulation down to the
shot noise level
Laser RIN is substantially eliminated
Error in ratiometric measurements is greatly reduced

The Bad News:

Everything else will be left behind

Everything depends on the correlation between signal and

comparison beam remaining high
You're going to learn things about your beams that you never
wanted to know: Coherence fluctuations, spatial side modes,
amplified spontaneous emission, polarization instability, vignetting,
and especially etalon fringes
 Applications Advice

System design
Etalon fringes:
Keep design simple, avoid perpendicular surfaces
Spontaneous emission:
Use an efficient polarizer right at the laser
Spatial decorrelation:
Don't vignette anything after the beam splitter
Path length imbalances:
Keep path lengths within ~ 10 cm of each other
Photodiode linearity:
Keep current density lowish & reverse bias highish
Transistor linearity: ID > 1 mA requires differential model or RE
compensation
Keep balance somewhere near 0 V (big negative voltages hurt)
 Applications Advice

System design
Temperature stability
Etalon fringes drift like crazy (>10% transmission change/K)
Photodiode windows a common culprit
Log ratio output proportional to TJ
Temperature-stabilize TJ using monolithic quad (MAT-04)
1 heater, 1 thermometer, 2 for diff pair
-5
~ 10 absorption stability in 1 hour
Care and feeding of photoelectrons:
Never put photodiodes on cables--put the amplifier right there
Photodiode electrical shielding often required
Alarm conditions:
Use a window comparator on the log ratio output to check for
fault conditions, e.g. no light
 Applications Advice

Setup & Testing

Shot noise is easy to verify & you get the frequency response free!
A flashlight generates a photocurrent with exactly full shot noise
A dc-measuring DVM is all you need to know iNshot
Source is white => Output Noise PSD == frequency response

Check cancellation behaviour

Block comparison beam to turn canceller into an ordinary TIA
Use a flashlight to replace Icomp in log ratio mode (∆Vbe constant)
Compare Icomp and Isig to ∆Vbe formula--do they agree?

Wiggle and poke things

Tapping components with the eraser end of a pencil will tell you
which ones are generating the fringes
 Measurement Physics

Laser noise depends on polarization, position, and time

Noise is spatially variable (interference with spontaneous
emission and weak spatial side modes):
Vignetting can destroy correlation

Etalon fringes demodulate everything

Mode partition noise, FM noise, weak longitudinal side modes,
and coherence fluctuations turn into AM
Polarizing cube has 2-5% p-p fringes if perpendicular to beam
-1
FSR is only 0.13 cm (fringes really demodulate everything)
Be paranoid about fringes

Spontaneous emission
Has different noise than laser light & will split differently
 Measurement Physics

Coherence fluctuations
All optical systems
are interferometers

Interferometer path imbalance of 1% of coherence length

=> 40 dB SNR in ∆ν, maximum (|ψ1| = |ψ2|)
Outside coherence length, fringes turn into noise
Full interference term becomes noise in bandwidth ~∆ν
Can easily dominate all other noise sources if ∆ν isn't >>> BW

Time delays
Delaying one arm reduces noise correlation due to phase shift
To get 40 dB cancellation, phase shift ω∆t < 0.01 rad
Summary: Low Frequency Front Ends

It isn't just about detectors

Good analogue design can give huge performance gains

bootstrapping
cascode TIAs

Careful system design prevents trouble:

Etalon fringe elimination
Believing your noise budget

Linear combinations--used intelligently--make hard things easier

Differential detection
Laser noise canceller
Cavity locking
 Ceiling
Footprints:
Concept

What Are My Customers Really Doing?

Quantitative Evaluation of Store Design

See Where Customers Go & What They Look At
Real-time Feedback On Store Ops
(To make it worth instrumenting every store)

Distribute Cheap Sensors In The Ceiling

Extract Trajectories Automatically
 RF Communications $10 Pyroelectric
MUX
CPU/Memory Camera
SENSOR
FILM
Signal
Conditioning

ANTENNA

CEILING TILE

Fresnel
Fascia
Lens

Array of Distributed Pyroelectric Sensors

Sensors Mounted In Ceiling
~ 100 pixels/sensor
100-1000 Sensors Per Store (100-200 sq ft each)
Base Manufacturing Cost: $50-100
 Pyroelectric
Free

+ + + + + + + + +
Charge
+
Effect
3 micron
- - - - - - - - - - Carbon Ink
Bound
Charge
Voc= 0 E free= - E bound 9 microns
Poled PVDF

+ + + + + + + + + +
- - - - - - - - - - 3 micron
Carbon Ink

Ferroelectric PVDF (fluorinated Saran Wrap)

Ferroelectric Has Frozen-In E
Like Remanent B In A Ferromagnet
Polarization drops ~ 1% / K
Free Charge q Flows To Zero Out Etotal, so ∆q gives ∆T
Very inexpensive
Inherently AC: Static Objects Disappear
 Multiplexed
BOTTOM TOP
Pyroelectric
Array

Footprints IR Sensor Photomask Rev C: POSITIVE TONE

Phil Hobbs, June 25, 1999

IR FPA sensitivity, porch-light cost

Free-Standing PVDF Film In Air
8 x 12 Array, 6 mm Pitch
(Tee-shirt Lithography)
Needs Fancy Multiplexer
 Optical
Design
Moulded Polyethylene Fresnel Lenses

7.5-13 µm
Slow is Beautiful
Thermal Design
-2
Signal Power ~ G
Gain
Johnson Noise Is Flat 1
Thermal Mass Limit Sampling Function
(Fluctuation PSD ~ G) (0.2 s Boxcar - Last Boxcar)

Bandwidth ~ G/Mth 0.1

Extra Signal
Johnson-Limited SNR ~ 1/G By Slowing Down!

Thermal Conduction
=> Insulate the Sensor & 0.01 Pixel Thermal Response

Filter Data To Recover BW

0.001
Overall Raw
Pixel Response
0.0001

1E-05

1E-06

1E-07
0.001 0.003 0.01 0.03 0.1 0.3 1 3 10
Frequency (Hz)
 Thermodynamic
50% Reflected
Incident
Thermal
Light 85% Reflected
Efficiency
Semitransparent Metal
75% Area Coverage Carbon Ink
Lattice:
188 Ohm/Sq
25% Area Sensitivity proportional to
Coverage
9 micron 42%
surface emissivity
PVDF Absorbed
in metal
Carbon ink is shiny at 10 µm
"Swiss-cheese" ink blanket
800 Ohm/Sq
Semitransparent Metal
0%
Transmitted
halves the thermal mass
75% Area Coverage Tuned metal coating
increases ∆T
Ink lattice on tuned metal
should give ~ 20 dB more
signal
 Sensor Design:
Multiplexer

∆Tpixel ~ 8 K (Human Crossing the Floor)

∆q/∆Tpixel = (3V/K)(160 pF) ~ 500 pC/K
BUT: ∆Tpixel / ∆TIFOV ~ 0.002, τ ~ 2 s (10 Frames)
Total Signal Available ~ 0.1 pC/pixel/frame

Multiplexer Leakage <= 5 pA

Charge Injection < 0.5 pC
Nothing like it is available commercially
 Diode
Switches

Nanoamp Leakage
Control And Data Paths Not Separate
Unidirectional And Nonlinear: Bias Required

1 mA IF : Si diode ~ 0.65 V, LED ~ 1.6 V

=> IS for a LED Should Be 10-16 That of Si
$0.05 LED has |IF|< 100 fA, -5 V < VF < +0.5 V
 Biasing Hack
Need 1-5 pA Bias Per Pixel, CPU Adjustable
10 Ω Resistors Don't Come in SMT
12

Use Photocurrent Instead

Bias
LED

~1 pA
Bias Pixel

LED Is a Photodiode Too

Use Diffused Light From CPU-Throttled LEDS
1 mA LED Drive => 1 pA Bias
Switch + Adjustable Bias = 1 LED @ $0.05/Pixel
 LED Mux
Schematic
Multiplexer Output Amp

Bias
10M
LED 1k Vbias
Strobes
~1 pA
Bias Pixel 100pF
100k
CS -
0 Switch
LED
+ Out
CS1 1/4 LMC
6034

CS
15
 Footprints
Data
(Raw data,
Person 2
1 sq ft pixels,
28 µm metallized
PVDF)
Person 1

Person 4 (Pseudo-integral,
1 sq ft pixels, 4 µm
carbon ink on 9 µm
Person 3
Person 2 PVDF)
Person 1
 Footprints
Data

Person 4 (Pseudo-integral,
1 sq ft pixels, 4 µm
carbon ink on 9 µm
Person 3
Person 2 PVDF)
Person 1
More if time permits....
 Going Faster: RF Techniques

TC reduction goes only so far

Impedance Transformation
Reactive networks
Transmission-line transformers
Constant-resistance T-coils

Low-noise RF amps
35K noise temperature: 9 dB
improvement vs 300K
Driving 50Ω
 Noise Figure & Noise Temperature

Ways of quoting low noise levels

Noise Figure
NF = 10 log[(SNR before)/(SNR after)] (300K source)
3 dB is garden-variety
< 0.4 dB is the state-of-the-art @ 1-2 GHz (Miteq)

Noise Temperature
Very low NFs awkward to use
TN = PN / (kB)
NF/10
TN = 300K(10 -1)
3 dB NF = 300K TN, 0.5 dB NF = 35K TN, LT1028 = 15K (@1kHz)
TN << Tambient! (F-D theorem doesn't apply to active circuits--or
refrigerators for that matter)
 Impedance Transformation

PD is a current source
Signal power proportional to Re{ZL}
Increasing ZL at the diode can improve SNR
Want all-reactive networks
Resistors in the matching network dissipate power uselessly
and add a 300 K noise source to a ~ 40 K system
Not an impedance matching problem for λ < 1.8 µm!
Available power not fixed for Si, InGaAs PDs
Source impedance poorly defined
IR diodes, e.g. InAs, InSb, HgCdTe have low shunt resistances:
Available power is fixed, so impedance matching is relevant
 Impedance Transformation

Low Noise Amps

PD is a nearly-pure reactance => almost noiseless
35K amp is 9 dB quieter than 300K amp for reactive source
BJT emitter ideally has TN = Tamb / 2,
ideal BJT base has TN = Tamb / (2β)--same noise voltage, β
times higher impedance
Connect PD straight into MMIC with no resistor or capacitor--fix
frequency funnies afterwards, at higher signal levels

Transformers
Quiet RF amps are all around 50 Ω (amps are typically 2:1
VSWR, so it might be 100Ω or 25Ω )
2
N:1 turns ratio gives N impedance change
Transform 50 Ω up for Si PD, or down for, e.g., InAs
 Bode Limit

How wide can we go?

Bode theorem specifies tradeoff between BW and insertion gain Γ

2
|Γ| is the return loss (fraction of power reflected from the load)
RC has 1.03 dB average passband loss (to 3 dB points)
2
Choose |Γ| = 0.21 (79% efficiency, or 1.03 dB signal loss)
BW increases 4x vs RC, for no net signal loss whatsoever
3 elements will usually get within 0.5 dB of this limit
Increasing mismatch gains bandwidth almost reciprocally
2
|Γ| = 0.5 gives 9x BW @ 3 dB loss
 L-Network or Series Peaking
Simplest Reactive Network

Moves RC bandwidth from DC to f0 (same BW, settling time doubled)

Q = X/R [at resonance, Q = 1/(ω0RC) (ratio of f0 to fRC)
Bandwidth BW3dB = ω0/Q
2
Transforms load impedance by a factor of Q +1
50 Ω , Q = 10 => effective RL = 5kΩ (pure resistance at ω0)
Can also be used at baseband for a 1.4x BW increase
 Constant-Resistance T-Coil
Tektronix Vertical Amplifier Secret

Doesn't waste current in R while there's C to charge

2.8x BW increase (at 3 dB points)
No overshoot or ringing
Design equations available
Best simple network for baseband use (lowpass characteristic)
Disadvantage: Load resistor and output are different nodes
Harder to get TN < 300K (may have to put active device in for R)
 Example: 5 pF PD, DC-50 MHz

Direct connection to 50 Ω
BW = 1/[2p(5pF)(50Ω)]=640 MHz
Shot noise limit: Iphot >= 1 mA
(300K), 370 µA (35K)
Wasteful

3:1 Turns Ratio Transformer (450Ω)

BW = 1/[2π(5pF)(450Ω)] = 70MHz
Shot noise limit: Iphot >= 115 µA
(300K), 13 µA (35K)
(DC current x AC resistance
> 50 mV (300K), > 6 mV (35K))
9 dB SNR improvement (Johnson
limit)
 Example: 5 pF PD, DC-50 MHz
Constant-Resistance T-Coil:
2.8x BW increase, resistive load
Can be used with 6:1 transformer
RL = 1800Ω
SN Limit: 29 µA (300K), 3.4 µA (35K)
Best step response
15 dB SNR improvement

Bode Limit:
4x BW increase, resistive load
RL = 2550 Ω
SN Limit: 20 µA (300K), 2.4 µA (35K)
17 dB SNR improvement
Beyond there, you have to trade off SNR
or reduce Cd
 Example: 5 pF PD, 250+-5 MHz

Put passband anywhere you like

Simple 81 nH series L, 5 Ω load
RL=3130 Ω (Q=25--no higher)
Use e.g. a cascode or 1:3 xfrmr
Can tune by changing Vbias
SN Limit: 16 µA (300K), 2 µA (35K)
17 dB SNR improvement vs 50 Ω

Bode Limit:
4x BW increase, resistive load
RL=12.8 kΩ
SN Limit: 4 µA (300K), 0.5 µA (35K)
24 dB SNR improvement vs 50 Ω