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Politecnico di Torino - ICT School

Analog and Telecommunication Electronics

B6 - Non-linear circuits» Nonlinear circuits taxonomy» Log amplifiers: Error sources» Ratiometric, bipolar circuits» Saturating amplifier chain, RSSI» Circuit example

AY 2015-16

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Lesson B6: Nonlinear circuits

• Nonlinear circuits taxonomy

• Logarithmic amplifiers– Parameters of a logarithmic transfer function– Circuits: Error sources, Design procedure– Exponential, ratiometric, bipolar circuits

• Saturating amplifier chain

• RSSI circuits

• References – Elettronica per Telecom.: 2.2.3 – Amplif logaritm. ed espon.– Design with Op Amp …: 13.1 - Log and antilog amplifiers

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Nonlinear circuits

• Approximation of nonlinear transfer function Errors– offset, gain– “nonlinearity”: deviation from the designed behaviour

• Piecewise approximation– Amplifiers with gain and offset related with specific

input voltage Vi ranges » Active diode» Wave shaper» RSSI circuits

• Continuous approximation– Need for a nonlinear element

» Multipliers: squaring, root, polynomial function» Semiconductor junction: log or exponential function

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Logarithmic amplifier: transfer function

• Generic log transfer function

– k2 and k3 represent the same parameter

• Two degrees of freedom + one “distortion”– Input offset k4

– Input slope k2

– Distortion: k1 and k3

Vo = k1 log(k2(Vi+k4)) + k3

X X+ +logVi Vo

k4 k2 k1 k3

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Log amplifier: transfer diagram (linear)

• Vo = k1 log(k2(Vi+k4)) + k3

• Representation on linear plot– X axis: Vi;

Y axis: Vo = log Vi– Fixed ratio on Vi fixed shift on Vo

– Vi = 0 ??– Hard to see

effects of k4

Vo

Vi

54321

1 2 4 8 16

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Log amplif.: transfer diagram (half-log)

• Vo = k1 log(k2(Vi+k4)) + k3

• Representation on semilog plot– X axis: log k2 Vi

– Straight line: y = k1 x + k3

– Changing k1modifies the slope (rotation)

– Changing k3 (or k2) causes a shift (translation)

– Changing k4 causes nonlinearity for low Vi

log k2Vi

Vo

k3

k2k1

k4Vo = k1 log(k2(Vi+k4)) + k3

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Effects of input offset k4

• Input additive constant input offset– The same offset (Δk4) corresponds to different shifts on the

logVi axis– The effect on output depends on the actual value of Vi

logVi

Vo

k4k4k4

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Logarithmic element

• Functional specification: Vu = K log Vi» Wide dynamic» Low errors» Wide band» ….

• Exploit the V(I) relation in a PN junction

– Set the current I

– Read the voltage V

VD =

V

I

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Logarithmic amplifier: circuit

• Use transconductance Op. Amp. circuit

– Set the current» I = I2 = I1 = Vi / R

– Read the voltage» VU = -VD

– Control the parameters» V I conversion at the input: k2 (and k3)

» Output gain: k1 (negative)– Correct temperature-related errors:

» Is: cancel with reference junction, constant current» Vt: correct with temperature-dependent gain element (NTC)

AO 1

Vi-+

VO

DR

Vd

I1

I2

I- VD

VD =

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Basic circuit for logarithmic amplifiers

• Logarithmic junction Reference junction

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Error sources

• Low input values:– Low V across R1 Op. Amp. Offset (Voffset)– Low I in the log junction Ioff and Ibias

• High input values: High currents– Additional voltage drop on junction intrinsic resistance rBB’

logVi

Vo

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[mV]

Total errors

• Overall transfer function (inverting)

Errors causedby Ib, Ioff, Voff

Error causedby rBB’

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Ratiometric logarithmic amplifier

• Log of voltage ratiolog (x) – log (y) = log (x/y)

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Bipolar logarithmic amplifier

• Twin junctions to handle bipolar Vi (bidirectional current)

• Compression transcaracteristic

• If R ↔ diodes expander transcaracteristic

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Applications of log amplifiers

• DC amplifiers:– lin-log conversion (dB, bode diagrams, …)– (Analog “computation”) – After AM demodulation

» Level measurement (IF chain, RSSI…),» Gain control (AGC)

• AC and bipolar amplifiers:– Dynamic range compression

• AC-DC log converters– Sequence of saturating stages – Wide dynamic range level measurement

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AC-DC log converters

• Piecewise approximation

• Sequence of amplifiers with breakpoint; two types – A/1 amplifiers:

» Gain A for Vi < E; 1 for Vi > E» Direct output from last stage

– A/0 amplifiers: » Gain A for Vi < E; 0 (Vu = S = E*A) for Vi > E» output = sum of input + single amplifier outputs

• Obtained with saturating amplifiers– Usually differential stages, with summation of currents– As the level increases, the number of saturated stages

increases

Vo

ViE

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Saturating chain

• Eeach stage has Gain = 2, and saturates at Vo = S– Stage 1 with gain for Vi < S/2; saturated for: Vo = 2 Vi, – Stage 2 with gain for Vi < S/4; saturated for: Vo = 4 Vi, – Stage 3 with gain for Vi < S/8; saturated for: Vo = 8 Vi, – Stage 4 with gain for Vi < S/16; saturated for: Vo = 16 Vi,

• Total gain– 0<Vi<S/8 active: 1, 2, 3, 4 G = 24 = 16– S/8<Vi<S/4 active: 1, 2, 3 saturated: 4 G = 23 = 8– S/4<Vi<S/2 active: 1, 2 sat.: 3, 4 G = 22 = 4– S/2<Vi<S active: 1 sat.: 2, 3, 4 G = 21 = 2

• Saturation = gain 0: sum of the outputs

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Chain with saturation

• Low Vi :all stages have gain

• High Vi:only first stages have gain

• Higher gain for lower Vi:16, 8, 4, 2, 1

• Compression

Vo

Vi

1 2 3 4Σ

VoVi

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Saturating logarithmic amplifiers

• Good for AC, can provide wide band and wide dynamic

• AC-DC conversion on each stage– Reduced dynamic on the single converter

• Applications: RF power measurement– AGC for LNA and IF amplifiers– Power control for PA

• RSSI (Received Signal Strength Indicator) output– Carrier detection– RF signal level– Squelch control

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Example of saturation log circuit

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Limiting amplifier + RSSI

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Log amplifiers

• Lab exercise:– Design a log amplifier from the assigned specs– Evaluate errors– Verify with simulation– Verify with measurements

• Specs:– Provided each year in the lesson

• Design procedure: Sect 2, 2.P2

• Lab experience: Sect 2, 2.L2

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Design procedure

• Selection of circuit configuration

• Definition of current dynamic range– Evaluation of error at upper range limit

» RBB’, maximum current– Evaluation of errors at lower range limit

» Op. Amp (Ib), minimum current– Selection of Op. Amp.– Current range selected to get balanced error at the dynamic

range extremes

• Positioning input & output constants and parameters– Gain and translation of Vu’ = log Vi

• Temperature compensation (if required)

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Lesson B6 – final test

• Which are the techniques to obtain nonlinear transfer functions?

• How can Op Amp be used to get nonlinear transfer functions?

• How many parameters describe a log transfer function?

• Describe an application for logarithmic amplifiers

• Draw the diagram of a basic log amplifier.

• Which are the main error sources at low end of input range?

• Which are the main error sources at high end of input range?

• Describe how to get nonlinear transfer functions using saturating amplifiers.

• Which is the meaning of the acronym RSSI?

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