EEE225: Analogue and Digital ElectronicsLecture VIII

James E. Green

Department of Electronic EngineeringUniversity of Sheffield

j.e.green@sheffield.ac.uk

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EEE225: Lecture 8

This Lecture

1 Non-Linear EffectsSlew Rate LimitingSlew Rate Limiting: SquareSlew Rate Limiting: SineSlew Rate Limiting: Triangle

2 Opamps with Frequency Dependent FeedbackIntegratorIntegrator: Frequency DomainIntegrator: Time DomainProblems with IntegratorsDifferentiator

3 Review

4 Bear

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EEE225: Lecture 8

Non-Linear Effects

Slew Rate Limiting

Slew Rate LimitingSlew rate limiting is non-linear, the ratio of vo and vi dependson the magnitude of vi . It is a limit on the maximum rate ofchange of output voltage.It is particularly prevalent in problems where large signals andhigh frequencies are in use.It is often caused by the differential pair and VAS currentsource’s inability to charge or discharge the compensationcapacitor sufficiently quickly.Manufacturers specify in V /µs. (TL081 8 V /µs). Specificopamps can manage 5000 V /µs.Opamp manufacturers artificially increase the value of ccb toobtain stability and a first order response. But increasing ccbincreases the current needed from the differential stage andVAS current source. It’s a compromise, greater stability (esp.at lower closed loop gain) comes at the expense of lower slewrate.

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EEE225: Lecture 8

Non-Linear Effects

Slew Rate Limiting: Square

0

1

2

3

4

5

6

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Time [µs]

Voltage[V

]

τ

k VinVout = k Vin

(1− exp

(−tτ

))

k Vin

Vout

The square input signal interacts with the (low pass) opamp as ifthe opamp was an RC network. The result is an exponential rise tomaximum of the form Vo = k Vin (1− exp t/τ) where t = 0 is therising edge of the square signal, k is the system gain and τ is thetime-constant of the opamp. Max rate of change = (k Vin)/τ . Ifthe initial rate of change was maintained the output waveformwould cross the setpoint at τ .

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EEE225: Lecture 8

Non-Linear Effects

Slew Rate Limiting: Sine

−5−4−3−2−1012345

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Time [µs]

Voltage[V

]

Limitingd Vout

dtthat the

amplifier can support

Vin

Vout

Max rate of change of a sinusoid,

Vin sin (ωt) =d (Vin sin (ωt))

dt

∣∣∣∣max

= Vin ω cos (ωt)|max (1)

Max when cos (ω t) = 1. Max dV /dt for sinusoid is Vin ω.

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EEE225: Lecture 8

Non-Linear Effects

Slew Rate Limiting: Triangle

−5−4−3−2−1012345

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Time [µs]

Voltage[V

]

SlopeVp

T/4=

4Vp

T

T/4 T

Vp

Vout

For the triangle the rate of change of voltage is constant. In thegraph above the amplifier must change it’s output voltage by Vp ina time, T/4 where T is the period. For example if Vp = 5 V andT = 1.6 µs the slew rate must be ≥ 4×5

1.6×10−6V/s or 12.5 V/µs

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Opamps with Frequency Dependent FeedbackPart II of the second section of the course...

Introduction of simple general opamp amplifier (Z1, Z2, notR1, R2)An analogue integrator

Freq domain analysisTime domain analysisProblems with integratorsAnalogue circuit to solve 1st order differential equation(printed notes)

An analogue differentiatorFreq domain analysisTime domain analysisProblems with differentiators

Pole-Zero Circuits.Description of first order circuits (HP, LP, PZ)Example with defined componentsExample of intrinsic freq response type problem

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Some Standard opamp circuits

−

+Av

Z1

ii

Z2 i f

vi vo

−

+

Z2

Z1

Av

vovi

Inverting design gain...

vovi

= −Z2

Z1(2)

Non-inverting design gain...

vovi

=Z1 + Z2

Z1(3)

Provided closed loop gain is not dependent on open loop gain (i.e.if Av →∞). Zn is an arbitrary impedance (could be R, L and C).

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Integrator

Opamp Integrator

−

+

R

ii

C i f

vi vo

Av

In the frequency domain.

vovi

= −Z2

Z1(4)

vovi

= −

(1

j ω C

)

R= − 1

j ω C R(5)

Integrators used in filters,instrumentation circuits andin control systems, but notoften implemented using anopamp.

Often j ω = s where ‘s’ isthe same as appears in theLaplace transform. So (5)becomes 1/(s C R).

As ω approached 0 (i.e. DC)the gain →∞. This can notactually happen as the gaincan not rise above Av

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Integrator: Frequency Domain

−20

0

20

40

60

80

100

120

10−6 10−5 10−4 10−3 10−2 10−1 100 101

Normalized Frequency, ω [rads−1]

Vo

ViMagnitude[dBV]

−280

−260

−240

−220

−200

−180

−160

−140

Vo

ViPhase

[◦]

deviation from idealcaused by Av 6= ∞

Realistic Magnitude [dBV]Realistic Phase [◦]Ideal Magnitude [dBV]Ideal Phase [◦]

The finite Av affects performance by moving the pole up from zerofrequency to some finite frequency. The graph above is normalisedi.e. C R = 100. The usable range of the integrator is about10−3 → 101 Hz normalised but it depends on the value of Av tosome extent. Care should be taken to avoid phase errors as well asmagnitude errors.

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Integrator: Time Domain

−

+

R

II

C IF

VI VO

Av

(VO − V−)

In the time domain (notice uppercase letters)

II + IF =VI − V−

R+

Cd (VO − V−)

dt= 0 (6)

Assuming V− ≈ 0

VI

R C= −dVO

dt(7)

integrating both sides,

VO = − 1

R C

∫VI dt + A (8)

A is a constant proportional tothe voltage across the capacitorprior to the start of theintegration.Integrators have a major problemhowever, called “wind-up”.

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Problems with Integrators

There is no DC feedback between output and input.

Any small voltage (offset of the opamp or offset of the signalsource) is integrated over time.

Eventually the integrator output will saturate against one ofthe power supply rails.

This can be avoided by providing DC feedback either as part of alarger system or more directly using a resistor.

−

+Av

RVI

C

RF

VO

The result of the DC pathway(RF ) is to change the gain of thecircuit from −A0 at DC to−RF/R. This moves the pole upin frequency, decreasing theuseful frequency range of theintegrator.

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Differentiator

Opamp Differentiator

−

+

C

ii

R i f

vi vo

In the frequency domain.

vovi

= −j ω C R = −s C R (9)

In the time domain,

VO = C RdVI

dt(10)

Key componentsinterchanged. Sameassumptions as forintegrator. Same style ofanalysis.

The capacitance andintrinsic frequency responseof the opamp (Av ) interactwith each-other forming (inthe case of first order opampassumptions) a second ordercircuit. This makes thedifferentiator unusable for afew decades of frequencyaround resonance.

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EEE225: Lecture 8

Opamps with Frequency Dependent Feedback

Differentiator

−20

0

20

40

60

80

100

120

10−1 100 101 102 103 104 105

Normalized Frequency, ω [rads−1]

Vo

ViMagnitude[dBV]

80

120

160

200

240

280

320

360

Vo

ViPhase

[◦]

Resonance caused byinteraction of opampfrequency dependenceand ideal differentiatorresponse

Semirealistic Magnitude [dBV]Semirealistic Phase [◦]Ideal Magnitude [dbV]Ideal Phase [◦]

−3

−2

−1

0

1

2

3

0 10 20 30 40 50 60 70 80 90 100

Time [ms]

Voltage[V

]

Vin

Vout

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EEE225: Lecture 8

Review

Review

Discussed slew rate limiting, a non-linear effect which dependson signal magnitude. Examples for square, sine and trianglegiven.

Introduced two opamp circuits for integration anddifferentiation of signals using resistors and capacitors as gainsetting components.

Considered some limitations and impracticalities of bothcircuits including the interaction between the intrinsicfrequency response of the opamp and the frequencydependent feedback.

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EEE225: Lecture 8

Bear