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    An example of positive feedback op amp circuit: Schmitt Trigger

    In real OA imperfections cause Vd to have a small value:

    If Vd> 0 VOIf Vd< 0 VO

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    Lets assume VSUPPLY= 5V

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    Op Amp Imperfections

    a.Voltage Supply Limits (Saturation)

    For a real op amp, clipping occursif the output voltage reaches

    certain limits.

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    b.Output Current Limit (Short-Circuit Output Current ISC) The current that an op amp can supply to a load is limited (typically

    +/-25 mA)

    If a small-value load draw a current outside the limit, the outputwaveform becomes clipped

    Example:

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    Ideally,

    This means, the op amp should provide 205 mA, which is higher thanthe 25mA the op amp can really issue.

    In practice:

    Thus, as soon as the output voltage reaches 2.44 V (which correspondsto an input voltage of 2.44/4 0.61 V) the output voltage gets clipped.

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    Example:

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    In this case the OA has no problem to provide the current required to

    produce an output voltage up to 20V.

    Unfortunately, as soon as the value of Vo gets larger than 12V (which

    correspond to an input voltage value of Vin =12/4 = 3V), the output isstill going to be clipped. This time the reason is not the OA current limit,

    but the OA supply limit.

    Because of that, the max current that the op amp is really going to

    provide is only:

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    c. CMRR

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    d.Non Linearity

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    e. DC offset voltageOffset is caused by the fact that the internal circuit of the op amp

    experiences fabrication mismatches.

    As a result the op amp is not perfectly balanced, i.e.:

    VO0 when Vin1=Vin2

    Offsets arise from input stage mismatch and cause the input-output

    characteristic to shift in either the positive or negative direction (the plotdisplays positive direction).

    We model the offset by a single voltage source placed in series with one

    of the inputs.

    Since offsets are random can be positive or negative, Vos can appear at

    either input with arbitrary polarity.

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    Why are DC offsets important?

    Can cause unacceptable accuracy errors Can cause saturation

    Example: Effects of DC offset (Accuracy Error)

    The op amp amplifies the input as well as the offset, creating errors.

    Lets assume we have a weighting station that uses an electronic

    pressure meter whose output is amplified by the circuit shown, and the

    pressure meter generates 20mV for every 100 Kg.

    If the op amp has an offset of Vos = 2mV, this correspond to an error of10Kg.

    Example: Effects of DC offsets (Saturation)

    ( )osinout VVR

    RV +

    +=

    2

    11

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    The first op amp has an offset of 2 mV. What happens at the output?

    A1 = 1 + 104/10

    2100A2 = 1 + 10

    4/10

    2100Av= Vout/Vin= A1A2104

    The first stage will amplify the offset by a factor of 100generating a DC

    level of 200 mV at node X.

    At this point, the second stage amplifies VXby another factor of 100,

    attempting to generate Vout=20V.

    Since the op amps have a supply voltage of 3V the Voutcannot exceed

    this value and the second op amp will saturate.

    Example: effects of DC offsets on the ideal integrator (saturation)

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    In other words, the circuit integrates the offset, generating an output that

    tends to +(in reality the positive supply voltage) or (in reality the

    negative supply voltage) depending on the sign of Vos.

    It is worth to notice that if we model the offset placing a voltage source

    in series with the negative input instead of the positive input the specific

    formulae may change but the physical effect is the same.

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    To avoid the saturation problem caused by the offset, we can modify the

    integrator circuit as follow:

    The idea is to place a resistor in parallel with the capacitor to absorbthe offset.

    Since Vosis a DC its effect at the output is given by:

    For example if Vos= 2mV and R2/R1=100, then Voutcontains a dc error

    of 202 mV, but at least does not saturate.

    However, the presence of R2also affects the integration function. Theclosed-loop transfer function has no longer has a pole at origin. Now the

    circuit contains a pole at 1/(R2C1)

    1

    1

    121

    2

    +=

    sCRR

    R

    V

    V

    in

    out

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    In summary: R2/R1must be sufficiently small to minimize the effect of the

    offset

    R2C1must be sufficiently large so as to negligibly impact the inputsignal frequencies of interest

    Commonly used offset voltage nulling circuits

    Universal offset voltage balancing schemes:It is possible to build many reasonable balancing circuits. Few examples

    follow.

    (a) Inverting Configuration

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    For R3>> R1:

    Thus:

    (b) Non Inverting configuration

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    Alternative scheme:

    (c) Mixed configuration

    In general, there are many reasonable balancing circuits that can be built.

    Ad-Hoc offset voltage balancing schemes:

    Many op-amp manufacturers provide extra terminal for offset nulling.

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    General Offset Nulling Procedure:

    1.build the circuit you want to implement2.put the input terminals to ground3.move the POT wiper until Voutis zero

    NOTE: offset balancing is usually done after the circuit has been

    working for a couple of hours!!!

    f. Input Bias (=DC) CurrentsIn order for the op amp to operate the two input transistors of differential

    stage have to be biased.

    As a result there may be dc currents flowing into the inputs of the opamp:

    For op amps implemented in bipolar technology there is always adc base current drawn (0.1-1 A) from each input

    For op amp implemented in MOS technology there almost no dccurrents.

    The input bias currents create inaccuracies in the circuits.

    The error due to the input bias currents appear similar to the dc voltageoffset effect (i.e., corrupt the output of the op amp). However, unlike the

    dc voltage offset, this phenomenon is not random.

    The effect of bipolar base currents can be modeled as current sources

    tied from the input to ground.

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    Usually two parameters are specified to describe this op amp

    imperfection:

    The average value of IB1and IB2

    The expected difference

    a. Effect of the dc input currents on the non inverting configuration

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    Setting Vin=0 (no signal applied) the circuit reduces as follows:

    Since the + terminal is at groundthen the node X is at virtual ground.As a result there is a zero voltage across R2, which means no current

    will flow through R2, so all current IB2flows through R1:

    Vout= IB2R1

    The error due to the bias current appears similar to the dc voltage offset

    effect.

    Then, a reasonably simple method for canceling this error seems to bethe application of a DC correction voltage in series with the positive

    terminal.

    12

    2

    11 RIR

    RVV Bcorrout +

    +=

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    In order to produce a zero output it must be:

    In other words:

    If we assume that nominally IB1=IB2obtaining Vcorrand canceling the

    effect of the bias currents for the non inverting amplifierbecomes

    quite simple:

    122

    110 RIR

    RV

    Bcorr

    +

    +=

    21 BB II =

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    b.Effect of the dc input currents on the inverting configuration

    Canceling the effect of the bias currents for the non inverting

    amplifier

    c. Effect of bias currents on the unity gain amplifier and its cancellation

    There is no effect on the non inverting configuration (thus no need for

    cancellation)

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    d.Effect of bias currents on the summing amplifier and its cancellation

    e. Effect of input bias currents on integrator

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    Since IB2is a DC current it will flow all through R1:

    Input bias current will be integrated by the integrator and eventuallysaturate the amplifier.

    If we assume that IB2=IB1by placing a resistor in series with the positive

    input, integrator input bias current can be cancelled.

    ( ) =t

    Bout dtRICR

    V0

    12

    11

    1

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    In reality, the output still saturates due to other effects such as the DCvoltage offset, mismatch between IB1and IB2, and mismatch between the

    resistors.

    To avoid saturation we can use the non ideal integrator circuit:

    In this case the presence of R2makes the op amp work in closed loop for

    the DC. As a result the negative input is at virtual ground, so all IB2will

    flow in R2.

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    * Detailed analysis of the error caused by the DC bias currents:

    A straightforward way of assessing the effect of the input bias currents is

    to find the output with all input signals set to zero.

    To analyze in detail the effect of the error caused by the DC biascurrents, lets consider the inverting configuration:

    Expand (**):

    2

    BNBPB

    III

    +=

    BNBPOS III =

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    Substitute (*) in:

    Lets now consider:

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    Lets now substitute in ():

    And if we impose RP = R1||R2the term involving IBwill be eliminated,leaving:

    Which is a quite small quantity since it is proportional to IOS

    (IOS= IPIN0)

    OSO IRRR

    RV

    += )||(1 21

    2

    1

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    g. Speed Limitations

    Finite Bandwidth (Small Signal Speed) Slew Rate (Large Signal Speed)

    Finite Bandwidth (Small Signal Speed)

    Because of its internal capacitances, the gain of the op amp begins to fall

    as the frequency of operation exceeds a certain break value.

    The internal circuitry of the op amp can be modeled (approximated) by a

    first-order (one pole) system:

    ( )

    1

    0

    21

    21

    )(

    )(

    f

    s

    A

    sV

    sVs

    VV

    V

    d

    out

    inin

    out

    +==

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    At frequencies well below the break frequency (s/1> 1):

    and since the op amp gain falls to unity at u:

    Having a loop around the op amp (inverting, non-inverting, etc) helpsto increase the bandwidth of the system, however, it also decreases the

    gain (Bandwidth and Gain Tradeoff)

    Non-inverting Amplifier Example: (Bandwidth Gain Tradeoff)

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    Graphically:

    In conclusion: we can trade gain for bandwidth and vice versa

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    Slew Rate (Large Signal Speed)

    Slew rate is a non linear phenomenon.

    Lets consider a non inverting configuration and its small signal voltage

    transfer function:

    Then for simplicity, lets make the configuration into a voltage follower

    (R2=):

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    Applying to the input of a follower a voltage step of sufficiently small

    amplitude VPwill result in an exponential response:

    The rate at which Voutchanges with time is highest at the beginning of

    the exponential transition.

    If we increase VPthe rate at which the output slews will have to increase

    accordingly.

    In practice, we observe that the rate at which the output voltage changes

    (i.e. the output slope) cannot exceed a certain limit called the slew rate(= SR).

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    If the input is small (linear region), when the input doubles then the

    output and the output slope also double. However, when the input is

    large, the op amp slews.

    Slew rate limiting is a non linear effect due to the limited abilityof the

    internal circuitry to charge or discharge the internal capacitance.

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    As long as the input step amplitude VPis sufficiently small, the op ampwill respond in proportion (i.e., linearly) and yield:

    However, if we overdrive the op amp applying a large input voltage, the

    Ioutwill saturate at ITAIL.

    Then the capacitor C will become current starved and the speed of theop amp is further limited.

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    The non linearity effect of slew rate can also be observed by applying a

    large sine wave to the circuit.

    +=2

    1

    max

    1R

    RV

    dt

    dVP

    out

    tR

    RV

    dt

    dVP

    outcos1

    2

    1

    +=

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    As long as the output slope is less than the slew rate, the op amp canavoid slewing.

    However, as operating frequency and/or amplitude is increased, theslew rate becomes insufficient and the output becomes distorted.

    Full-Power Bandwidth (FP)

    To determine the maximum frequency before op amp slews, we assumethe op amp produces its maximum allowable voltage swing without

    saturating (worst case scenario).

    If the op amp provides a slew rate of SR:

    Then in conclusion:

    2sin

    2

    minmaxminmax VVtVV

    Vout+

    +

    =

    2minmax VV

    SRFP

    =

    2

    minmax

    max

    VV

    dt

    dVSR out ==

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    h. Finite Gain, Finite Input Impedance and non-zero Output

    Impedance

    Actual op amps do not provide infinite gain, infinite input impedance

    (MOS based op amps have a very high input impedance at lowfrequencies), and non-zero output impedance.

    The effect of these limitations is to increase the gain error of the

    circuit.

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    Lets consider the effects of finite gain, finite input impedance, and non-

    zero output impedance on a non-inverting configuration:

    Voltage Gain Derivation:

    Nid vvv =

    21 R

    v

    R

    vv

    r

    vv NNO

    d

    Ni =

    +

    1

    0

    R

    vv

    r

    vAv NO

    O

    dO =

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    Input Resistance Derivation:

    Matlab Code:

    clear all; close all; clc;syms VNRdRoR1R2Ao;% Ax = b ; x(1) = Vt, x(2) = ItA = [1 -Rd ; 1/Rd+Ao/(R1+Ro) 0];b = [VN ; VN*(1/Rd+1/R2+Ao/(R1+Ro)+1/(R1+Ro))];x = A\b;Rin = x(1)/x(2);disp('Rin = ');pretty(Rin);IdealRin = limit(Rin,Ao,inf);disp('IdealRin = ')pretty(IdealRin);

    R2 R1 + R2 Ro + Rd R1 + Rd Ro + Ao Rd R2 + Rd R2

    Rin = ------------------------------------------------

    R1 + Ro + R2

    signum(Rd) signum(R2) Inf

    IdealRin = -------------------------

    signum(R1 + Ro + R2)

    i

    iin

    i

    vR =

    0)(

    1

    0

    2

    =+

    +

    o

    NNiN

    d

    Ni

    rR

    vvvA

    R

    v

    r

    vv

    d

    Nii

    r

    vvi

    =

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    Output Resistance Derivation:

    0)0

    1

    =

    =o

    NxdNxx

    r

    vvvA

    R

    vvi

    )||(||

    2

    21

    RrRrR

    vvv d

    d

    NxN +

    =

    Nd vv =

    x

    xout

    i

    vR =

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    Matlab Code:

    clear all; close all; clc;syms VdVNIxVxRoR1RpAo

    s = solve('Vd=-VN', 'Ix +(-Ao*VN-Vx)/Ro-(Vx-VN)/R1 = 0', 'VN=Vx*Rp/(R1+Rp)',

    'Vd', 'Ix', 'Vx');Vx = s.Vx;Ix = s.Ix;Rout=Vx/Ix;fprintf('Rout = ');pretty(Rout);IdealRout = limit(Rout,Ao,inf);disp('IdealRout = ');pretty(IdealRout);

    Rout =(R1 + Rp) Ro

    --------------------

    R1 + Rp Ao + Rp + Ro

    IdealRout =

    0

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    Example 8.16

    A student has an OA with Ao=10000, Rout=1and want to design aninverting amplifier with R1=50and R2=10. The amplifier fails to

    provide the output swing of 2V we need. Why?

    _____

    40 mA is a quite big current and many op amps provide only a very

    small output current

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    Example 8.17

    Design an inverting amplifier with a nominal gain of 4 a gain error of

    0.1% and a nominal input impedance of at least 10K.

    ______

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    In conclusion:

    Thus:

    Just for the record, it can be shown that the |% Gain Error| for the non

    inverting configuration is also given by (1+R1/R2)/A0

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    Example 8.18

    Design a non-inverting amplifier with the following spec.: closed loop

    gain = 5, gain error = 1%, closed loop BW = 50 MHz. Assume the op

    amp has IBIAS=0.2A.Determine the required open loop gain and BW of the OA.

    _______

    The choice of R1 and R2 depends on the driving capability(output

    resistance) of the OA.

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    The BW is given by:

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    Example 8.19

    Design an integrator for a unity gain frequency 10MHz and an input

    impedance of 20K.If the OA provides a slew rate of 0.1V/ns, what is the largest peak-to-

    peak sinusoidal swing at the input at 1MHz that produces an output free

    from slewing?

    ______

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    For a sinusoidal input Vin = Vpcos t:

    Then in order for the OA to do not slew it must be SR VP/(R1C1):