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Analog Electronics(Course Code: EE314)

Lecture 31‐32: OPAMP and Frequency Response

Indian Institute of Technology Jodhpur, Year 2018

Frequency Response

Course Instructor: Shree Prakash TiwariEmail: sptiwari@iitj.ac.in

Webpage: http://home iitj ac in/~sptiwari/Webpage: http://home.iitj.ac.in/~sptiwari/

Course related documents will be uploaded on  http://home.iitj.ac.in/~sptiwari/EE314/

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Note: The information provided in the slides are taken form text books for microelectronics (including Sedra & Smith,  B. Razavi), and various other resources from internet, for teaching/academic use only

Basic OPAMP

Op amp is a circuit that has two inputs and one output. It amplifies the difference between the two inputs.

210 ininout VVAV

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Unity Gain Amplifier

0

0

0

1

)(

AA

VV

VVAV

in

out

outinout

Op Amp with Supply Rails

To explicitly show the supply voltages, VCC and VEE are shown.

In some cases, VEE is zero.

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Noninverting Amplifier (Infinite A0)

2

11RR

VV

in

out

A noninverting amplifier returns a fraction of output signal thru a resistor divider to the negative input.

With a high Ao, Vout/Vin depends only on ratio of resistors, which is very precise.

Noninverting Amplifier (Finite A0)

The error term indicates the larger the closed-loop gain, the less accurate the circuit becomes.

02

1

2

1 1111

ARR

RR

V

V

in

out

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Extreme Cases of R2 (Infinite A0)

If R2 is zero, the loop is open and Vout /Vin is equal to the intrinsic gain of the op amp.

If R2 is infinite, the circuit becomes a unity-gain amplifier and Vout /Vin becomes equal to one.

Inverting Amplifier

21

0

RV

R

V

R

V inout

Infinite A0 forces the negative input to be a virtual ground.

2

1

R

R

V

V

in

out

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Another View of Inverting Amplifier

Inverting Noninverting

Gain Error Due to Finite A0

2

1

02

1 11

1RR

ARR

VV

in

out

The larger the closed loop gain, the more inaccurate the circuit is.

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Integrator

sCRVV

in

out

11

1 dtV

CRV inout

11

1

Integrator with Pulse Input

tCR

VdtV

CRV inout

11

1

11

1 bTt 0

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Comparison of Integrator and RC Lowpass Filter

The RC low-pass filter is actually a “passive” approximation to an integrator.

With the RC time constant large enough, the RC filter output approaches a ramp.

Lossy Integrator

sCRAA

V

V

in

out

11

11

1

1

When finite op amp gain is considered, the integrator becomes lossy as the pole moves from the origin to -1/[(1+A0)R1C1].

It can be approximated as an RC circuit with C boosted by a factor of A0+1.

AA 00

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Differentiator

RV

dtdV

CRV inout 11

sCR

sC

RV

V

in

out11

1

1

1

Differentiator with Pulse Input

)(111 tVCRVout

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Comparison of Differentiator and High‐Pass Filter

The RC high-pass filter is actually a passive approximation to the differentiator.

When the RC time constant is small enough, the RC filter approximates a differentiator.

Lossy Differentiator

11

11

11

sCRsCR

VV

in

out

When finite op amp gain is considered, the differentiator becomes lossy as the zero moves from the origin to –(A0+1)/R1C1.

It can be approximated as an RC circuit with R reduced by a factor of (A0+1).

00 AA

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Voltage Adder

VV

If R R R

21

2

2

1

1

VVRR

V

RV

RV

RV

Fout

Fout

Ao

If Ao is infinite, X is pinned at ground, currents proportional to V1 and V2 will flow to X and then across RF to produce an output proportional to the sum of two voltages.

If R1 = R2=R

Precision Rectifier

When Vin is positive, the circuit in b) behaves like that in a), so the output follows input.

When Vin is negative, the diode opens, and the output drops to zero. Thus performing rectification.

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Inverting Precision Rectifier

When Vin is positive, the diode is on, Vy is pinned around VD,on, and Vx at virtual ground.

When Vin is negative, the diode is off, Vy goes extremely negative, and Vx becomes equal to Vin.

Logarithmic Amplifier

inVVV ln

S

inTout IR

VV1

ln

By inserting a bipolar transistor in the loop, an amplifier with logarithmic characteristic can be constructed.

This is because the current to voltage conversion of a bipolar transistor is a natural logarithm.

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Square‐Root Amplifier

THin

out VW

VV

2TH

oxn

out

RL

WC 1

By replacing the bipolar transistor with a MOSFET, an amplifier with a square-root characteristic can be built.

This is because the current to voltage conversion of a MOSFET is square-root.

Op Amp Nonidealities:  DC Offsets

Offsets in an op amp that arise from input stage mismatch cause the input-output characteristic to shift in either the positive or negative direction (the plot displays positive direction).

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Effects of DC Offsets

osinout VVRR

V

2

11

As it can be seen, the op amp amplifies the input as well as the offset, thus creating errors.

Saturation Due to DC Offsets

Since the offset will be amplified just like the input signal, output of the first stage may drive the second stage into saturation.

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Offset in Integrator

11

121

2

sCRRR

VV

in

out

A resistor can be placed in parallel with the capacitor to “absorb” the offset. However, this means the closed-loop transfer function no longer has a pole at origin.

Input Bias Current

The effect of bipolar base currents can be modeled as current sources tied from the input to ground.

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Effects of Input Bias Current on NoninvertingAmplifier

It turns out that IB1 has no effect on the output and IB2

affects the output by producing a voltage drop across R1.

21

2

122 BBout IR

RR

IRV

Input Bias Current Cancellation

12

2

11 RIRR

VV Bcorrout

We can cancel the effect of input bias current by inserting a correction voltage in series with the positive terminal.

In order to produce a zero output, Vcorr=-IB2(R1||R2).

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Correction for  Variation 

21 BB II

Since the correction voltage is dependent upon , and varies with process, we insert a parallel resistor combination in series with the positive input. As long as IB1= IB2, the correction voltage can track the variation.

Effects of Input Bias Currents on Integrator

dtRIV1

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

dtRICR

V Bout 12

11

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Integrator’s Input Bias Current Cancellation

By placing a resistor in series with the positive input, integrator input bias current can be cancelled.

However, the output still saturates due to other effects such as input mismatch, etc.

Speed Limitation

1

0

21 1s

As

VV

V

inin

out

Due to internal capacitances, the gain of op amps begins to roll off.

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Bandwidth and Gain Tradeoff

Having a loop around the op amp (inverting, noninverting, etc) helps to increase its bandwidth. However, it also decreases the low frequency gain.

Slew Rate of Op Amp

In the linear region, when the input doubles, the output and the output slope also double. However, when the input is large, the op amp slews so the output slope is fixed by a constant current source charging a capacitor.

This further limits the speed of the op amp.

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Comparison of Settling with and without Slew Rate 

As it can be seen, the settling speed is faster without slew rate (as determined by the closed-loop time constant).

Slew Rate Limit on Sinusoidal Signals

tRR

Vdt

dVout cos12

10

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

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

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Maximum Op Amp Swing

ii VVVV SR

To determine the maximum frequency before op amp slews, first determine the maximum swing the op amp can have and divide the slew rate by it.

2sin

2minmaxminmax VV

tVV

Vout

2minmax VVFP

Nonzero Output Resistance

1

0

1

RRR

RA

RR

v

vout

out

In practical op amps, the output resistance is not zero. It can be seen from the closed loop gain that the nonzero

output resistance increases the gain error.

2

10

2

2 1RR

ARRRv outin

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Review: Sinusoidal Analysis

• Any voltage or current in a linear circuit with a sinusoidal source is a sinusoid of the same frequency ().

W l d t k t k f th lit d d h h– We only need to keep track of the amplitude and phase, when determining the response of a linear circuit to a sinusoidal source.

• Any time‐varying signal can be expressed as a sum of sinusoids of various frequencies (and phases).

Applying the principle of superposition: Applying the principle of superposition:

– The current or voltage response in a linear circuit due to a time‐varying input signal can be calculated as the sum of the sinusoidal responses for each sinusoidal component of the input signal.

High Frequency “Roll‐Off” in Av

• Typically, an amplifier is designed to work over a limited range of frequencies.

At “hi h” f i th i f lifi d– At “high” frequencies, the gain of an amplifier decreases.

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Av Roll‐Off due to CL • A capacitive load (CL) causes the gain to decrease at high frequencies.

Th i d f C d hi h f i h– The impedance of CL decreases at high frequencies, so that it shunts some of the output current to ground.

LCmv Cj

RgA1

||

Frequency Response of the CE Stage

• At low frequency, the capacitor is effectively an open circuit, and Av vs. is flat.  At high frequencies, the impedance of the capacitor decreases and hence theimpedance of the capacitor decreases and hence the gain decreases.  The “breakpoint” frequency is 1/(RCCL).

222 Cm

v

RgA

1222 LCCR

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Amplifier Figure of Merit (FOM)• The gain‐bandwidth product is commonly used to benchmark amplifiers.  – We wish to maximize both the gain and the bandwidthWe wish to maximize both the gain and the bandwidth.

• Power consumption is also an important attribute.– We wish to minimize the power consumption.

CCC

LCCm

VI

CRRg

1

nConsumptioPower

BandwidthGain

LCCT

CCC

CVV

VI

1

nConsumptioPower

Operation at low T, low VCC, and with small CL superior FOM

Bode Plot

• The transfer function of a circuit can be written in the general form

jj

• Rules for generating a Bode magnitude vs. frequency plot:

21

210

11

11

)(

pp

zz

jj

jj

AjH

A0 is the low-frequency gainzj are “zero” frequenciespj are “pole” frequencies

g g g q y p

– As  passes each zero frequency, the slope of |H(j)| increasesby 20dB/dec.

– As  passes each pole frequency, the slope of |H(j)| decreases by 20dB/dec.

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Bode Plot Example

• This circuit has only one pole at ωp1=1/(RCCL); the slope of |Av|decreases from 0 to ‐20dB/dec at ωp1. 

LCp CR

11

• In general, if node j in the signal path has a small‐signal resistance of Rj to ground and a capacitance Cj to ground, then it contributes a pole at frequency (RjCj)

‐1

• The impedance of CL decreases at high frequencies, so that it shunts some of the output current to ground.

Av Roll‐Off due to CL

0LD

p CR

10

LDmv Cj

RgA1

||

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Pole Identification Example 1

0

inGp CR

11

LDp CR

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Pole Identification Example 2

0 0

inm

G

p

Cg

R

1||

11

LDp CR

12

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Pole Identification Example 3

inSp CR

11

LCp CR

12

High‐Frequency BJT Model

• The BJT inherently has junction capacitances which affect its performance at high frequencies.Collector junction depletion capacitance CCollector junction: depletion capacitance, CEmitter junction:  depletion capacitance, Cje, and also 

diffusion capacitance, Cb.

jeb CCC

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Next

• Frequency response