1

Fully Understanding CMRR in DAs, IAs, and OAs

Pete Semig

Analog Applications Engineer-Precision Linear

2

Outline

• Definitions– Differential-input amplifier

– Common-mode voltage

– Common-mode rejection ratio (CMRR)

– Common-mode rejection (CMR)

• CMRR in Operational Amplifiers

• CMRR in Difference Amplifiers

• CMRR in Instrumentation Amplifiers

• CMRR in ‘Hybrid’ Amplifiers

3

Differential Input Amplifier

• Differential input amplifiers are devices/circuits that can input and amplify differential signals while suppressing common-mode signals– This includes operational amplifiers, instrumentation amplifiers, and

difference amplifiers

Difference

Amplifier

Instrumentation AmplifiersOperational

Amplifier

4

Common-Mode Voltage

• For a differential input amplifier, common-mode voltage is defined as the average of the two input voltages. [2]

-

++

Vp

+

Vn-

+Vo

Vcm=Vp+Vn

2

5

Common-Mode Voltage (Alternate defn.)

• For a differential amplifier, common-mode voltage is defined as the average of the two input voltages. [2]

gain mode-Common

gain mode-alDifferenti

where

cm

dm

cmcmiddmout

A

A

VAVAV

-

+Vout

Vcm

-

+Vid

Vid/2Vid/2

-

+

IOP1

Vcm=Vp+Vn

2

w here

Vp=Vcm+V id

2

Vn=Vcm-V id

2

6

Common-Mode Voltage

• Ideally a differential input amplifier only responds to a differential input voltage, not a common-mode voltage.

V-

V+V-

V+

Vb 0 Va 0

-

+Vid

Vs+ 5

Vs- 5

Vcm 0

-

+ +3

2

6

74 OP1

-

+Vo 0V

0V

V-

V+

Vb 0 Va 0

-

+Vid

Vcm 1

-

+ +3

2

6

74 OP1

-

+Vo 0V

0V

V-

V+

Vb 0 Va 1m

-

+Vid

Vcm 1

-

+ +3

2

6

74 OP1

-

+Vo 3.826745V

1000uV

7

CMRR and CMR

• Common-Mode Rejection Ratio is defined as the ratio of the differential gain to the common-mode gain

• CMR is defined as follows [2]:

• CMR and CMRR are often used interchangeably

cm

dm

A

ACMRR

CMRRdBCMR 10log20

8

Ideal Differential Amplifier CMRR

• What is the CMRR of an ideal differential input amplifier (e.g. op-amp)?

• Recall that the ideal common-mode gain of a differential input amplifier is 0.

• Voltage Amplifier Model [1]

• Also recall the differential gain of an ideal op-amp is infinity.

• So

cm

dm

cm

dmOAideal A

A

A

ACMRR

Vs

Rs

Ri-

+

-

+

VCVS

-

+Vi Vi

Ro

Rload-

+Vo

Source Amplifier

Adm->Infinity

Load

9

Real Op-Amp CMRR

• In an operational amplifier, the differential gain is known as the open-loop gain.

• The open-loop gain of an operational amplifier is fixed and determined by its design

10

Real Op-Amp CMRR

• However, there will be a common-mode gain due to the following– Asymmetry in the circuit

• Mismatched source and drain resistors• Signal source resistances• Gate-drain capacitances• Forward transconductances• Gate leakage currents

– Output impedance of the tail current source

– Changes with frequency due to tail current source’s shunt capacitance

• These issues will manifest themselves through converting common-mode variations to differential components at the output and variation of the output common-mode level. [4]

11

Resistor Mismatch

• Let’s look at the case of a slight mismatch in drain resistances [4] in the input stage (diff-in, diff-out) of an op-amp

• What happens to Vx and Vy as Vin,cm changes?

• Assuming M1 and M2 are identical, Vx and Vy will change by different amounts:

• This imbalance will introduce a differential component at the output

• So changes in the input common-mode can corrupt the output signal

12

Transistor Mismatch

• What about mismatches with respect to M1 and M2?

– Threshold mismatches– Dimension mismatches

• These mismatches will cause the transistors to conduct slightly different currents and have unequal transconductances.

• We find the conversion of input common mode variations to a differential error by the following factor [4]

121

SSmm

DmDMCM Rgg

RgA

13

Tail Current Source Capacitance

• As the frequency of the CM disturbance increases the capacitance shunting the tail current source will introduce larger current variations. [4]

OPA333

14

Modeling CMRR

• Now that we understand what CMRR is and what affects it in operational amplifiers, let’s see how it can affect a circuit.

• First, however, we need to understand the model

• To be useful, CMRR needs to be referred-to-input (RTI)

• We can therefore represent it as a voltage source (aka offset voltage) in series with an input. The magnitude (RTI) is Vcm/CMRR [2]

-

+Vo

Vcm/CMRR

+

Vn

+

Vp

-

+

15

OA CMRR Error

• Example: non-inverting buffer

CMRRV

V

A

ACMRR

A

V

V

CMRRAVAV

CMRR

AVAVAVV

CMRR

VVVAV

VVCMRR

VVV

VnVpAV

p

O

p

O

pO

pOpO

pOpO

Ocm

cmOn

O

11

As

1

11

111

that Note

Vn

-

+Vo

Vcm/CMRR

+

Vp

-

+

A

16

Real CMRR Example

• To understand the effects CMRR can have at the output of a device, let’s look at an example.

• OPA376 PDS– Notice the Vcm is specified at the top of the page

– Deviation from this value will induce an offset error

– Remember CMRR is RTI

17

Real CMRR Example

• Remember

• In reality, CMRR is measured by changing the input common-mode voltage and observing the output change.

– For an operational amplifier, this is usually done with a composite amplifier

• It is then referred-to-input by dividing by the gain and can be though of as an offset voltage

• From reference [3], in TI datasheets CMRR is defined as follows so that the value is positive

)(log20)( 10 CMRRdBCMR

os

cm

V

VCMRR

18

Real CMRR Example

• For the OPA376, CMRR(min)=76dB. Note this is really CMR!

uVV

V

V

V

V

VdB

CMRRdBCMR

os

os

cm

os

cm

5.1585.6309

1

modecommon in change 1V aFor

5.630910

log2076

)(log20)(

20

76

10

10

19

CMRR of Difference Amplifiers

• A difference amplifier is made up of a differential amplifier (operational amplifier) and a resistor network as shown below.

• The circuit meets our definition of a differential amplifier

• The output is proportional to the difference between the input signals

-

+

R3 R4

R1 R2

+

V1

+

V2

-

+Vo

Ri1

Ri2

Ro

20

DA CMRR

• Let’s replace V1 and V2 with our alternate definition of the inputs (in terms of differential-mode and common-mode signals)

• It is readily observed that an ideal difference amplifier’s output should only amplify the differential-mode signal…not the common-mode signal.

dmo

dmcm

dmcmo

o

dmcm

dmcm

VR

RV

VV

VV

R

RV

VVR

RV

VVV

VVV

1

2

1

2

121

2

2

1

22

2

2

-

+

R1 R2

R1 R2

-

+Vo

Vcm+

Vdm/2

+Vdm/2

21

DA CMRR

• This assumes that the operational amplifier is ideal and that the resistors are balanced.

• Keeping the assumption that the operational amplifier is ideal, let’s see what happens when an imbalance factor (ε) is introduced.

-

+

R1 R2

R1

-

+Vo

Vcm+

Vdm/2

+Vdm/2 R2(1-)

22

DA CMRR

• Using superposition we find that

• After some algebra we find that [1]

• As expected, an imbalance affects the differential and common-mode gains, which will affect CMRR!

• As the error->0, Adm->R2/R1 and Acm->0.

1

11

2

1

2 21

2

21

2

1

2

RR

R

RR

RVV

R

RVVV dm

cmdm

cmo

21

2

21

21

1

2

2

21

where

RR

RA

RR

RR

R

RA

VAVAV

cm

dm

cmcmdmdmo

23

DA CMRR

• Since we have equations for Acm and Adm, let’s look at CMR

• If the imbalance is sufficiently small we can neglect its effect on Adm

• With that and some algebra we find [1]

21

2

21

21

1

2

1010

22

1

log20log20)(

RRR

RRRR

RR

A

AdBCMR

cm

dm

1

2

10

1

log20)(RR

dBCMR

24

DA CMRR

• This equation shows two very important relationships

– As the gain of a difference amplifier increases (R2/R1), CMR increases– As the mismatch (ε) increases, CMR decreases

• Please remember that this just shows the effects of the resistor network and assumes an ideal amplifier

1

2

10

1

log20)(RR

dBCMR

25

-

+

R1 R2

R1 R2

-

+Vo

Vcm+

Vdm/2

+Vdm/2

DA CMRR

• Another possible source for CMRR degradation is the impedance at the reference pin.

• So far we have connected this pin to low-impedance ground.

• Placing and impedance here will disturb the voltage divider we come across during superposition analysis.

• This will negatively affect CMR

26

Real DA CMRR Example (INA149 PDS)

27

Why not make our own DA?

• If a DA is simply an operational amplifier and 4 resistors, I can save money by making my own, right?

• Should be well-matched

• Should have low temperature drift

-

+

R2 25kR1 25k

R3 25k R4 25k -

+Vout

+

Vcm

0% 0%

0% 0%

T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-319.09

-319.09

-319.09

28

Why not make our own DA?

• Let’s assume an ideal amplifier and just look at resistor mismatches using TINA (only changing R2)

• Monte Carlo analysis

• Gaussian distribution (6σ), 100 cases

• Values are negative due to TINA

Assuming 0% tolerance for R1, R3, and R4 and only 0.1% tolerance for R2 this network can degrade CMRR to 66dB (calculated), 69.16dB (simulated).

-

+

R2 25kR1 25k

R3 25k R4 25k -

+Vout

+

Vcm

0% 0.1%

0% 0%T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-140.00

-120.00

-100.00

-80.00

-60.00

29

-

+

R2 150kR1 150k

R3 150k R4 150k -

+Vout

+

Vcm

0.1%

0.1%0.1%

0.1%

T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-125.86

-93.35

-60.84

Why not make our own DA?

• What if all resistors are 0.01% or 0.1%?

T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-119.74

-100.84

-81.93

-

+

R2 25kR1 25k

R3 25k R4 25k -

+Vout

+

Vcm

0.01% 0.01%

0.01% 0.01%

Worse performance than all of our DAs

30

T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-100.00

-80.00

-60.00

-40.00T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-120.00

-100.00

-80.00

-60.00

-40.00

Why not make our own DA?0.5%: 52dB (calc), 53.64dB (sim) 1.0%: 46dB (calc), 46.85dB (sim)

5.0%: 32dB (calc), 33.34dB (sim)T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-100.00

-80.00

-60.00

-40.00

-20.00

31

0402

06030805

1206

SO-8

Why not make our own DA?

• 80dB: Lowest cost of one 0.01%, 10ppm/C resistor (1k pricing)– 1206 package: $0.45 ($1.80 total cost)– 0805 package: $0.53 ($2.12 total cost)– 0603 package: $0.53 ($2.12 total cost)– 0402 package: $0.50 ($2.00 total cost, 10k pricing!)

• 60dB: Lowest cost 4-pack 0.1%, 25ppm/C resistor (1k pricing)– SO-8 package: $0.98 ($0.98 total cost)

• Footprint size comparison:

4 required

1 required

(need op amp)

32

Why not make our own DA?

• Now that we understand how the resistor matching can affect CMRR and the related cost, what about an integrated solution?

• TI can trim resistors to within 0.01% relative accuracy

• INA152 – CMR(min)=80dB– GE=10ppm/˚C (max)– On-chip resistors will drift together– MSOP-8– 1k price on www.ti.com: $1.20– Includes amplifier!

• Some DA’s can give CMR(min)=74dB @ $1.05!

• Customer will require 2 suppliers (1 for OA, 1 for precision resistors)

Op amp included!

MSOP-8

SO-8

33

DA Gain

• We learned that the gain of a difference amplifier is set by R2 and R1.

• What if we wanted variable gain?

• We would have to adjust 2 resistors due to the topology.– To retain good CMR they would have to be tightly matched, too.– This is difficult and expensive

• Alternately, you could use an external operational amplifier (with very low output impedance so as not to degrade CMR) to drive the reference pin as shown below [4]

1231

2 vvRR

RRv Go

34

DA Gain

• But, R3 should be a precision resistor. Its error will be seen as a gain error.

• You also need to purchase an external operational amplifier and potentiometer.

• If you need variable gain, there are better options– Instrumentation amplifiers (IAs) usually have an external resistor that can be used to

set the gain– Programmable Gain Amplifiers (PGAs) can be programmed (either with pin settings

or digitally) with a particular gain

• In summary, difference amplifiers are typically manufactured with a set gain so as to preserve CMR and since there are alternate (better) solutions for variable gain

• Since difference amplifiers come with a fixed gain, you will only see 1 CMR curve in the datasheet

35

Difference Amplifiers-Summary

• Pros:– Difference amplifiers amplify differential signals and reject common-mode signals– The common-mode rejection is based mainly resistor matching– Making your own difference amplifier will not yield the same performance– Difference amplifiers can be used to protect against ground disturbances

• Cons:– Externally changing the gain of a difference amplifier is not worthwhile– The input impedance is finite

• This means that a difference amplifier will load the input signals• If the input signal source’s impedances are not balanced, CMR could be degraded

• Is there a way we can amplify differential signals, change the gain, retain high CMR, and not load our source?

• Yes! Buffer the inputs…this creates an Instrumentation Amplifier (IA).

36

Instrumentation Amplifier

• There are 2 common types of instrumentation amplifiers– 2 op-amp (e.g. INA122)

– 3 op-amp (e.g. INA333)

37

Instrumentation Amplifier

• Notice both have gain equations so you can vary the gain

• Notice the input impedance is that of the non-inverting terminal of a non-inverting amplifier

Difference Amp

High-Z Nodes

High-Z Nodes

Variable Gain

38

IA CMRR

• So, what is the CMRR of an instrumentation amplifier?

• Instrumentation amplifiers reject common-mode signals (Acm->0)

• Recall

• CMRR is directly related to differential gain. Since we can change the differential gain of an IA, we also change the CMRR.

cm

dm

A

ACMRR

39

INA826 CMRR Model Verification

+V 15

++

-Rg

Rg

RefU1 INA826

V1 15

-

+Vout

Rg 1k

+

Vcm

T

G1

G10

G100

G1000

Frequency (Hz)

10 215 5k 100k

Ga

in (

dB

)

0

20

40

60

80

100

120

140

160

G1

G10

G100

G1000

40

INA826-Effects of Rg Tolerance on CMRR

• Now that we see our INA826 model is accurate, let’s look at the effects of Rg’s tolerance on CMRR

• Set G=100, 6σ resistors, 100 cases.– Note that due to the number of cases, no post-processing was performed

– Normally this would be Gain/Waveform. Therefore we have to mentally subtract 20dB from this cluster of waveforms.

T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-88.13

-81.16

-74.19

-87.97dB<CMR<-88.13dBAdjusted for gain:-107.97dB<CMR<-108.13dB

5% Resistor

T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-88.08

-81.13

-74.19

-88.04dB<CMR<-88.07dBAdjusted for gain:-108.04dB<CMR<-108.07dB

1% Resistor

T

Frequency (Hz)

10.00 1.00k 100.00k

Gai

n (d

B)

-88.07

-81.13

-74.19

-88.065531dB<CMR<-88.06869dBAdjusted for gain:-108.065531dB<CMR<-108.06869dB

0.1% Resistor

Notice the gain setting resistor tolerance does

not significantly affect the CMR.

41

2-OA Instrumentation Amplifiers

• What are the properties of 2-OA Instrumentation Amplifiers?

• Pros– Lower cost (only 2 op-amps), less trimming– High impedance input– Can be placed in a smaller package

• Cons– Compare signal path to Vo for Vin+ and Vin-– Vin+ has a shorter path than V-– This delay does not allow the common-mode

components to cancel each other as well as frequency increases

– Therefore CMR degradation occurs earlier in frequency than the 3-OA designs

Since we can change the differential gain, the CMR also

changes.

42

‘Hybrid’ Difference Amplifiers

• Some devices have unique topologies (e.g. INA321).

• How do we determine whether CMRR will change with the ‘gain’ of this device?

2OA Instrumentation

Amp

Op-amp (has fixed differential

gain)

43

‘Hybrid’ Difference Amplifiers

• Depends on what ‘gain’ you’re talking about.

• With respect to CMRR, it’s all about the differential gain since the common-mode gain of all differential amplifiers is ideally 0.

• When you place resistors for R1 and R2, are you changing the differential gain?

44

• No. The differential gain of the device is set internally!

• If you can’t change the differential gain of the device, the CMRR will not change with gain.

• Remember the differential gain of an op-amp (A3) is fixed (it’s the open-loop gain)

‘Hybrid’ Differential Amplifiers

45

Real IA CMR Competitive Analysis

46

Summary

• A ‘differential amplifier’ amplifies differential signals, not common-mode signals– Examples include operational amplifiers, difference amplifiers, and

instrumentation amplifiers

• CMRR is defined as the ratio of differential gain to common-mode gain

• All differential amplifiers have an ideal common-mode gain of 0

• To determine if a circuit’s CMRR is going to change with gain, you must look at the differential gain. Remember an op-amp’s differential gain is fixed.

• If you can change the differential gain of the device/circuit, the CMRR will also change

47

References

• [1] Franco, “Design with Operational Amplifiers and Analog Integrated Circuits”, 3rd Edition, McGraw-Hill, 2002.

• [2] Tobey, Graeme, Huelsman, “Operational Amplifiers: Design and Applications”, McGraw-Hill, 1971.

• [3] Karki, “Understanding Operational Amplifier Specifications”, White Paper: SLOA011, Texas Instruments, 1998.

• [4] Razavi, “Design of Analog CMOS Integrated Circuits”, McGraw-Hill, 2001.

48

Questions?