### A Deeper Look into Difference Amplifiers

```A Deeper Look into
Difference Amplifiers
In a 1991 article, Ramón Pallás-Areny and John Webster showed
that the common-mode rejection, assuming a perfect op amp, is
By Harry Holt
where Ad is the gain of the difference amplifier and t is the resistor
tolerance. Thus, with unity gain and 1% resistors, the CMRR
is 50 V/V, or about 34 dB; with 0.1% resistors, the CMRR is
500 V/V, or about 54 dB—even given a perfect op amp with infinite
common-mode rejection. If the op amp’s common-mode rejection
is high enough, the overall CMRR is limited by resistor matching.
Some low-cost op amps have a minimum CMRR in the 60 dB to
70 dB range, making the calculation more complicated.
CMRR ≅
Introduction
The classic four-resistor difference amplifier seems simple, but
many circuit implementations perform poorly. Based on actual
encountered with discrete resistors, filtering, ac common-mode
rejection, and high noise gain.
College electronics courses illustrate applications for ideal op
amps, including inverting and noninverting amplifiers. These are
then combined to create a difference amplifier. The classic four
resistor difference amplifier, shown in Figure 1, is quite useful
and has been described in textbooks and literature for more than
40 years.
R2
V1
R1
U1
V2
R3
R4
0.1
1%
V2
R1
5k
V
R4
150k
Figure 1. Classic difference amplifier.
R1 + R2 
R2

 ×V2 –
V1
R1
 R1 
(1)
With R1 = R3 and R2 = R4, Equation 1 simplifies to  R2 
VOUT =   (V2 − V1)
 R1 
(2)
This simplification occurs in textbooks, but never in real life, as the
resistors are never exactly equal. In addition, other modifications
of the basic circuit can yield unexpected behavior. The following
examples come from real application questions, although they have
been simplified to show the essence of the problem.
CMRR
An important function of the difference amplifier is to reject signals
that are common to both inputs. Referring to Figure 1, if V2 is
5 V and V1 is 3 V, for example, then 4 V is common to both. V2
is 1 V higher than the common voltage, and V1 is 1 V lower. The
difference is 2 V, so the “ideal” gain of R2/R1 would be applied to
2 V. If the resistors are not perfect, part of the common-mode voltage
will be amplified by the difference amplifier and appear at VOUT as
a valid difference between V1 and V2 that cannot be distinguished
from a real signal. The ability of the difference amplifier to reject this
is called common-mode rejection (CMR). This can be expressed
as a ratio (CMRR) or converted to decibels (dB).
Analog Dialogue 48-02, February (2014)
VOUT
U1
VOS = 1.25mV OVER TEMPERATURE
VOL = 35mV MAX OVER TEMPERATURE
Figure 2. Low-side sensing with high noise gain.
The transfer function of this amplifier is
 R4 
 ×
VOUT = 
R3 + R4
R2
150k
R3
5k
R1 – R4 ARE 0.5%
V
(3)
Low Tolerance Resistors
R5
V3
4t
The first suboptimal design, shown in Figure 2, was a low-side
current sensing application using an OP291. R1 through R4 were
discrete 0.5% resistors. From the Pallás-Areny paper, the best
CMR would be 64 dB. Luckily, the common-mode voltage is very
close to ground, so CMR is not the major source of error in this
application. A current sense resistor with 1% tolerance will cause
1% error, but this initial tolerance can be calibrated or trimmed.
The operating range was more than 80°C, however, so the
temperature coefficient of the resistors must be taken into account.
VOUT
For very low value current shunts, use a 4-terminal, Kelvin sense
resistor. With a high-accuracy 0.1-Ω resistor, make the connections
directly to the resistor, as a few tenths of an inch of PCB trace can
easily add 10 mΩ, causing more than 10% error. But the error gets
worse; the copper trace on the PCB has a temperature coefficient
greater than 3000 ppm.
The value of the sense resistor must be chosen carefully. Higher
values develop larger signals. This is good, but power dissipation
(I 2R) increases, and could reach several watts. With smaller
values, in the milliohm range, parasitic resistance from wires or
PCB traces can cause significant errors. To reduce these errors,
Kelvin sensing is usually employed. A specialized 4-terminal
resistor (Ohmite LVK series, for example) can be used, or the PCB
layout can be optimized to use standard resistors, as described in
“Optimize High-Current Sensing Accuracy by Improving Pad
Layout of Low-Value Shunt Resistors.” For very small values, a
PCB trace can be used, but this is not very accurate, as explained
in “The DC Resistance of a PCB Trace.”
Commercial 4-terminal resistors, such as those from Ohmite
or Vishay, can cost several dollars or more for 0.1% tolerance
with very low temperature coefficients. A complete error budget
analysis can show where the accuracy can be improved for the
least increase in cost.
One complaint regarding a large offset (31 mV) with no current
through the sense resistor was caused by a “rail-to-rail” op amp
that couldn’t swing all the way to the negative rail, which was tied
to ground. The term rail-to-rail is misleading: the output will
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1
get close to the rail—a lot closer than classical emitter follower
output stages—but will never quite reach the rail. Rail-to-rail op
amps specify a minimum output voltage, VOL , of either VCE(SAT)
or R DS(ON) × I LOAD, as described in “MT-035: Op Amp Inputs,
Outputs, Single-Supply, and Rail-to-Rail Issues.” With a
1.25-mV offset and a noise gain of 30, the output will be
1.25 mV × 30 = ±37.5 mV due to VOS, plus 35 mV due to VOL .
Depending on the polarity of VOS, the output could be as big as
72.5 mV with no load current. With a max VOS of 30 µV and a
maximum VOL of 8 mV, a modern zero-drift amplifier, such as the
AD8539, would reduce the total error to the point that the error
due to the sense resistor would dominate.
Single Capacitor Roll-Off
The example shown in Figure 5 is a little more subtle. So far, all
of the equations focused on the resistors; but, more correctly, the
equations should have referred to impedances. With the addition
of capacitors, either deliberate or parasitic, the ac CMRR depends
on the ratio of impedances at the frequency of interest. To roll off
the frequency response in this example, capacitor C2 was added
across the feedback resistor, as is commonly done for inverting
op amp configurations.
C2
R1
10k
Another Low-Side Sensing Application
R2
60k
R1
5k
VPLUS
R3
5k
0.5%
RSENSE
0.1
U1
R1 – R3 ARE 0.1%
VOUT
R4
60k
Figure 3. Low-side sensing, example 2.
V1
R3
10k
C4
R4
10k
VOUT
V3
V2
V
V
Figure 5. Attempt to create a low-pass response.
To match the impedance ratios Z1 = Z3 and Z2 = Z4, capacitor C4
must be added. It’s easy to buy 0.1% or better resistors, but even
0.5% capacitors can cost more than \$1.00. At very low frequencies
the impedance may not matter, but a 0.5-pF difference on the two
op amp inputs caused by capacitor tolerance or PCB layout can
degrade the ac CMR by 6 dB at 10 kHz. This can be important
if a switching regulator is used.
or AD8276, have much better ac CMRR because the two inputs
of the op amp are in a controlled environment on the die, and
the price is often lower than that of a discrete op amp and four
precision resistors.
Capacitor Between the Op Amp Inputs
High Noise Gain
The design shown in Figure 4 attempts to measure high-side
current. The noise gain is 250. The OP07C op amp specifies
150-µV max VOS. The maximum error is 150 µV × 250 = 37.5 mV.
To improve this, use the ADA4638 zero-drift op amp, which
specifies 12.5 µV offset from –40°C to +125°C. With high noise
gains, however, the common-mode voltage will be very close to the
voltage across the sense resistor. The input voltage range (IVR)
for the OP07C is 2 V, meaning that the input voltage must be at
least 2 V below the positive rail. For the ADA4638, IVR = 3 V.
R1
1k
R2
250k
R3
1k
U1
To roll off the response of the difference amplifier, some designers
attempt to form a differential filter by adding capacitor C1 between
the two op amp inputs, as shown in Figure 6. This is acceptable for
in-amps, but not for op amps. VOUT will move up and down to close
the loop through R2. At dc, this isn’t a problem, and the circuit
behaves as described in Equation 2. As the frequency increases, the
reactance of C1 decreases. Less feedback is delivered to the op amp
input, so the gain increases. Eventually, the op amp is operating
open loop because the inputs are shorted by the capacitor.
R2
3.3k
R1
1k
C1
VOUT
R4
250k
Figure 4. High-side current sensing.
2
U1
VIN
The next example, shown in Figure 3, had a lower noise gain, but
it used a low-precision quad op amp, with 3-mV offset, 10-µV/°C
offset drift, and 79 dB CMR. An accuracy of ±5 mA over a 0-A
to 3.6-A range was required. With a ±0.5% sense resistor, the
required ±0.14% accuracy cannot be achieved. With a 100-mΩ
resistor, ±5 mA through creates a ±500-µV drop. Unfortunately,
the op amp’s offset voltage over temperature is ten times greater
than the measurement. Even with VOS trimmed to zero, a 50°C
change would consume the entire error budget. With a noise gain
of 13, any change in VOS will be multiplied by 13. To improve
performance, use a zero-drift op amp, such as the AD8638,
precision sense resistor.
270pF
R2
10k
R3
1k
100pF
U1
VOUT
R4
3.3k
Figure 6. Input capacitor decreases high-frequency feedback.
Analog Dialogue 48-02, February (2014)
On a Bode plot, the open-loop gain of the op amp is decreasing
at –20 dB/dec, but the noise gain is increasing at +20 dB/dec,
resulting in a –40 dB/dec crossing. As taught in control systems
class, this is guaranteed to oscillate. As a general guideline: never
use a capacitor between the inputs of an op amp. (There are very
few exceptions, but they won’t be covered here.)
Conclusion
The four-resistor difference amplifier, whether discrete or
monolithic, is widely used. To achieve a solid, production
worthy design, carefully consider noise gain, input voltage range,
impedance ratios, and offset voltage specifications.
References
Kitchin, Charles and Counts, Lew. A Designer’s Guide to
Instrumentation Amplifiers, 3rd edition. 2006. Page 2-1.
Miller, Eric M. The DC Resistance of a PCB Trace.
O’Sullivan, Marcus. “Optimize High-Current Sensing Accuracy
by Improving Pad Layout of Low-Value Shunt Resistors.”
Analog Dialogue, Volume 46, Number 2, 2012.
Analog Dialogue 48-02, February (2014)
Pallás-Areny, Ramón and Webster, John G. Common Mode
Rejection Ratio in Differential Amplifiers. IEEE Transactions
On Instrumentation and Measurement, Volume 40, Number 4,
August 1991. Pages 669–676.
MT-035 Tutorial. Op Amp Inputs, Outputs, Single-Supply,
and Rail-to-Rail Issues.
Author
Harry Holt [[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */] is a
senior applications engineer in ADI’s Central
Applications Group. Previously, he worked
five years in the Precision Amplifiers Group,
following 27 years in both field and factory
applications at National Semiconductor, where
he was responsible for a variety of products,
including data converters, op amps, references,
audio codecs, and FPGAs. Harry has a BSEE degree from San
Jose State University; he is a Life Member of Tau Beta Pi and a
Senior Member of the IEEE.
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