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ib, iboff, Aol
The advanced op amp tackles errors beyond the basic that can be critical for your amplifier design.
We'll perform an analysis for each error. You'll get
For tutorials on Key Concepts and other circuits, goto EBA Series
Get a refresh of the Basic Amplifier.
We'll start with basic error definitions of an amplifier block. What are Offset and Gain Errors?
The max budget (target spec) for amplifier has been chosen as:
This topic walks through critical errors highlighted in yellow below.
Note: For errors voff, R1_Tol & R2_Tol, check out analysis already covered in the Basic Amplifier.
Description Initial Drift OFFSET ERRORS voff, Input Offset Voltage
ib, Input Bias Current
iboff, Input Offset Current0.1 mV
5 nA
5 nA10 uV / C
1 nA / C
1 nA / CGAIN ERRORS R2_Tol, Resistor Tolerance
R1_Tol, Resistor Tolerance
U1 Aol, Open-Loop Gain0.1 %
0.1 %
80dB, 10000 (V/V)100 ppm / C
100 ppm / C
Temperature
Amplifier
Errors
While the steps may seem more detailed than needed for simpler errors, the value of creating a systematic approach will pay off when analyzing more complex, multi-stage designs.
Modern op amps achieve low bias currents using a technique called "input bias current cancellation". The current into the pos / neg inputs has two components
ibp = ib + iboff / 2
ibn = ib - iboff / 2
where ib is the average of two currents
ib = (ibp + ibn) / 2
and iboff is the offset between the two currents
iboff = ibp - ibn
The output due to both ibp and ibn can be written
vo = ibp*Rs*(R2/R1+1) - ibn*R2
Applying some math, we can find the Gain (Sensitivity) of vo to ib only
S = vo/ib = Rs*(R2/R1+1) - R2
Note, if you can make Rs = R1||R2, then S = 0 and vo = 0V. Yes that's good news for older devices where ib >> iboff.
However, for newer devices where ib has a similar magnitude to iboff, the output error could be worse for Rs = R1||R2. Best to keep Rs small as possible.
Similary, we can find the Gain (Sensitivity) of vo to iboff only
S = vo/ib = ½ [ Rs*(R2/R1+1) + R2 ]
Note that R2 carries a + sign. No way to get S = 0 or vo = 0V.
The quick theory refresh (above) showed the Gain (Sensitivity) of vo to ib only
S = vo/ib = Rs*(R2/R1+1) - R2
Let's walk through the error analysis.
Description | Initial Errors | Drift Errors |
Error Source: e | ib = 5nA | ib_TC = 1nA/C |
Pick Analysis Node: Va | vo | vo |
Calc Sensitivity: S How does e impact Va? |
S = vo / ib = Rs*(R2/R1+1)-R2 = -8000 |
S = -8000 |
Calc Offset Error at Analysis Node Initial: ∆Voffset = e * S Drift: ∆Voffset = e * ∆T * S |
∆Voffset = 5nA * -8000 = -0.4mV |
∆Voffset = 1nA/C*30C*-8000 = -2.4mV |
Calc Gain from Input to Analysis Node: Ka = Va / Vin |
Ka = vin/vo = R2/R1+1 = 10 |
Ka = 10 |
Calc Error RTI (Referred-to-Input): ∆voffset_RTI = ∆voffset / Ka |
∆voffset_RTI = -0.40mV / 10 = -0.04mV |
∆voffset_RTI = -2.4mV / 10 = -0.24mV |
The quick theory refresh (above) showed the Gain (Sensitivity) of vo to iboff only
S = vo/iboff = ½ [ Rs*(R2/R1+1) + R2 ]
Let's walk through the error analysis.
Description | Initial Errors | Drift Errors |
Error Source: e | iboff = 5nA | ibp_TC = 1nA/C |
Pick Analysis Node: Va | vo | vo |
Calc Sensitivity: S How does e impact Va? |
S = vo / ib = ½ [Rs*(R2/R1+1) + R2] = 50000 |
S = 50000 |
Calc Offset Error at Analysis Node Initial: ∆Voffset = e * S Drift: ∆Voffset = e * ∆T * S |
∆Voffset = 5nA * 50000 = 0.25mV |
∆Voffset = 1nA/C*30C*50k = 1.5mV |
Calc Gain from Input to Analysis Node: Ka = Va / Vin |
Ka = vin/vo = R2/R1+1 = 10 |
Ka = 10 |
Calc Error RTI (Referred-to-Input): ∆voffset_RTI = ∆voffset / Ka |
∆voffset_RTI = 0.25mV / 10 = 0.025mV |
∆voffset_RTI = 1.5mV / 10 = 0.15mV |
Gain errors often require more effort when calculating the Sensitivity S.
You need to write the gain equation and then apply calculus (Difference
Method) to find S.
Ideally, we expect the gain of the Non-Inverting Amplfier to be
K = vo/vin = R2/R1+1
However, the actual gain can be found by
K' = Aol / (1+Aol*B)
where Aol is the Open Loop Gain of the Op Amp and B is the feedback factor
B = vn/vo = R1 / (R1+R2)
If A is large, then the equation approaches the familiar equation
K ≈ 1/B = (R1+R2) / R1
What is the error between the actual K' and ideal K? With some basic math, you can find the error as
∆K/K = (K' - K) / K ≈ 1 / (Aol*B)
Let's apply our theory refresh (above) to finding the impact of Aol on Gain Error.
Description | Initial Errors |
Error Source: e Aol = 80 (dB) = 1080/20 (V/V) = 10000 (V/V) B = vn/vo = R1/(R1+R2) = 10k/(10k+90k) = 0.1 |
1/(Aol*B) = 1/(10000*0.1) = 0.001 |
Pick Analysis Node: Va | vo |
Calc Sensitivity: S How does e impact Gain K? |
S = 1.0 The op amp gain, directly impacts the gain at vo |
Calc Gain Error at Analysis Node ∆K/K = e * S |
∆K/K = 0.001*1.0 = 0.001 = 0.1% |
Normailzed gain errors can be referred to input as-is, no RTI calc needed. |
An Excel file was created to implement the error budget analysis.
3 Worksheets
Worksheet Enter Calculate CIRCUIT CALC Circuit values Signal gains, levels and error Sensitivities OFFSET Offset error sources Offset errors and totals GAIN Gain error Sources Gain errors and totals
While 3 worksheets seems over-the-top for smaller circuits, you'll find a big advantage when analyzing more complex circuits or multi-stage systems!
Check out the easy entry (BLU col) and calculations (RED col) on the Offset Error sheet.
Explore the hands-on spreadsheet!
For tutorials and other examples, goto EBA Series