Library / Circuits, electronics and noise / ELN-26
Circuits, electronics and noise
Weighted inverting summing amplifier
Output of an ideal inverting amplifier whose input resistors meet at the inverting node, with the non-inverting input grounded.
- Ref
- ELN-26
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- Vo [V]
With the non-inverting input at ground the inverting node is a virtual ground, so each input resistor carries a current Vk/Rk into it and all the currents flow through Rf. The output is Vo = −Rf(V₁/R₁ + V₂/R₂). Each input is amplified by its own gain −Rf/Rk, with no interaction between inputs, which makes it ideal as a mixer. More inputs add more terms.
Output of an ideal inverting amplifier whose input resistors meet at the inverting node, with the non-inverting input grounded. Each input is scaled by Rf over its own resistor and the results are added and inverted.
Calculator
- Gain from input 1, −Rf/R₁−2 V/V
- Gain from input 2, −Rf/R₂−4 V/V
- Total current into the feedback resistor4×10−5 A
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The two inputs are weighted by −2 and −4, so 0.2 V and 0.1 V combine to −0.8 V.
- The weights are set only by resistors, so a DAC or mixer can be given binary-weighted gains by using resistors in a 1 : 2 : 4 ratio.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R1 Input resistor 1 | 10 to 1,000,000 Ω | −400.4 | −0.404 | 400 | +10% | −4.55 % |
| R2 Input resistor 2 | 10 to 1,000,000 Ω | −200.4 | −0.402 | 200 | +10% | −4.55 % |
| V2 Input voltage 2 | −5 to 5 V | −20.4 | 19.6 | 40 | +0.2 V | +100 % |
| Rf Feedback resistor | 10 to 1,000,000 Ω | −40 | −4e-4 | 40 | +10% | +10 % |
| V1 Input voltage 1 | −5 to 5 V | −10.4 | 9.6 | 20 | +0.2 V | +50 % |
Sweep of Input voltage 1
| Input voltage 1 (V) | Output voltage (V) |
|---|---|
| −5 | 9.6 |
| −3.75 | 7.1 |
| −2.5 | 4.6 |
| −1.25 | 2.1 |
| 0 | −0.4 |
| 1.25 | −2.9 |
| 2.5 | −5.4 |
| 3.75 | −7.9 |
| 5 | −10.4 |
Sweep one variable
Variables and how each one contributes
Output voltage
What the equation gives you. Everything else feeds it.
Feedback resistor
The feedback resistor, in Ω. Scales every gain, so the output size rises in direct proportion. Raising it by 10 % (to 22,000) moves the result by +10 %, from −0.8 to −0.88 V.
Input voltage 1
The first input voltage, in V. Moves the output by −Rf/R₁ times its change. Raising it by 10 % (to 0.22) moves the result by +5 %, from −0.8 to −0.84 V.
Input resistor 1
The first input resistor, in Ω. A larger R₁ reduces the gain from input 1, in inverse proportion. Raising it by 10 % (to 11,000) moves the result by −4.55 %, from −0.8 to −0.7636 V.
Input voltage 2
The second input voltage, in V. Moves the output by −Rf/R₂ times its change. Raising it by 10 % (to 0.11) moves the result by +5 %, from −0.8 to −0.84 V.
Input resistor 2
The second input resistor, in Ω. A larger R₂ reduces the gain from input 2, in inverse proportion. Raising it by 10 % (to 5,500) moves the result by −4.55 %, from −0.8 to −0.7636 V.
Worked example with the values above
- Input currents: V₁/R₁ = 2×10−5 A and V₂/R₂ = 2×10−5 A.
- Sum: 4×10−5 A.
- Vo = −20,000 × 4×10−5 = −0.8 V.
Assumptions and limits
- An ideal op amp with infinite gain and no input current, in its linear range with negative feedback.
- Output limited only by the supply, and signals within the useful bandwidth.
Common mistakes
- Forgetting the minus sign of an inverting stage.
- Using resistors that are not matched in a difference amplifier, which gives a common-mode error.
- Leaving out the bandwidth-limiting resistor in a differentiator, so that high-frequency noise is amplified.
Statistics and uncertainty
Resistor tolerances set the gain accuracy, so use 1 % parts or better for precision, and a matched network for the difference amplifier, where mismatch converts common-mode voltage into error. Real op amps add offset, bias current and finite gain.
Where it is used
- Signal mixing, scaling, integration and current-to-voltage conversion in instrumentation.
- Photodiode, sensor and ramp-generator circuits.
Maths you need
- Virtual ground: the inverting input sits at the non-inverting voltage
- Negative feedback: the output acts to cancel the input difference
- Kirchhoff’s current law: currents at a node add to zero