Summing amplifier calculator: inverting and non-inverting summer

What sets a summing amplifier's output

A summing amplifier adds voltages. In its usual, inverting form it is an inverting amplifier with more than one input resistor: every input drives a current through its own resistor into the op amp's inverting input, and the op amp holds that node at ground by pulling the same total current out through the feedback resistor. TI's Handbook of Operational Amplifier Applications (SBOA092) puts it in one line: "Current in the feedback loop is the algebraic sum of the current due to each input. Each source, E1, E2, etc., contributes to the total current, and no interaction occurs between them" (p. 23). The output is that current times R_F, inverted, so each input arrives at the output multiplied by −R_F/R_k, its own gain, set by its own resistor. The formula in the reference section above is the whole of it.

Three properties follow, and all three are what make the circuit useful. The gains are independent: changing R₁ changes input 1's weight and nothing else. AN-20, TI's applications guide for op amps, calls this the circuit's advantage: "there is no interaction between inputs and operations such as summing and weighted averaging are implemented very easily" (p. 6). The input impedance of each input is its own resistor; SBOA092 p. 24: "Each input “sees” its respective input resistor as the input resistance." And the op amp's inputs never move: SBOA272, TI's inverting-summer cookbook circuit, notes that "The common-mode voltage of an inverting amplifier is equal to the voltage connected to the non-inverting node, which is ground in this design" (p. 1), so the common-mode range of the op amp is never in question.

The price is the inversion, and one thing that is easy to miss: the input resistors are all, in effect, in parallel at the summing node, and that parallel combination is what the op amp's loop sees. It sets the noise gain, and with it the bandwidth, for every input at once. The next sections come back to it.

The non-inverting summer turns the arrangement round. The inputs meet at the non-inverting input through their resistors, which average them, weighted by the conductances 1/R_k; the op amp then amplifies that average by the non-inverting gain 1 + R_F/R_G. AN-31, TI's amplifier circuit collection, prints the two-input case in Figure 1-6 (p. 5), with every resistor 10 kΩ, so the average (V₁ + V₂)/2 is doubled and the output is the plain sum. It keeps the sign, at the cost of every property of the inverting circuit listed above: each input's weight depends on every other input resistor, the input impedance of one input depends on the others, and the op amp's inputs follow the weighted average, so its common-mode range matters.

Noise gain and bandwidth: why adding an input slows all of them

An op amp with a constant gain-bandwidth product closes its loop at GBW divided by the noise gain, and for the inverting summer the noise gain is 1 + R_F/(R₁ ∥ … ∥ R_n). SBOA272 is explicit about which gain to use: "Be sure to use the noise gain (NG), or non-inverting gain, of the circuit. When calculating the noise gain note that R1 and R2 are in parallel." AN-20 says it for three inputs, bandwidth "may be calculated as in the inverting amplifier shown in Figure 1 by assuming the input resistor to be the parallel combination of R1, R2, and R3" (p. 6), where the inverting amplifier's bandwidth "is equal to the unity-gain frequency divided by one plus the closed-loop gain" (p. 3).

Because the noise gain does not depend on which input is driven, every input has the same bandwidth, whatever its own gain. SBOA272's simulation shows it: its inputs have gains of 0.98 V/V and 9.76 V/V, and "the bandwidth is the same for either input. This is because the bandwidth depends on the noise gain of the circuit, not the signal gain of each input" (p. 4). The table runs the formula for SBOA092's adder, every resistor 10 kΩ, and its averager, R_F = R/n, on the 1.2 MHz op amp SBOA272 uses.

A unity-gain adder of n inputs has a noise gain of n + 1: four inputs at unity gain cost as much bandwidth as a single inverting stage with a gain of −4. The averager, whose feedback resistor shrinks with the number of inputs, stays at a noise gain of 2. An input that is wired but unused, its resistor grounded, still counts: add a third, unity-gain input to SBOA272's circuit and its noise gain goes from 11.73 to 12.73, its bandwidth from 102 kHz to 94.3 kHz. For the non-inverting summer, the input network sits outside the loop and the noise gain is the stage gain 1 + R_F/R_G, AN-20's non-inverting bandwidth, "the amplifier unity-gain frequency divided by the closed-loop gain" (p. 4).

Worked example: TI SBOA272's two-input summer

SBOA272 designs an inverting summer for two signals of very different size: input 1 spans −2.5 V to 2.5 V, input 2 spans −250 mV to 250 mV, and the output is to use −4.9 V to 4.9 V on ±5 V rails, at up to 10 kHz, with an OPA170 (1.2 MHz, 0.4 V/µs, rail-to-rail output). Step 1 is to "Select a reasonable resistance value for R3", 20 kΩ. The rest follows from one decision in step 2: "For this design, half of the output swing is devoted to each input." Each gain is then half the output span over the input's span, each input resistor is R_F over its gain, and the calculator's design mode, whose defaults are these goals, reproduces every step.

step 2   |G1| = ((4.9 − (−4.9))/2) / (2.5 − (−2.5))   = 0.980 V/V, −0.175 dB
step 3   R1 = 20 kΩ / 0.98                            = 20.4 kΩ → 20.5 kΩ (E96)
step 4   |G2| = ((4.9 − (−4.9))/2) / (0.25 − (−0.25)) = 9.80 V/V, 19.82 dB
step 5   R2 = 20 kΩ / 9.8                             = 2.04 kΩ → 2.05 kΩ (E96)
gains    −20 kΩ/20.5 kΩ  ·  −20 kΩ/2.05 kΩ            = −0.9756  ·  −9.756 V/V
step 6   NG = 1 + 20 kΩ / (20.5 kΩ ∥ 2.05 kΩ = 1.86 kΩ) = 11.732 V/V, 21.39 dB
         BW = 1.2 MHz / 11.732                        = 102 kHz
step 7   SR > 2π × 10 kHz × 4.9 V                     = 0.308 V/µs
step 8   1 / (2π × 6 pF × 1.70 kΩ)                    = 15.6 MHz
output   full scale: −2.44 V + −2.44 V                = −4.88 V

SBOA272 prints 0.98 V/V and −0.175 dB, 20.4 kΩ ≈ 20.5 kΩ (Standard Value), 9.8 V/V and 19.82 dB, and 2.04 kΩ ≈ 2.05 kΩ. Its step 6 gives "NG = 1 + 20 kΩ / 1.86 kΩ = 11.75 V/V = 21.4 dB" and "BW = 1.2 MHz / 11.75 V/V = 102 kHz". The small difference in the noise gain is rounding: SBOA272 rounds R₁ ∥ R₂ to 1.86 kΩ before dividing, and 1 + 20/1.86 is 11.753, where the unrounded1.86 kΩ gives 11.732. Both are 21.4 dB and both round to 102 kHz and 102 kHz, SBOA272's 102 kHz. "This requirement is met because the closed-loop bandwidth is 102kHz and the design goal is 10kHz."

Step 7 prints "SR > 2 × π × 10 kHz × 4.9 V = 307.87 kV/s = 0.31 V/µs"; unrounded it is 307.876 kV/s, so the printed figure is truncated rather than rounded, and the OPA170's 0.4 V/µs meets it. Step 8 checks that the zero the input resistors make with the op amp's input capacitance lies above the bandwidth, with C_cm and C_diff of 3 pF each and R₁ ∥ R₂ ∥ R₃ = 1.7 kΩ, giving "15.6 MHz > 102 kHz"; the calculator has 15.6 MHz. The numeric line of that equation prints the capacitance as "3 pF × 3 pF"; its symbolic line has C_cm + C_diff, and only the sum, 6 pF, gives 15.6 MHz.

SBOA272's simulations check the rounded design. Sweeping either input over its range with the other at 0 V, "The output is inverted and ranges from –2.44V to 2.44V" (p. 3): 2.44 V to −2.44 V for input 1 through 20.5 kΩ and 2.44 V to −2.44 V for input 2, each a little inside the 2.45 V the unrounded resistors would give, because rounding both resistors up lowered both gains. The AC simulation reports 0.98 V/V (−0.21 dB) and 9.76 V/V (19.79 dB), which are 0.976 (−0.21 dB) and 9.756 (19.79 dB) from the standard values, and a bandwidth of 114.86 kHz against 102 kHz calculated, which SBOA272 says "correlate well". With both inputs at full scale the output reaches −4.88 V, inside the ±4.9 V goal: −4.88 V to 4.88 V over every combination of the two ranges. That is the default the calculator opens with in its analysis mode.

Adders, averagers and the non-inverting summer: the classic circuits

SBOA092's section on "Summing and averaging amplifiers" (pp. 64–66) opens: "Voltages are summed by applying the signals to the same input of the amplifier. Amplifying, averaging, etc., may be accomplished by input resistor scaling. Inputs are effectively isolated from each other. Any number of inputs may be used in each of these circuits." Its circuits, with AN-31's two, run through the calculator as follows. The output column puts 1 V on every input and ignores the rails, to show the scale.

Two of the printed lines need a word. SBOA092's scaling adder writes its result as "−100(100E1 + 10E2 + E3)", but its own first form, R₀/R₁·E₁ + …, with 100 kΩ over 1, 10 and 100 kΩ, is −(100E1 + 10E2 + E3), which is what the calculator gives; the leading 100 does not follow from the circuit. Its weighted average sets the feedback resistance to R₁ ∥ R₂ ∥ R₃ (5.45 kΩ, a 5.1 kΩ resistor with a 1 kΩ trimmer) so that the weights add to one, and prints them rounded: 16.4/30, 8.2/30 and 5.4/30 against 0.545, 0.273, 0.182 exactly.

AN-20 describes the gain of an input to its three-input summer as "the ratio of the appropriate input resistor to the feedback resistor, R4" (p. 6). Its Figure 4 on the same page has it the other way up, V_OUT = −R4(V1/R1 + V2/R2 + V3/R3), and the equation is the one that agrees with SBOA272, AN-31 and SBOA092.

The direct-addition circuit is the non-inverting summer with a third input resistor taken to ground. Grounding it is not incidental: it sets the average to (E₁ + E₂)/3, which the gain of 3 turns back into the sum. The same page gives the input impedance, "Zin = 3/2 R2 = 15 kΩ for each input": 10 kΩ in series with the other two 10 kΩ legs in parallel, which the calculator reports as 15.0 kΩ. Add the same grounded 10 kΩ leg to AN-31's Figure 1-6 without changing its gain and its output for 1 V on each input drops from 2.00 V to 1.33 V: in the non-inverting summer, every resistor on the + input is part of every input's weight.

Source impedance: where the two summers differ most

SBOA272's design description says the inputs "typically come from low-impedance sources because the input impedance of this circuit is determined by the input resistors", and its note 2 asks for input resistors "large when compared to the output impedance of the source". The reason is that a source's output impedance adds to its input resistor. What it does next depends on the circuit.

Take two 1 V inputs through 10 kΩ each and put 1 kΩ of source impedance in series with input 1 (a derived example). In the inverting adder with R_F = 10 kΩ, input 1's gain falls from −1.000 to −0.909, −9.1 %, and input 2's does not move:−1.000. In AN-31's non-inverting summer the same 1 kΩ moves both: input 1's gain goes to 0.952 and input 2's to 1.048, because the weights are shares of one total and must still add to one. An error in one source's impedance shows up as a gain error on a different input. AN-20's "no interaction between inputs" is a property of the inverting summer only.

Where the model stops being valid

• Open-loop gain. The ideal gains hold while the noise gain is small against the op amp's open-loop gain; AN-20 gives the inverting gain "when this ratio is small compared with the amplifier open-loop gain" (p. 3). Near the bandwidth the gain falls away, and the bandwidth is the noise-gain figure above for every input.
• Output swing. SBOA272 note 1: "Use the op amp in a linear operating region. Linear output swing is usually specified under the AOL test conditions." When the sum reaches the swing limit the output stops and the stage is no longer linear; the calculator flags it and holds the output at the limit.
• Slew rate. SBOA272 note 6: "Large signal performance may be limited by slew rate." The check is its step 7, SR > 2π f V_p, for a sine; the slew rate calculator shows what the output becomes when it fails.
• Large resistors and input capacitance. SBOA272 note 3: "Using high-value resistors can degrade the phase margin of the circuit and introduce additional noise in the circuit." Its step 8 is the quantitative form, and note 5 the remedy: "Adding a capacitor in parallel with R3 will also improve stability of the circuit if high-value resistors are used."
• Capacitive loads. SBOA272 note 4: "Avoid placing capacitive loads directly on the output of the amplifier to minimize stability issues." The op amp capacitive load calculator covers the isolation resistor.
• Offset and bias current. AN-20 notes for the inverting amplifier that "there will be an offset voltage at the amplifier output equal to closed-loop gain times the offset voltage at the amplifier input" (p. 3), and gives the resistor that balances bias current for the summer, R5 = R1 ‖ R2 ‖ R3 ‖ R4. None of that is in the output figure here, which is ideal; the op amp error calculator works it through.
• Common-mode range, non-inverting summer only. Its op amp inputs sit at the weighted average of the inputs. SBOA272 lists the OPA170's input range as "(Vee-0.1V) to (Vcc-2V)" (p. 5); an average above V+ − 2 V would be outside it, and the calculator warns when the + input leaves that window on the rails entered.

Common mistakes

• Taking the bandwidth from the signal gain. An input with a gain of 0.98 V/V in SBOA272's design has the same 102 kHz as the one with 9.76 V/V, not the 1.2 MHz/0.98 its own gain would suggest.
• Forgetting the unused inputs. A grounded or spare input resistor still sits in parallel at the summing node, raises the noise gain and costs bandwidth; in a non-inverting summer it also changes every other input's weight.
• Designing for each input alone. With half the swing each, SBOA272's two inputs take the output to its limit only when both are at full scale at once; give each input the whole swing and the sum clips as soon as two of them are large. Design mode's share weights split the swing and report the range the rounded resistors actually reach.
• Centring a unipolar input. The inverting summer maps 0 V in to 0 V out, so a 0 V to 5 V input fills only one half of a ±4.9 V output. Centring it takes an offset: another input, driven from a reference.
• Driving the summer from a high-impedance source, and reading the source's output impedance as part of the input resistor only in the inverting circuit. In the non-inverting one it moves every weight.
• Checking the slew rate at the input's amplitude. It is the output that slews: SBOA272 uses the 4.9 V output peak, not either input's.

Further reading

• TI SBOA272, Analog Engineer's Circuit: Inverting Summer Circuit — the design goals (p. 1), the eight design steps with the noise gain, bandwidth, slew rate and input-capacitance checks (p. 2), and the DC, AC and transient simulations (pp. 3–4).
• TI SNLA140, AN-31 Amplifier Circuit Collection — the inverting summing amplifier of Figure 1-5 and the non-inverting one of Figure 1-6 (p. 5).
• TI SNOA621, AN-20 An Applications Guide for Op Amps — the inverting and non-inverting amplifiers with their bandwidths and bias-current resistors (pp. 3–5) and the summing amplifier (p. 6).
• TI SBOA092, Handbook of Operational Amplifier Applications — the voltage adder and scaling summer (pp. 23–24), the summer derived from the summing-point restraints (p. 63), and the adders and averagers (pp. 64–66).
• Op amp gain calculator — the single-input inverting and non-inverting stages the two summers are built on, with noise gain and bandwidth.
• Instrumentation amplifier gain calculator — when the job is the difference of two inputs rather than their sum.
• E-series resistor calculator — the nearest standard value and the best pair for a ratio, where design mode's single-resistor rounding is not close enough.
• Slew rate calculator — full-power bandwidth and what a slew-limited output looks like.