Designing with op amps: Single-formula technique keeps it simple

Dieter Knollman, PhD, Lucent Technologies -March 02, 1998

Numerous texts and application notes describe the design of op-amp circuits. Most designers use the RF/RI formula for inverting amplifiers and the 1+RF/RI formula for noninverting amplifiers, where RF and RI are the feedback and input resistors, respectively. These two formulas have been the basis of op-amp design for many years. Why learn a new procedure? One reason is that the new procedure is simple in that it uses the same formula--RF divided by the magnitude of the gain--for both positive and negative gains.

Second, the procedure is universal and works for all sets of gains. Although you can use classic methods to design op-amp circuits with multiple negative-gain inputs and a single positive-gain input, circuits with multiple positive-gain inputs are difficult to design using classic methods. Another reason to learn this new procedure is that it creates designs that always exhibit minimal bias-current error.

The design procedure relies on two new circuit concepts: a gain formula and a circuit theorem (see box "Theoretical background") and is based on the following definitions and assumptions:
• The procedure applies to single op-amp circuits with any number of positive- and negative-gain inputs, and the op amp is ideal.
• All inputs are ideal voltage sources.
• Ground acts as an input and is a voltage source of value zero.
• If the input has a source impedance, the source impedance is part of the circuit.
• Gain is defined as the gain from the ideal voltage source to the op-amp output.
• The feedback resistor, RF, connects from the op-amp output to the inverting input.
• The input resistor, Ri, connects from the ideal voltage source input to the op-amp input. Positive-gain inputs connect to the noninverting op-amp input terminal, and negative-gain inputs connect to the inverting op-amp input terminal. This definition differs from standard usage. In classic circuit analysis, the input resistor always connects to the inverting op-amp input. The Ri notation emphasizes the difference.
• The circuit's input impedance is equal to Ri only for negative inputs. For positive inputs, Ri is the impedance that connects the input to the noninverting op-amp terminal.
To implement the design, you must first specify the desired circuit gains. You then must calculate the ground gain, as follows:

Ground gain=1-the sum of the positive and negative gains from the desired circuit gains.

In other words, ground gain+ circuit gains=1. You must now choose a feedback-resistor value. For example, let RF=100 kiloohms. Next, calculate the resistor values for each input, including the ground-gain resistor, using Ri,G=RF/|gain|.

You can have any number of positive-gain and negative-gain inputs, and each input must have a defined gain. Ground gain=1-sum of gains. Some op-amp circuits need a resistor to ground from the op amp's inverting input, and others need a resistor to ground on the noninverting input. The sign of the ground gain determines where to place the ground resistor. For positive ground gains, you connect the resistor to the op amp's noninverting terminal; for negative ground gains, you connect the resistor to the inverting terminal. If the desired gains add up to one, a ground resistor is unnecessary.

The feedback resistor you choose can theoretically be any value, but op-amp and circuit considerations usually limit the range of acceptable values. Selecting a feedback-resistor value that is a multiple of the gains avoids fractional values for the input resistors. Calculate the input resistor values as RF divided by the magnitude of the gain for both positive and negative gains.

For those op-amp experts who have their doubts, consider an example: Design an inverting amplifier with a gain of 4. Thus, the circuit needs a ground gain equal to +5. Selecting RF=100 kiloohms yields an input-resistor value, Ri, of 25 kiloohms and a ground-gain input-resistor value, RG, of 20 kiloohms (Figure 1). The 4-gain input resistor connects to the op amp's inverting input. The +5 ground-gain resistor connects to the noninverting input. The design of this inverting amplifier is similar to that of a classic inverting amplifier. The new procedure adds a ground resistor at the noninverting op-amp input. This ground resistor is necessary only to minimize bias-current error.

To design an inverting amplifier with a gain of 4 and an RF of 100kΩ, you connect an Ri of 25 kΩ to the inverting input and an RG of 20kΩ to the noninverting input.

The noninverting op-amp input terminal sees a resistance of 20 kiloohms looking into the circuit. The resistance at the inverting op-amp input is the parallel combination of the 100-kiloohms feedback resistance and the 25-kiloohms input resistance. Because the parallel resistance of 25-kiloohms and 100-kiloohms is 20-kiloohms, both op-amp terminals see the same resistance looking into the circuit. The voltages that the op-amp bias current produces at the op-amp inputs are equal. The bias current creates only a common-mode input, which the op-amp rejects. Thus, the 20-kiloohms grounded input resistor minimizes the bias-current error for this example.

In another example, consider the design of a noninverting op-amp circuit with a gain of +3. Thus, the circuit needs a ground gain of 2. With RF=100 kiloohms, the input resistor, Ri, is 100 kiloohms/3=33.3 kiloohms, and the grounded input resistor, RG, is 100 kiloohms/2=50 kiloohms (Figure 2). The 33.3-kiloohms resistor is necessary only to minimize the bias-current error. Note that 33.3 kiloohms is equal to the parallel resistance of 50 and 100 kiloohms.

A noninverting amplifier with a gain of +3 and an RF of 100 kiloohms requires an Ri of 33.3 kiloohms connected between the input and the noninverting terminal and an RG of 50 kiloohms connected between the inverting terminal and ground.

Classic design states that the gain of a noninverting amplifier is 1+RF/RI. In this case, RI of the classic case is equivalent to the ground-gain resistor, RG. Thus, the ground gain of Figure 2 is equal to RF/RG. Adding the input gain (+3) and ground gain (2) yields the gain sum of 1. For noninverting amplifiers, the positive-gain input resistor, Ri, is necessary only to minimize the bias-current error.

In another design example, consider a differential amplifier with gains of +1 and 1, corresponding to inputs VIN1 and VIN2, respectively. Thus, the circuit needs a ground gain of +1. With RF=100 kiloohms, all resistor values--Ri1, Ri2, and RG--are 100 kiloohms (Figure 3). In this circuit, RG acts as an attenuator for the positive-gain input, VIN1. Note that the equivalent resistance at each op-amp input terminal is the same.

A differential amplifier with gains of +1 and –1 requires a ground gain of +1. With RF=100 kiloohms, all input resistors, including the ground resistor, are 100 kiloohms.

You can check this design using the classic formulas. The gain from the op amp's inverting input is 100 kiloohms/100 kiloohms=1. The gain from VIN1 to the op amp's noninverting input is 0.5. The gain from the op amp's noninverting input to the output is 1+RF/Ri2=2. Thus, the gain from VIN1 to the output is 0.5×2=1.

In another example, design a summing amplifier with gains of +3, 5, and +7. The presence of two positive gains makes this example difficult to design using classic methods. The input gains are +3, +7, and 5, which add up to +5. Thus, a ground gain of 4 is necessary. With RF=420 kiloohms, positive-gain input-resistor Ri1 is 420 kiloohms/3=140 kiloohms, and positive-gain input-resistor Ri2 is 420 kiloohms/7=60 kiloohms. The negative-gain input resistor, Ri3, is 420 kiloohms/5=84 kiloohms. The ground-gain input resistor is 420 kiloohms/4=105 kiloohms (Figure 4).

The procedure makes it easy to design an op-amp circuit with multiple positive-gain inputs.

Note that the op amp's input terminals see the same equivalent resistance: 140 kiloohms||60 kiloohms=105 kiloohms||84 kiloohms||420 kiloohms. The gain from VIN1 to the op amp's noninverting input is 60 kiloohms/(60 kiloohms+140 kiloohms)=0.3. The gain from the noninverting input to the output is equal to 1+RF/(Ri3||RG)=1+420 kiloohms/(84 kiloohms||105 kiloohms)=10. Thus, the gain from VIN1 to the output is 0.3x10=3. You can verify the gains from VIN2 and VIN3 to the output in a similar manner.

 Theoretical background The design technique uses a new op-amp gain formula, Plato's gain formula, and a new circuit theorem, Daisy's theorem.Plato's gain formula isgain=pxRF/Ri,where RF is the feedback resistance, Ri is the resistor that connects the input to the op amp, and p is a circuit constant for positive-gain inputs and always equal to −1 for negative-gain inputs. For negative gains, Plato's gain formula is simply the classic RF/RI gain formula.For positive gains, this formula differs from classic analysis. In classic analysis, the positive gain is the gain from the source to the noninverting op-amp terminal times 1+RF/RI.Alternatively, Plato's gain formula states that you can express positive gains as the product of a circuit-constant p and RF/Ri. The value of p is the same for all positive gains in a given circuit. Different circuits may have different values for p. Don't assume p=1 for existing designs. For new designs, setting p=1 simplifies the formula and creates a better design. The only value that minimizes bias-current error is p=1.Daisy's theorem, which you use to determine the ground gain, is as follows:The sum of the gains=1.The design procedure assumes that the circuit inputs are ideal voltage sources and that ground is a voltage input of value 0. The overall gain is the gain from the ideal voltage source to the op-amp output. For new op-amp circuits, Daisy's theorem specifies whether a ground resistor is necessary at the op amp's noninverting or inverting input and specifies the magnitude of the ground gain.

Author's biography
Dieter Knollman, PhD, is a distinguished member of the technical staff of Lucent Technologies (Denver), where he has worked for 33 years. In his current position, he designs PBX port circuits. Knollman earned a BSEE from the Virginia Polytechnic Institute and State University (Blacksburg, VA) an MSEE from the University of Illinois--Urbana/Champaign, and a PhD from New York University (New York).