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Some time ago I found a very useful EDN article of Dieter Knollman called "Designing with op amps: Single-formula technique keeps it simple". The article describes a method that greatly simplifies the design of complex single OPAMP configurations, where multiple voltages are summed together or subtracted. The method can also be used for the simple single OPAMP configurations such as the difference amplifier, the non-inverting and the inverting amplifier. The method works as long as the output of the OPAMP is fed back to the inverting input, meaning that there is negative feedback.
Because I think it is a very interesting method, I want to give it some extra attention here.
The method is based on 1 simple rule : GROUND gain = 1 - (positive gain + negative gain)
We only need to select a negative feedback resistor value. The rest of the resistor values and where to connect them is calculated using the rule above.
Note that the method only works for amplifiers using negative feedback.
Sounds fair enough, but what does it mean ? What is ground gain ? To understand ground gain, we will treat ground as an input of zero volts. So, ground gain is the gain of the input that is connected to ground. The positive gain is the gain of the non-inverting input, and the negative gain is the gain of the inverting input.
An example will make things more clear :
Suppose we want an OPAMP configuration that gives us an overall gain of -9 using 1 input, and we want to use a feedback (output to inverting input) resistor of 9K. The negative overall gain means that the OPAMP will be used as an inverting amplifier.
The above ground gain formula can now be used to determine the values of R1 and R2 and how R1 and R2 will be connected.
The method states that : GROUND gain = 1 - (positive gain + negative gain).
Since we only want a negative gain of -9, there is no positive gain, so the positive gain is 0.
This means that GROUND gain = 1 - (0 - 9) = 10, which is a positive number. The fact that the GROUND gain is a positive number, means that we need to connect the ground to the non-inverting input using a resistor that gives us a gain of 10 → Rf / 10 = 9K / 10 = 900E.
Because we want an overall negative gain of -9 using 1 input, we need to connect the inverting input to the input signal that we want to amplify using a resistor that gives us a gain of 9 → Rf / 9 = 9K / 9 = 1K.
Using the method, we constructed an inverting OPAMP amplifier with a gain of -9. Note that the method even takes care of the impedance matching for both OPAMP inputs, so the input bias currents are balanced.
Another example :
Suppose we want an OPAMP configuration that gives us an overall gain of +3 using 1 input, and we want to use a feedback (output to inverting input) resistor of 10K. The positive gain means that the OPAMP will be used as a non-inverting amplifier.
The method states that : GROUND gain = 1 - (positive gain + negative gain)
Since we only want a positive overall gain of 3, there is no negative gain, so the negative gain is 0.
This means that GROUND gain = 1 - (3 - 0) = -2, which is a negative number. The fact that the GROUND gain is a negative number, means that we need to connect the ground to the inverting input using a resistor that gives us a gain of 2 → Rf / 2 = 10K / 2 = 5K.
Because we want an overall positive gain of +3, we need to connect the non-inverting input to the input signal that we want to amplify using a resistor that gives us a gain of 3 => Rf / 3 = 10K / 3 = 3K33.
Using the method, we constructed a non-inverting OPAMP amplifier with a gain of 3. Note that the method even takes care of the impedance matching for both OPAMP inputs, so the input bias currents are balanced.
A more difficult example :
Suppose we want an OPAMP configuration for a differential amplifier with a gain of 10 and we want to use a feedback resistor of 100K.
Since we want to subtract 2 signals, we need 2 input signals. This implies that we need 2 separate gains, besides the ground gain. We need a positive gain of 10 and a negative gain of -10, so the 2 input signals are subtracted and amplified.
The method states that : GROUND gain = 1 - (positive gain + negative gain)
Since we have 2 inputs with respectively a positive and a negative gain of 10, the GROUND gain = 1 - (10 - 10) = 1, which is a positive number. The fact that the GROUND gain is a positive number, means that we need to connect the ground to the non-inverting input using a resistor that gives us a gain of 1 → Rf / 1 = 100K / 1 = 100K.
The positive gain of 10 means that we need to connect the non-inverting input to the first input signal using a resistor of Rf / 10 = 100K / 10 = 10K.
The negative gain of -10 means that we need to connect the inverting input to the second signal using a resistor of Rf / 10 = 100K / 10 = 10K
Using the method, we constructed a differential amplifier with a gain of 10.
A more complex example :
Suppose we want an OPAMP configuration with a gain of 1 that subtracts 2 voltages from another voltage and using a feedback resistor of 100K.
Lets says we want to subtract 3V and 1V from 8V, so the output of the circuit is 8V - 3V - 1V = 4V.
For the 8V input, we need a positive gain of +1.
For the 3V and the 1V input, we need a negative gain of -1.
The method states that : GROUND gain = 1 - (positive gain + negative gain)
We need a gain of -1 for the 3V, a gain of -1 for the 1V input, and we need a gain of +1 for the 8V input. So the GROUND gain will be = 1 - (1 - 1 - 1) = +2, which is a positive number. The fact that the GROUND gain is a positive number, means that we need to connect the ground to the non-inverting input using a resistor that gives us a gain of 2 → Rf / 2 = 100K / 2 = 50K.
Since we have 3 signal inputs and the gain for each input needs to be 1 or -1, the resistors for each of the inputs will be Rf / 1 = 100K.
The inputs that need a negative gain are connected each via a 100K resistor to the inverting input of the OPAMP and the inputs that need a positive gain are connected each via a 100K resistor to the non-inverting input of the OPAMP.
The examples are kept rather simple for the sake of clarity, but the method allows calculating the resistors for more complex configurations with different gains for each input and without needing pages full of calculations. Especially when different gains for the individual signal sources are needed, calculating it in a normal way can be quite tedious and error-prone.
With this method, it becomes a piece of cake.
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