Let's design a circuit architecture that performs the square root function. The output voltage should be equal to the square root of the input voltage.
Available blocks for creating our circuit are only the four arithmetic operators.
This might seem an academic exercise, and may be it is, but it will show an approach that proved to be efficient in many cases and it will question the specification, a very common situation.
For simplicity, let's assume that input and output voltages are positive and let's ignore the considerations on input and output resistances and voltage ranges.
To be exact, for units consistency reasons, we should write:
Where A is a dimensional constant. A = 1 V ^(1/2)
As VIN is supposed to be positive, we can write:
A very common approach when a circuit must achieve such a condition is to rewrite the equation in another way:
When a difference must be equal to zero, a very common implementation is to create a loop with a large gain so that the input difference is constantly kept very small.
Of course, for the loop to operate properly, it has to be DC and AC stable, what we will assume first and demonstrate later on when the design is viable.
To go one step further, let's define that amplifier has gain G and offset VIO.
The loop equation the writes:
Solving for VOUT brings:
Which can be simplified, assuming VOUT is positive:
It can be checked that if G is large and VIO is small:
This circuit has only two parameters to be sized;
These two parameters affect circuit accuracy. In turn they can be defined related to the circuit accuracy specification that is then required. Accuracy can be specified in terms of acceptable error in the circuit operation
Error can be expressed as the difference between actual output and target
As we have two parameters to size, we need two equations to solve in order to calculate the two sizable parameters. So our specification so far is not sufficient to size the circuit. We need two specification items related to accuracy.
The error can be mentally separated in two terms, one that applies when input is zero, one that depends on input voltage.