Open-circuit time constant method
The open-circuit time constant method is an approximate analysis technique used in electronic circuit design to determine the corner frequency of complex circuits. It also is known as the zero-value time constant technique. The method provides a quick evaluation, and identifies the largest contributions to time constants as a guide to the circuit improvements.
The basis of the method is the approximation that the corner frequency of the amplifier is determined by the term in the denominator of its transfer function that is linear in frequency. This approximation can be extremely inaccurate in some cases where a zero in the numerator is near in frequency.
The method also uses a simplified method for finding the term linear in frequency based upon summing the RC-products for each capacitor in the circuit, where the resistor R for a selected capacitor is the resistance found by inserting a test source at its site and setting all other capacitors to zero. Hence the name zero-value time constant technique.
Figure 1 shows a simple RC low-pass filter. Its transfer function is found using Kirchhoff's current law as follows. At the output,
where V1 is the voltage at the top of capacitor C1. At the center node:
Combining these relations the transfer function is found to be:
The linear term in jω in this transfer function can be derived by the following method, which is an application of the open-circuit time constant method to this example.
In effect, it is as though each capacitor charges and discharges through the resistance found in the circuit when the other capacitor is an open circuit.
The open circuit time constant procedure provides the linear term in jω regardless of how complex the RC network becomes. For a complex circuit, the procedure consists of following the above rules, going through all the capacitors in the circuit. A more general derivation is found in Gray and Meyer.
So far the result is general, but an approximation is introduced to make use of this result: the assumption is made that this linear term in jω determines the corner frequency of the circuit.
That assumption can be examined more closely using the example of Figure 1: suppose the time constants of this circuit are τ1 and τ2; that is:
Comparing the coefficients of the linear and quadratic terms in jω, there results:
One of the two time constants will be the longest; let it be τ1. Suppose for the moment that it is much larger than the other, τ1 >> τ2. In this case, the approximations hold that:
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