The general time- and transfer-constants (TTC) analysis[1] is the generalized version of the Cochran-Grabel (CG) method,[2] which itself is the generalized version of zero-value time-constants (ZVT), which in turn is the generalization of the open-circuit time constant method (OCT).
[3] While the other methods mentioned provide varying terms of only the denominator of an arbitrary transfer function, TTC can be used to determine every term both in the numerator and the denominator.
Its denominator terms are the same as that of Cochran-Grabel method, when stated in terms of time constants (when expressed in Rosenstark notation[4]).
Transfer constants are low-frequency ratios of the output variable to input variable under different open- and short-circuited active elements.
In general, a transfer function (which can characterize gain, admittance, impedance, trans-impedance, etc., based on the choice of the input and output variables) can be written as: The first denominator term
can be expressed as the sum of zero value time constants (ZVTs): where
is the time constant associated with the reactive element
Setting a capacitor value to zero corresponds to an open circuit, while a zero-valued inductor is a short circuit.
This is the essence of the ZVT method, which reduces to OCT when only capacitors are involved.
) is a capacitor, the time constant is given by and when element
The second-order denominator term is equal to: where the second form is the often-used shorthand notation for a sum that does not repeat permutations (e.g., only one of the permutations
, is simply the time constant associated with the reactive element
In general, any denominator terms can be expressed as: where
) are infinite valued (shorted capacitors and opened inductors).
Usually, the higher-order time constants involve simpler calculations, as there are more infinite valued elements involved during their calculations.
The major addition in the TTC over the Cochran-Grabel method is its ability to calculate all the numerator terms in a similar fashion using the same time constants used for the denominator calculation in conjunction with transfer constants, denoted as
Transfer constants are low-frequency gains (or, in general, ratios of the output to input variables) under different combinations of reactive elements being zero and infinite valued.
The notation uses the same convention, with all the elements whose indexes appear in the superscript of
, being infinite valued (shorted capacitors and opened inductors) and all unlisted elements zero-valued.
The zeroth order transfer constant
denotes the ratio of the output to input when all elements are zero-valued (hence the superscript of 0).
In particular: which is the transfer constant when all elements are zero-valued (e.g., dc gain).
The first order numerator term can be expressed as the sum of the products of first-order transfer constants
Similarly, the second-order numerator term is given by where, again, transfer constant,
th numerator term is given by: This allows for full calculation of any transfer function to any degree of accuracy by generating a sufficient number of numerator and denominator terms using the above expressions.