Open-circuit time constant method

Figure 1: Simple RC circuit and auxiliary circuits to find time constants

Figure 1 shows a simple RC low-pass filter. Its transfer function is found using Kirchhoff's current law as follows. At the output,

${\displaystyle {\frac {V_{1}-V_{O}}{R_{2}}}=j\omega C_{2}V_{O}\ ,}$

where V₁ is the voltage at the top of capacitor C₁. At the center node:

${\displaystyle {\frac {V_{S}-V_{1}}{R_{1}}}=j\omega C_{1}V_{1}+{\frac {V_{1}-V_{O}}{R_{2}}}\ .}$

Combining these relations the transfer function is found to be:

${\displaystyle {\frac {V_{O}}{V_{S}}}={\frac {1}{1+j\omega \left(C_{2}(R_{1}+R_{2})+C_{1}R_{1}\right)+(j\omega )^{2}C_{1}C_{2}R_{1}R_{2}}}}$

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.

1. Set the signal source to zero.
2. Select capacitor C₂, replace it by a test voltage V_X, and replace C₁ by an open circuit. Then the resistance seen by the test voltage is found using the circuit in the middle panel of Figure 1 and is simply V_X / I_X = R₁ + R₂. Form the product C₂ ( R₁ + R₂ ).
3. Select capacitor C₁, replace it by a test voltage V_X, and replace C₂ by an open circuit. Then the resistance seen by the test voltage is found using the circuit in the right panel of Figure 1 and is simply V_X / I_X = R₁. Form the product C₁ R₁.
4. Add these terms.

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. This was originally developed and proven by calculating the co-factors of the admittance matrix by Thornton and Searle.[2] A more intuitive inductive proof of this (and other properties of TTC) was later developed by Hajimiri.[3]

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.[4]

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:

${\displaystyle \left(1+j\omega {\tau }_{1})(1+j\omega {\tau }_{2}\right)=1+j\omega \left(C_{2}(R_{1}+R_{2})+C_{1}R_{1}\right)+(j\omega )^{2}C_{1}C_{2}R_{1}R_{2}}$

Comparing the coefficients of the linear and quadratic terms in jω, there results:

${\displaystyle \tau _{1}+\tau _{2}=C_{2}(R_{1}+R_{2})+C_{1}R_{1}\ ,}$

${\displaystyle \tau _{1}\tau _{2}=C_{1}C_{2}R_{1}R_{2}\ .}$

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:

${\displaystyle \tau _{1}+\tau _{2}\approx \tau _{1}\ ,}$

and

${\displaystyle \tau _{2}={\frac {\tau _{1}\tau _{2}}{\tau _{1}}}\approx {\frac {\tau _{1}\tau _{2}}{\tau _{1}+\tau _{2}}}\ .}$

In other words, substituting the RC-values:

${\displaystyle \tau _{1}\approx {\hat {\tau _{1}}}=\ \tau _{1}+\tau _{2}=C_{2}(R_{1}+R_{2})+C_{1}R_{1}\ ,}$

and

${\displaystyle \tau _{2}\approx {\hat {\tau _{2}}}={\frac {\tau _{1}\tau _{2}}{\tau _{1}+\tau _{2}}}={\frac {C_{1}C_{2}R_{1}R_{2}}{C_{2}(R_{1}+R_{2})+C_{1}R_{1}}}\ ,}$

where ( ^ ) denotes the approximate result. As an aside, notice that the circuit time constants both involve both capacitors; in other words, in general the circuit time constants are not decided by any single capacitor. Using these results, it is easy to explore how well the corner frequency (the 3 dB frequency) is given by

${\displaystyle f_{3dB}={\frac {1}{2\pi {\hat {\tau _{1}}}}}\ ,}$

Figure 2: Comparison of OCTC estimates with actual pole positions

as the parameters vary. Also, the exact transfer function can be compared with the approximate one, that is,

${\displaystyle {\frac {1}{(1+j\omega \tau _{1})(1+j\omega \tau _{2})}}\ }$ ${\displaystyle \ }$   with   ${\displaystyle \ }$ ${\displaystyle \ {\frac {1}{(1+j\omega {\hat {\tau _{1}}})(1+j\omega {\hat {\tau _{2}}})}}\ .}$

Of course agreement is good when the assumption τ₁ >> τ₂ is accurate.

Figure 2 illustrates the approximation. The x-axis is the ratio τ₁ / τ₂ on a logarithmic scale. An increase in this variable means the higher pole is further above the corner frequency. The y-axis is the ratio of the OCTC (open-circuit time constant) estimate to the true time constant. For the lowest pole use curve T_1; this curve refers to the corner frequency; and for the higher pole use curve T_2. The worst agreement is for τ₁ = τ₂. In this case τ^{^}₁ = 2 τ₁ and the corner frequency is a factor of 2 too small. The higher pole is a factor of 2 too high (its time constant is half of the real value).

In all cases, the estimated corner frequency is closer than a factor of two from the real one, and always is conservative that is, lower than the real corner, so the actual circuit will behave better than predicted. However, the higher pole always is optimistic, that is, predicts the high pole at a higher frequency than really is the case. To use these estimates for step response predictions, which depend upon the ratio of the two pole frequencies (see article on pole splitting for an example), Figure 2 suggests a fairly large ratio of τ₁ / τ₂ is needed for accuracy because the errors in τ^{^}₁ and τ^{^}₂ reinforce each other in the ratio τ^{^}₁ / τ^{^}₂.

The open-circuit time constant method focuses upon the corner frequency alone, but as seen above, estimates for higher poles also are possible.

Application of the open-circuit time constant method to a number of single transistor amplifier stages can be found in Pittet and Kandaswamy.[5]