Open-circuit time constant method

The open-circuit time constant method is a 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 the largest contributions to time constants as a guide to 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: Simple RC circuit and auxiliary circuits to find time constants

Example: Simple RC network

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

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

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

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

Combining these relations the transfer function is found to be:

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

The linear term in jω in this transfer function can be derived by the following method.

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 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 R₁. Form the product C₁ R₁.
4. Add these terms.

In effect, it is as though each capacitor charge and discharges through the resistance of the circuit found 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 this example: suppose the time constants of this circuit are τ₁ and τ₂; that is:

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

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

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

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

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:

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

and

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

In other words, substituting the RC-values:

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

and

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

where ( ^ ) denotes the approximate result. Using these results, it is easy to explore how well the corner frequency (the 3dB frequency) is given by

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

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

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

Of course agreement is good when the assumption τ₁ >> τ₂ is accurate. The worst case is for τ₁ = τ₂. In this case τ^{^}₁ = 2 τ₁ and the corner frequency is a factor of 2 too small. In all other 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.

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