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# Schmitt Trigger Astable Multivibrator

*Created: Sep, 2015*

Schmitt trigger inverter can be used to generate square waves as follows (picture taken from this post):

The question is what is the frequency of this astable multivibrator?

Assume we have a Schmitt trigger inverter with output high voltage of V_OH, output low voltage of V_OL, rising edge threshold voltage of V_T+, and falling edge threshold voltage of V_T-. In the steady state, one cycle of oscillation goes as follows:

- Initial condition: V_out is at V_OL, and then input voltage V_in falls to V_T-
- V_out becomes V_OH, which charges the input capacitor towards V_T+
- V_in rises to V_T+ again, making V_out switch to V_OL again

From State 1 to 2, input voltage V_in can be written as

\[ V_{in} = V_{OH} - (V_{OH} - V_{T-}) \exp(-\frac{t}{\tau}) \]

This process ends when V_in reaches V_T+. The duration T_1 can be got by solving

\[ V_{T+} = V_{OH} - (V_{OH} - V_{T-}) \exp(-\frac{T_1}{\tau}) \]

yielding

\[ T_1 = \tau \ln \frac{V_{OH} - V_{T-}}{V_{OH} - V_{T+}}\]

From State 2 to 3, input V_in is

\[ V_{in} = V_{OL} + (V_{T+} - V_{OL}) \exp(-\frac{t}{\tau}) \]

The process ends when V_in becomes V_T-. The duration T_2 can be solved from

\[ V_{T-} = V_{OL} + (V_{T+} - V_{OL}) \exp(-\frac{t}{\tau}) \]

leading to

\[ T_2 = \tau \ln \frac{V_{T+} - V_{OL}}{V_{T-} - V_{OL}} \]

It is easy to see that the generated square wave can be asymmetric, and the period is

\[ T = T_1 + T_2 = \tau (\ln \frac{V_{OH} - V_{T-}}{V_{OH} - V_{T+}} + \ln \frac{V_{T+} - V_{OL}}{V_{T-} - V_{OL}}) \]

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