**Expression for the time period:**

An expression for the time period of the output waveform can be determined as follows. The equation for the capacitor charging may be written in the following general form.

V_{c} (t ) = A + B e ^{- t /( RA + RB ) C}

where A and B are constants which may be resolute from the following conditions.

at t = 0, V_{c} (t ) = V_{cc} / 3

at t = ∞ , V_{c} (t) = V_{cc}

Substituting in Eq. (97), we determine

V_{cc} = A + B and V_{cc} = A

from which it is found that

∴ B = 2V_{cc} /3

Eq. (97) now becomes

V_{c } (t) = V_{cc} - (2/3) V_{cc} e^{- t /( RA + RB ) C }

Now at t = T_{H}, VC (t) =(2/3)V_{cc} ,therefore, we get

(2 /3)V cc = V cc - (2/3) V cc e ^{(- tH/( RA + RB ) C)}

(2/3) e ^{- tH/ ( RA + RB ) C} = 1/3

e ^{tH/ ( RA + RB ) C} = 2

from which the needed time period is given by

T_{H} = (R_{A} + R_{B} ) C ln 2

likewise, one can determine that considering the equation for discharge of the capacitor with time constant CR_{B}, one may find T_{L} as

T_{L } = R_{B} C ln 2

The frequency of the output waveform is, thus, given by

f = 1 / (R_{A} + R_{B} ) C ln 2

and the duty cycle is given by

f = (R_{A} + R_{B} ) (R_{A} + 2R_{B} )

Note down that even if one selects R_{A} = R_{B}, the duty cycle may not be made 50%. However, the duty cycle may be made 50% by R_{B} shunting by a diode as illustrated in Figure

**Figure: Astable Multivibrator with 50% Duty Cycle**

This makes T_{H} = R_{A} C ln 2 and T_{L}= R_{B} C ln 2, so that the duty cycle is given by

δ= R_{A} /R_{A} + R_{B}

which would become 50% for R_{A }= R_{B}.