Square Wave Generator or Astable Multivibrator

Definition

Astable Multivibrator is also called as Free running multivibrator or relaxation oscillator with no stable states. It is a square wave generator and has two unstable states.It oscillates back and forth between these two states when the circuit is given power supply.

Circuit Diagram

Astable Multivibrator

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Design equations

The Time period of Square wave

T = 2*R*C*ln((1+β)/(1-β))

Assume R1=R2 then T= R*C*ln(3), in which the values of R and C can have any combination based on availability of capacitor and resistor but should satisfy the time period equation.

Circuit analysis

The output of opamp is +Vccif V2 >> V1 and is –Vcc if V2<< V1.Assume that the output initially is at positive saturation value of +Vcc. By voltage divider rule the voltage at non inverting terminal of opamp is Vcc*R1/(R2+R1). The capacitor starts charging through R with time constant R*C and the voltage across capacitor is given by Vc = Vcc*(1-exp(-R*C*t)). When the voltage across the capacitor is just more than Vcc*R1/(R2+R1), at that instant the output of opamp changes to –Vcc .Now the Vc = -Vcc*R1/(R2+R1),the capacitor has to discharge through R till it reaches to a value less than          -Vcc* R1/(R2+R1). At that instant when V1 << -Vcc* R1/(R2+R1),  output will be +Vcc. During the charging and discharging time voltage across the capacitor will be Vc=-Vcc*exp(-R*C*t)). Hence the voltage across capacitor switches between -Vcc* R1/(R2+R1) and +Vcc* R1/(R2+R1) and output switches between +Vcc and -Vcc. The voltage acroos the capacitor during charging time is given by Vc = Vcc*(1-(1+β)exp(-t/(R*C)) where β = R1/(R2+R1).

Let us assume that the voltage across capacitor at t= 0 s is equal to -Vcc..At = T/2  output transits from -Vcc to +Vcc.

Hence at t=T/2 Vc=β*Vcc, substituting in Vc = Vcc*(1-(1+β)exp(-t/(R*C)) we get
β*Vcc= Vcc*(1-(1+β)exp(-T/(2*R*C)))

T = 2*R*C*ln((1+β)/(1-β))

Hence the frequency of oscillation of square wave is f=1/T.

Remarks

This circuit can be used for generating square waves with frequencies of the Kilo hertz order. The slew rate of opamp poses a limitation in generating still higher frequency square waves.