in Inverting Schmitt Trigger, Given R2=10kΩ, Vref=4.5V and the output saturation voltage of the op-amp is 9V. Determine the value of R1 to achieve a hysteresis band of 3.0V.
The hysteresis band is the voltage difference between the upper threshold voltage (VUT) and the lower threshold voltage (VLT).
To find the upper threshold voltage (VUT), we can use the formula:
VUT = Vref + (R1 / (R1 + R2)) * (Vsat - Vref)
Where:
- Vref is the reference voltage, given as 4.5V
- R1 is the resistance connected to the inverting terminal, which we are trying to find
- R2 is the resistance connected to the non-inverting terminal, given as 10kΩ
- Vsat is the output saturation voltage of the op-amp, given as 9V
Substituting the values into the formula:
VUT = 4.5V + (R1 / (R1 + 10kΩ)) * (9V - 4.5V)
Next, we need to find the lower threshold voltage (VLT). Since the hysteresis band is 3.0V, we can simply subtract 3.0V from the upper threshold voltage:
VLT = VUT - 3.0V
Now we can equate VLT to the formula for VUT and solve for R1:
VLT = Vref + (R1 / (R1 + 10kΩ)) * (9V - Vref) - 3.0V
Simplifying the equation:
VLT = Vref + (R1 / (R1 + 10kΩ)) * (9V - Vref) - 3.0V
VLT - Vref + 3.0V = (R1 / (R1 + 10kΩ)) * (9V - Vref)
((R1 + 10kΩ) / R1) * (VLT - Vref + 3.0V) = 9V - Vref
((R1 + 10kΩ) / R1) * (VLT - Vref + 3.0V) + Vref = 9V
((R1 + 10kΩ) / R1) * (VLT - Vref + 3.0V) = 9V - Vref
(R1 + 10kΩ) * (VLT - Vref + 3.0V) = R1 * (9V - Vref)
R1 * (VLT - Vref + 3.0V) + 10kΩ * (VLT - Vref + 3.0V) = R1 * (9V - Vref)
R1 * (VLT - Vref + 3.0V - 9V + Vref) = 10kΩ * (VLT - Vref + 3.0V)
R1 * (VLT - 6.0V) = 10kΩ * (VLT + 3.0V - Vref)
R1 = (10kΩ * (VLT + 3.0V - Vref)) / (VLT - 6.0V)
Using Vref = 4.5V and VLT = VUT - 3.0V, we can substitute these values and solve for R1:
R1 = (10kΩ * ((VUT - 3.0V) + 3.0V - 4.5V)) / ((VUT - 3.0V) - 6.0V)
Simplifying and rearranging the equation:
R1 = (10kΩ * (VUT - 4.5V)) / (VUT - 9.0V)
Therefore, the value of R1 to achieve a hysteresis band of 3.0V is given by the equation:
R1 = (10kΩ * (VUT - 4.5V)) / (VUT - 9.0V)
To determine the value of R1 in order to achieve a hysteresis band of 3.0V in an Inverting Schmitt Trigger with R2 = 10kΩ, Vref = 4.5V, and an output saturation voltage of 9V, we can use the following formula:
Vsat = R2 * (Vo / R1)
where Vsat is the output saturation voltage, Vo is the hysteresis voltage (which is equal to 3.0V in this case), and R1 is the unknown resistor value that we want to find.
Rearranging the formula, we can solve for R1:
R1 = R2 * (Vo / Vsat)
Substituting the given values, we have:
R1 = 10kΩ * (3.0V / 9V)
R1 = 10kΩ * 0.333
R1 = 3.33kΩ
Therefore, to achieve a hysteresis band of 3.0V, the value of R1 should be approximately 3.33kΩ.