In a Non-Inverting Schmitt Trigger circuit has
r1 = 45kΩ
r2 = 7kΩ
Vcc = 12v
vee = -12v
and in a Inverting Integrator has
C=6nF
Vcc = 12v
vee = -12v
both Non-Inverting Schmitt Trigger circuit and Inverting Integrator circuit are connected by R3.
Determine the value of R3 to set the frequency of the output waveforms to 2.7kHz.
Express your answer in kohms, with two decimal places. Show your steps.
To determine the value of R3, we need to calculate the time constant of the integrator circuit and equate it to the desired frequency.
For the inverting integrator circuit, the time constant (τ) is given by the formula:
τ = R2 * C
Since R2 is the resistance connected to the capacitor (R3 in this case), we can rewrite the formula as:
τ = R3 * C
We can rearrange the formula to solve for R3:
R3 = τ / C
To set the frequency of the waveform to 2.7kHz, we need the time period (T) to be:
T = 1 / f = 1 / 2.7kHz = 0.3704 ms
The time constant (τ) is equal to 1/α, where α is the parameter determining the output voltage level at which the Schmitt Trigger switches from low to high or high to low. For a Non-Inverting Schmitt Trigger, α is given by:
α = (r2 / (r1 + r2))
Given that r1 = 45kΩ and r2 = 7kΩ, we can plug in the values to calculate α:
α = (7kΩ / (45kΩ + 7kΩ)) = 0.1346
Since α is the ratio of the time spent at each voltage level, we can calculate the time spent at each level:
time_high = τ * α
time_low = τ * (1 - α)
The time spent at each level must sum up to the time period (T), so we can equate the two:
time_high + time_low = T
Substituting the formulas for time_high and time_low:
(τ * α) + (τ * (1 - α)) = T
Simplifying the equation:
τ * α + τ - τ * α = T
τ = T
Since τ = R3 * C, we can replace τ in the equation:
R3 * C = T
Rearranging the equation for R3:
R3 = T / C
Plugging in the values:
R3 = 0.3704 ms / 6nF = 61.73 kΩ
Therefore, the value of R3 to set the frequency of the output waveforms to 2.7kHz is 61.73 kΩ.
To determine the value of R3 to set the frequency of the output waveforms to 2.7kHz, we need to consider the time constant of the RC circuit formed by R3 and C in the Inverting Integrator.
The time constant (τ) of an RC circuit is given by the product of resistance (R) and capacitance (C), which can be expressed as:
τ = R * C
In the Inverting Integrator circuit, the output frequency (f) is related to the time constant by the equation:
f = 1 / (2π * τ)
Given that the desired output frequency is 2.7kHz, we can rearrange the equation to solve for τ:
τ = 1 / (2π * f)
Substituting the values:
f = 2.7kHz (converted to Hz: 2.7 * 10^3 Hz)
π ≈ 3.1416
τ = 1 / (2 * 3.1416 * 2.7 * 10^3 Hz)
= 1 / (16.9744 * 10^3 Hz)
= 5.89096 * 10^(-5) s
Now we can rearrange the equation for the time constant to solve for R3:
τ = R3 * C
Rearranging:
R3 = τ / C
Substituting the values:
τ = 5.89096 * 10^(-5) s (time constant)
C = 6nF (converted to Farads: 6 * 10^(-9) F)
R3 = (5.89096 * 10^(-5) s) / (6 * 10^(-9) F)
= 9.81826 * 10^(3) Ω
= 9.82kΩ (rounded to two decimal places)
Therefore, the value of R3 to set the frequency of the output waveforms to 2.7kHz is approximately 9.82kΩ.