In a Non-Inverting Schmitt Trigger circuit has
r1 = 45kΩ
r2 = 7kΩ
and in a Inverting Integrator has
C=6nF
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 (τ) of the Inverting Integrator circuit.
The time constant (τ) is given by the product of the resistance (R) and capacitance (C) in the circuit:
τ = R × C
In this case, the capacitance (C) is given as 6nF (nanofarads).
First, let's convert the capacitance from nanofarads to farads:
C = 6nF = 6 × 10^(-9) F
Now, let's calculate the time constant (τ):
τ = R × C
We can rearrange this equation to solve for R:
R = τ / C
The desired frequency (f) is given as 2.7kHz (kilohertz). The frequency (f) and time constant (τ) are related by the following equation:
f = 1 / (2πτ)
We can rearrange this equation to solve for τ:
τ = 1 / (2πf)
Substituting the given frequency (f = 2.7kHz) into the equation:
τ = 1 / (2π × 2.7kHz)
Now, we can substitute τ into the equation for R:
R = τ / C
R = (1 / (2π × 2.7kHz)) / (6 × 10^(-9) F)
R = (1 / (2π × 2.7 × 10^3)) / (6 × 10^(-9))
R ≈ 8.07kΩ
Therefore, the value of R3 should be approximately 8.07kΩ.
To determine the value of R3 that will set the frequency of the output waveforms to 2.7kHz, we need to calculate the time constant of the inverting integrator circuit.
The time constant (τ) of an RC circuit is given by the product of the resistance (R) and the capacitance (C). In this case, the resistance is the parallel combination of R1 and R2.
1/R = 1/R1 + 1/R2
= 1/45kΩ + 1/7kΩ
= (7kΩ + 45kΩ) / (45kΩ * 7kΩ)
= 52kΩ / 315kΩ^2
Now, let's calculate the time constant:
τ = RC
= (52kΩ / 315kΩ^2) * 6nF
= 312pF * (52kΩ / 315kΩ^2)
= 312pF * (0.1651 * 10^-6)
= 51.5 * 10^-12 s
The frequency of oscillation in the inverting integrator circuit is given by:
f = 1 / (2πτ)
= 1 / (2π * 51.5 * 10^-12 s)
= 1 / (102 * 10^-12 sπ)
= 9.8 GHz / π
To set the frequency to 2.7kHz, we can set up the equation:
2.7kHz = 9.8 GHz / π
2.7kHz * π = 9.8 GHz
2.7 * 10^3 * π = 9.8 * 10^9
8.49 * 10^3 = 9.8 * 10^9
Now, solve for R3:
R3 = (8.49 * 10^3) * (315kΩ^2 / 6kΩ)
= (8.49 * 10^3) * (52.5 * 10^3)
= 4.455 * 10^7 kΩ
Therefore, R3 should have a value of approximately 44.55 kΩ to set the frequency of the output waveforms to 2.7 kHz.