**Differentiator Circuit Output Analysis**
**Given Parameters:**
- Input Signal: Sine wave, 1V peak at 1000 Hz
- Feedback Resistor (RF) = 1 kΩ
- Input Capacitor (C1) = 0.33 μF
- Circuit Type: Op-amp differentiator
**Differentiator Function:**
A differentiator circuit produces an output proportional to the rate of change (derivative) of the input signal with respect to time.
**Transfer Function:**
Vout = -RF × C1 × (dVin/dt)
**Output Voltage Calculation:**
For a sinusoidal input: Vin = A × sin(2πft)
Where A = 1V (peak amplitude), f = 1000 Hz
The derivative of sine is cosine:
dVin/dt = A × 2πf × cos(2πft) = 1 × 2π × 1000 × cos(2πft) = 6283.19 × cos(2πft)
**Peak Output Voltage:**
Vout(peak) = -RF × C1 × dVin/dt(peak)
Vout(peak) = -1000 × 0.33 × 10^-6 × 6283.19
Vout(peak) = -330 × 10^-6 × 6283.19
Vout(peak) = -2.073V
Vout(peak) ≈ 2.07V (ignoring polarity)
**Output Waveform Characteristics:**
1. **Waveform Shape**: The output will be a COSINE waveform (90° phase shift from input sine wave)
2. **Peak Amplitude**: Approximately 2.07V
3. **Frequency**: Same as input frequency (1000 Hz)
4. **Phase Relationship**: The output leads the input by 90° (cosine is 90° ahead of sine)
5. **Polarity**: Inverted due to the negative sign in the transfer function
**Detailed Waveform Description:**
- When the input sine wave has maximum positive slope (crossing zero going upward), the output is at maximum negative peak (-2.07V)
- When the input is at positive peak, the output crosses zero
- When the input has maximum negative slope (crossing zero going downward), the output is at maximum positive peak (+2.07V)
- When the input is at negative peak, the output crosses zero again
**Time Domain Expression:**
Vout(t) = -2.07 × cos(2π × 1000 × t) V
Or in terms of sine:
Vout(t) = -2.07 × sin(2π × 1000 × t + 90°) V
**Practical Observations:**
- The output is a pure cosine waveform with 2.07V peak amplitude
- The frequency remains 1000 Hz (same as input)
- The output oscillates between -2.07V and +2.07V
- The derivative relationship ensures that rapid changes in input produce large output voltages
- This circuit is ideal for detecting the rate of change of signals in applications like zero-crossing detection and high-frequency emphasis filters
**Circuit Behavior Summary:**
The differentiator converts the input 1V peak sine wave at 1000 Hz into a 2.07V peak cosine waveform, demonstrating the classic property of differentiators to perform time differentiation and produce a 90° phase lead in the output signal.