#35 Operational Amplifiers #35 — Integrator and differentiator

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Integrator circuit

   

   Op-amp integrator

   Input path: V₁→R→v₋

   Feedback element: C

   $v_{+}=0$

   Under negative feedback, v₋≈0

   Ideal input current: i₋=0

   The input current flows into C

   Narration transcript

   Now let's look at the op-amp integrator circuit. Notice the key difference from circuits we've seen before — the feedback element is a capacitor, not a resistor. The input signal V1 passes through resistor R to the inverting input, while capacitor C provides the feedback path. Since the non-inverting input is grounded, we still have a virtual ground at the inverting node.

2. 2. Integrator derivation

   

   Integrator equation

   Z_C=1/(sC)

   $i_{\mathrm{R}}=V_{1}/R=i_{\mathrm{C}}$

   V_o(s)/V₁(s)=−1/(sRC)

   v_o(t)=−(1/RC)∫v₁(t)dt

   This form assumes zero initial condition

   Minus sign: inverting connection

   Narration transcript

   Let's derive the transfer function. The capacitive impedance is X C equals 1 over s C in Laplace notation. The current through R is V1 over R. This same current flows through the feedback capacitor, producing an output voltage. Working through the math, we get V out over V1 equals negative 1 over s C R. Converting to the time domain, the output equals negative 1 over R C times the integral of the input. The op-amp integrator literally performs mathematical integration.

3. 3. Step example

   

   Step input → ramp output

   $v_{1}=1 V$

   R=1 MΩ, C=1 µF ⇒ RC=1 s

   dv_o/dt=−v₁/(RC)=−1 V/s

   R=100 kΩ ⇒ RC=0.1 s

   dv_o/dt=−10 V/s

   Smaller RC ⇒ steeper ramp

   Narration transcript

   Let's see this in action with a step input. When V1 is a constant 1 volt and R is 1 megohm, C is 1 microfarad, the scale factor 1 over RC equals 1. The output is a negative ramp, growing linearly over time. Now if we change R to 100 kilohms, 1 over RC becomes 10, and the output ramp is ten times steeper. The RC product controls how fast the integrator ramps.

4. 4. Integrator summary

   

   Integrator summary

   R at the input, C in feedback

   Step → ramp

   Slope: −v₁/(RC)

   Accumulates area over time

   A practical integrator adds a resistor across C to limit DC gain

   Purpose: delay saturation

   Narration transcript

   So the integrator converts a step into a ramp. The output is the negative integral of the input, scaled by 1 over RC. This is fundamental to analog computing and control systems.

5. 5. Differentiator circuit

   

   Op-amp differentiator

   Input path: V₁→C→v₋

   Feedback element: R

   $v_{+}=0$

   Under negative feedback, v₋≈0

   R and C swap relative to the integrator

   Output measures the input rate of change

   Narration transcript

   The differentiator circuit is essentially the mirror image of the integrator. Here, the capacitor is in the input path and the resistor provides feedback. The input signal passes through capacitor C to the inverting node, and resistor R connects from output back to this node.

6. 6. Differentiator derivation

   

   Differentiator equation

   i_C=C·dv₁/dt

   $i_{\mathrm{C}}=i_{\mathrm{R}}$

   v_o=−Ri_R

   v_o(t)=−RC·dv₁(t)/dt

   Scale factor: −RC

   Minus sign: inverting connection

   Narration transcript

   For the differentiator, the input current through the capacitor is C times the derivative of the input voltage. This current flows through the feedback resistor R, giving us the output. The result is: v out equals negative RC times the derivative of v1 with respect to time. The scale factor is negative RC.

7. 7. Ramp and sine

   

   Reading differentiator signals

   Derivative of a ramp = constant

   Positive slope ⇒ negative constant output

   $v_{1}=V_{\mathrm{m}} \sin (\omegat)$

   v_o=−ωRCV_m cos(ωt)

   Magnitude grows with frequency

   Output represents rate of change

   Narration transcript

   When we apply a ramp input to the differentiator, the derivative of a ramp is a constant, so the output is a steady DC voltage scaled by negative RC. For a sinusoidal input, the derivative of sine is cosine, so the output is a cosine waveform. The differentiator produces the rate of change of the input signal.

8. 8. Comparison

   

   Integrator ↔ differentiator

   Integrator: R input, C feedback

   Differentiator: C input, R feedback

   Integrator accumulates area

   Differentiator measures slope

   Both are inverting

   Two building blocks of analog computing

   Narration transcript

   Comparing the two circuits: the integrator has R in the input and C in the feedback, while the differentiator swaps them. The integrator smooths signals by accumulating area under the curve. The differentiator highlights rapid changes by computing the slope. Together, they form the building blocks of analog computing.

9. 9. Method summary

   

   Method summary

   v_o,int=−(1/RC)∫v_i dt

   v_o,diff=−RC·dv_i/dt

   Step → ramp; ramp → constant

   Ideal equations require negative feedback and no saturation

   A practical differentiator limits high-frequency gain

   Final check: bandwidth, noise, and rails

   Narration transcript

   To summarize: The integrator output equals negative 1 over RC times the integral of the input — it converts steps to ramps. The differentiator output equals negative RC times the derivative — it converts ramps to steps. These two circuits are fundamental to signal processing and control system design.