Calculate RC Integrator

Calculator and formulas for calculating an RC integrator

Calculate Integrator

RC Integrator

This function allows you to calculate the properties of an RC integrator. The function calculates the capacitor, resistor, or the period/frequency.

Results
Resistor:
Capacitor:
Frequency:
Period:

Integrator

Pulse Shaper Stage

The integrator works as a pulse shaper stage. The RC circuit generates a ramp-shaped output from a square wave input.

Square pulse
T = Period, t1 = Pulse
RC Integrator
Time Constant τ (tau)

The time constant of an RC circuit is the product of R · C. The unit is seconds. The symbol is the Greek letter τ (tau).

After 5τ, the charge is about 99.3%.

RC Integrator - Theory and Formulas

How the Integrator Works

The integrator works as a pulse shaper stage. The RC circuit generates a ramp-shaped voltage at the output from a square wave input. Integration occurs through the slow charging of the capacitor.

Pulse Shapes for Different Time Constants

t1 = 5τ Optimal Integration
5 tau pulse shape
\[t1 = 5 \cdot R \cdot C\] \[R = \frac{t1}{5 \cdot C}\] \[C = \frac{t1}{5 \cdot R}\]

If the length of the square pulse (t1) is five times the time constant τ, an optimal ramp shape is produced.

t1 > 5τ Steeper Ramps
\[t1 > 5 \cdot R \cdot C\] \[R = \frac{t1}{n \cdot C}\] \[C = \frac{t1}{n \cdot R}\]

If the pulse duration of the input voltage is greater than 5τ, steeper ramp shapes are produced.

Time Constant and Charging Behavior

The Time Constant τ (tau)
\[\tau = R \times C\]
  • The time constant is the product of R × C
  • Unit: seconds (s)
  • Symbol: τ (Greek letter tau)
  • After 5τ, the charge is about 99.3%
  • Determines the speed of ramp formation
  • For good integration: t1 ≥ 5τ

Practical Applications

Signal Processing:
• Ramp generators
• Triangle oscillators
• Sawtooth generators
• Timing circuits
Measurement Technology:
• Analog integrators
• Averaging
• Signal smoothing
• Low-pass filters
Control Engineering:
• PI controllers (integral part)
• Delay elements
• Compensation
• Stabilization

Design Guidelines

Optimal Dimensioning
  • For good integration: t1 ≥ 5τ
  • For linear ramps: t1 ≥ 10τ
  • Output amplitude: Depends on the time constant
  • Ramp slope: Proportional to 1/τ
  • Loading: High input resistance of the following stage
  • Frequency range: Depends on τ and desired accuracy

Mathematical Relationships

Basic Formulas
\[\tau = R \times C\] \[t1 = n \times \tau\]

Where n is the factor (typically 5 or greater)

Conversions
\[R = \frac{t1}{n \times C}\] \[C = \frac{t1}{n \times R}\]

Calculation of components for a given t1

Integrator vs. Differentiator

Differences
Integrator (RC):
  • Output at the capacitor
  • Generates ramps from square waves
  • Slow changes
  • Low-pass characteristic
Differentiator (CR):
  • Output at the resistor
  • Generates spikes from square waves
  • Fast changes
  • High-pass characteristic

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Capacitor functions

Series connection with capacitors  •  Series connection with 2 capacitors  •  Reactance Xc of a capacitor  •  Time constant of an R/C circuit  •  Capacitor charging voltage  •  Capacitor discharge voltage  •  R/C for the charging voltage  •  Series circuit R/C  •  Parallel circuit R/C  •  Low pass-filter R/C  •  High pass-filter R/C  •  Integrator R/C  •  Differentiator R/C  •  Cutoff-frequency R,C  •  R and C for a given impedance  •