Calculate R and C for Impedance

Calculator and formulas for calculating R and C for a given impedance and cutoff frequency

RC Impedance Calculator

RC Impedance Calculation

With this function, the resistor and capacitor of an RC series circuit (high pass / low pass) can be calculated for a given impedance and cutoff frequency.

Results
Capacitor C:
Resistor R:

RC Impedance Theory

Cutoff Frequency Condition

At the cutoff frequency, the reactance of the capacitor is identical to the ohmic resistance. This condition enables the calculation of RC components for a given impedance.

Basic Formulas
\[Z^2 = R^2 + X_C^2\]

At cutoff frequency: R = XC, therefore:

\[Z = \sqrt{2} \cdot R\]
Calculation Formulas
Resistor: \[R = \frac{Z}{\sqrt{2}}\]
Capacitor: \[C = \frac{1}{2\pi fR}\]

From given impedance Z and frequency f, R and C are calculated.


RC Impedance - Theory and Applications

Impedance at Cutoff Frequency

At the cutoff frequency of an RC circuit, the ohmic resistance R and the capacitive reactance XC are equal in magnitude. This special condition allows the calculation of corresponding component values from a desired impedance.

Mathematical Derivation

General Impedance
\[Z = \sqrt{R^2 + X_C^2}\]

Impedance of an RC series circuit according to Pythagoras.

At Cutoff Frequency
\[R = X_C \Rightarrow Z = \sqrt{2R^2} = \sqrt{2} \cdot R\]

Simplification at f = fc, where R = XC.

Calculation Steps

Step 1: Resistance
\[R = \frac{Z}{\sqrt{2}}\]

The resistance is calculated from the given impedance.

Step 2: Reactance
\[X_C = R\]

At cutoff frequency, R and XC are equal.

Step 3: Capacitance
\[C = \frac{1}{2\pi f \cdot X_C}\]

The capacitance follows from XC and the frequency.

Practical Applications

Filter Design:
• Impedance matching
• Cutoff frequency design
• Audio filters
• Anti-aliasing
Oscillators:
• RC oscillators
• Phase shifters
• Wien bridge
• Frequency determination
Impedance Converters:
• Line matching
• Transformers
• Attenuators
• RF technology

Important Characteristics

Characteristic Values
  • Impedance minimum: At f = 0, Z = R (purely resistive)
  • Cutoff frequency: At f = fc, Z = √2 · R
  • Phase angle: At fc, φ = ±45°
  • Frequency dependence: Z increases with frequency
  • Application range: Mainly optimal at cutoff frequency

Design Guidelines

High Pass Application
  • Capacitor in series
  • Output across resistor
  • High frequencies passed
  • AC coupling possible
Low Pass Application
  • Resistor in series
  • Output across capacitor
  • Low frequencies passed
  • Smoothing possible

Calculation Example

Example: 600Ω at 1kHz

Given: Z = 600Ω, f = 1kHz

\[R = \frac{600}{\sqrt{2}} = 424.3Ω\]
\[C = \frac{1}{2\pi \cdot 1000 \cdot 424.3} = 375nF\]

Result: R ≈ 424Ω, C ≈ 375nF

Tolerances and Practical Considerations

Important Design Aspects
  • Component tolerance: Standard components have ±5% to ±20% tolerance
  • Temperature influence: Capacitances and resistances are temperature dependent
  • Frequency stability: Exact impedance only at cutoff frequency
  • Loading: Following circuits affect the impedance
  • Standard values: Use of E-series values required





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  •