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Electrical
Critical resistance of a series RLC circuit
Resistance at which a series RLC circuit is critically damped: it returns to rest as fast as possible without oscillating.
- Ref
- ELE-46
- Rev
- v1
- Field
- Electrical
- Result
- Rc [Ω]
R_c = 2√(L/C) is the resistance at which the loop is critically damped (ζ = 1). Below R_c the circuit rings; above it the response is overdamped and creeps back. Critical damping gives the quickest return without overshoot.
Resistance at which a series RLC circuit is critically damped: it returns to rest as fast as possible without oscillating. Below it the circuit rings (underdamped); above it the response is slow and overdamped.
Calculator
- Characteristic impedance, √(L/C)31.62 Ω
- Undamped angular frequency, 1/√(LC)3,162 rad/s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With this L and C, a series resistance below 63.25 Ω makes the circuit ring; above it, the response is overdamped.
- The critical resistance depends only on the ratio L/C: raising L or lowering C raises R_c.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Ω) | Local step | Change in result |
|---|---|---|---|---|---|---|
| C Capacitance | 1e-9 to 0.01 F | 2 | 6,325 | 6,323 | +10% | −4.65 % |
| L Inductance | 1e-6 to 10 H | 0.6325 | 2,000 | 1,999 | +10% | +4.88 % |
Sweep of Inductance
| Inductance (H) | Critical resistance (Ω) |
|---|---|
| 1e-6 | 0.6325 |
| 7.499e-6 | 1.732 |
| 5.623e-5 | 4.743 |
| 4.217e-4 | 12.99 |
| 0.003162 | 35.57 |
| 0.02371 | 97.39 |
| 0.1778 | 266.7 |
| 1.334 | 730.3 |
| 10 | 2,000 |
Sweep one variable
Variables and how each one contributes
Critical resistance
What the equation gives you. Everything else feeds it.
Inductance
The inductance, in H. Raises the critical resistance in proportion to the square root of L. Raising it by 10 % (to 0.011) moves the result by +4.88 %, from 63.25 to 66.33 Ω.
Capacitance
The capacitance, in F. Lowers the critical resistance in inverse proportion to the square root of C. Raising it by 10 % (to 1.1×10−5) moves the result by −4.65 %, from 63.25 to 60.3 Ω.
Worked example with the values above
- Ratio: L/C = 0.01 / 1×10−5 = 1,000 Ω².
- Square root: √(L/C) = 31.6228 Ω.
- R_c = 2 × 31.6228 = 63.2456 Ω.
Assumptions and limits
- A series RLC loop with ideal L and C.
- R is the total series resistance, including the source and winding resistance.
Common mistakes
- Using the parallel-circuit formula, R_c = ½√(L/C), which differs.
- Leaving out winding resistance in R.
- Treating critical damping as the best design in every case.
Statistics and uncertainty
R_c is proportional to √(L/C), so tolerances of ±10 % on L and C each shift R_c by about 5 %. The circuit must also overcome the inductor’s winding resistance, so choose a damping resistor with a margin.
Where it is used
- Choosing a damping resistor.
- Designing a critically damped step response in a filter or driver.
Maths you need
- Series RLC: R, L and C
- Damping: under-, critical and over-damped