Electrical engineers and hobbyists frequently search for a ti-nspire calculator online free version because the physical TI-Nspire CX II CAS hardware costs over $150, and the desktop software requires a paid license. However, Texas Instruments does not offer a full, unrestricted, permanently free web-based CAS engine. For complex AC phasor math and impedance calculations, the most reliable free online alternative is the GeoGebra CAS web app, which handles complex numbers and symbolic algebra natively.

But relying on a black-box calculator without understanding the underlying derivations leads to catastrophic unit errors in AC circuit design. Below is the complete derivation, strict unit-tracked worked examples, and a concrete component selection path for Series RLC Resonance—the exact math you would use a CAS calculator to solve.

Core Formulas: Series RLC Resonance and Q-Factor

These formulas apply to ideal, linear series RLC circuits driven by a sinusoidal AC source. Assumptions: The inductor and capacitor are purely reactive (parasitics are lumped into R), the core material is not saturated, and the frequency is well below the self-resonant frequency (SRF) of the physical components.

Formula Symbol Definition Standard Unit Realistic Magnitude
fr = 1 / (2π√(LC)) fr Resonant Frequency Hertz (Hz) 1 kHz to 5 GHz
L Inductance Henrys (H) 1 nH to 10 mH
C Capacitance Farads (F) 1 pF to 1 mF
Q = (1/R)√(L/C) Q Quality Factor (Dimensionless) None 5 (power) to 200 (RF)
R Series Resistance (incl. ESR) Ohms (Ω) 0.01 Ω to 50 Ω
BW = fr / Q BW -3dB Bandwidth Hertz (Hz) 10 Hz to 50 MHz

Rearranged Forms for Component Sizing

When designing filters or matching networks, you rarely solve for fr. You are given a target frequency and must find the required L, C, or R. Use these rearranged forms to isolate your unknown variable:

  • Solve for L: L = 1 / (4π2 × fr2 × C)
  • Solve for C: C = 1 / (4π2 × fr2 × L)
  • Solve for R (Target Q): R = (1 / Q) × √(L / C)
  • Solve for L (Target Q & R): L = R2 × Q2 × C

Worked Examples with Strict Unit Tracking

A common failure mode when using a TI-Nspire or any CAS tool is entering '47' instead of '47e-6'. The calculator will happily output a mathematically correct but physically impossible answer. Always track units through intermediate steps.

Problem 1: DC-DC Converter Output Filter (Low Frequency, Low Q)

Given: A buck converter output filter has L = 47 μH, C = 220 μF, and the combined inductor DCR and capacitor ESR is R = 50 mΩ. Find fr and Q.

Step 1: Convert to base SI units.

  • L = 47 × 10-6 H
  • C = 220 × 10-6 F
  • R = 0.05 Ω

Step 2: Calculate Resonant Frequency (fr).

  • fr = 1 / (2π × √(47 × 10-6 × 220 × 10-6))
  • fr = 1 / (2π × √(1.034 × 10-8))
  • fr = 1 / (2π × 1.0168 × 10-4)
  • fr = 1 / (6.389 × 10-4) = 1565 Hz (1.56 kHz)

Step 3: Calculate Quality Factor (Q).

  • Q = (1 / 0.05) × √((47 × 10-6) / (220 × 10-6))
  • Q = 20 × √(0.2136)
  • Q = 20 × 0.462 = 9.24

Problem 2: 433 MHz RF Bandpass Filter (High Frequency, High Q)

Given: Design a series resonant trap for a 433 MHz ISM band receiver. Target fr = 433 MHz. You select a standard C0G capacitor C = 2.2 pF. Find the required L, and the maximum allowable series resistance R to maintain a bandwidth (BW) of 8.66 MHz.

Step 1: Convert and find target Q.

  • fr = 433 × 106 Hz
  • C = 2.2 × 10-12 F
  • Q = fr / BW = (433 × 106) / (8.66 × 106) = 50

Step 2: Calculate Required Inductance (L).

  • L = 1 / (4π2 × (433 × 106)2 × 2.2 × 10-12)
  • L = 1 / (39.478 × 1.8749 × 1017 × 2.2 × 10-12)
  • L = 1 / (1.628 × 107) = 6.14 × 10-8 H = 61.4 nH

Step 3: Calculate Maximum Allowable Resistance (R).

  • R = (1 / 50) × √((61.4 × 10-9) / (2.2 × 10-12))
  • R = 0.02 × √(27909)
  • R = 0.02 × 167.06 = 3.34 Ω
⚠ Safety & Hardware Warning: In high-Q series resonant circuits (like Problem 2), the voltage across the inductor and capacitor is magnified by the Q-factor (VL = Q × Vsource). A 1V RF signal with Q=50 will generate 50V across the capacitor dielectric. Ensure your capacitor's voltage rating exceeds Q × Vin, or the dielectric will avalanche and short.

Unit Mistakes That Break the Math

When plugging values into any online CAS tool, these three unit errors account for 90% of failed physical prototypes:

  1. The Angular vs. Cyclic Trap: The formula fr = 1 / (2π√(LC)) yields Hertz (cycles per second). If your design spec requires angular frequency (ω in radians/second), you must multiply the result by 2π. The formula for ωr is simply 1 / √(LC). Mixing these up shifts your filter response by a factor of 6.28.
  2. The Micro vs. Pico Prefix Error: In RF design, capacitance is in picofarads (10-12). In power design, it is in microfarads (10-6). Entering '2.2' instead of '2.2e-12' in a calculator will yield an inductance value 12 orders of magnitude off, resulting in a physical impossibility.
  3. Ignoring ESR in Q Calculations: The 'R' in the Q formula is the total series resistance. Beginners often use only the load resistance, forgetting to add the inductor's DC Resistance (DCR) and the capacitor's Equivalent Series Resistance (ESR). At high frequencies, skin effect and core losses further increase the effective R, lowering your real-world Q below the calculated ideal.

Component Selection Decision Path

Use this decision tree to terminate your math into a concrete physical component selection. Do not leave your design at the theoretical L and C values; map them to real manufacturer part numbers based on your operating regime.

Condition (If...) Then Select Component Type... Concrete Part Number Pick
fr < 1 MHz AND Current > 1A (Power Filtering) Ferrite drum core inductor (high saturation current) + Aluminum Polymer capacitor (low ESR, high capacitance). Murata LQH32CN470K53L (47 μH, 1.1A sat) paired with a Panasonic OS-CON 220 μF.
1 MHz < fr < 100 MHz AND Q > 20 (IF Filters / Antenna Matching) Multilayer ceramic chip inductor (high SRF) + C0G/NP0 ceramic capacitor (zero voltage coefficient). Coilcraft 0603HP-47NXJRW (47 nH, Q=45 at 500MHz) paired with KEMET C0603C NP0 series.
fr > 100 MHz AND Q > 50 (UHF / Microwave Traps) Wirewound air-core or ultra-low-loss ceramic inductor + Thin-film RF capacitor (tight tolerance, ±0.1pF). Coilcraft 0402HP-61NXJRW (61 nH, Q=65 at 300MHz) paired with AVX UQCFWB2R2 (2.2pF thin-film).

For further reading on the physical behavior of these components at resonance, consult the All About Circuits guide on Series Resonance, which details the phase-angle shifts that CAS calculators abstract away. By mastering the raw math and mapping it to specific parasitics, you eliminate the need to hunt for proprietary calculator software and ensure your first physical prototype matches your simulation.