The Core Formula: AC Impedance in Complex Form
When analyzing alternating current (AC) circuits, resistance alone is insufficient to describe how components oppose current flow. We must use impedance (Z), a complex number that accounts for both energy dissipation (resistance) and energy storage (reactance). While many basic calculators force you to calculate magnitude and phase angle separately using trigonometry, the online TI-36 calculator (and its physical counterpart, the TI-36X Pro) features a native Complex (CMPLX) mode. This allows you to enter, add, and convert AC phasors directly.
The foundational formula for the total complex impedance of a series RLC circuit is:
Z = R + j(XL - XC) or Z = R + j(2πfL - 1 / 2πfC)
| Symbol | Definition | Standard Unit | Typical Magnitude |
|---|---|---|---|
| Z | Total Complex Impedance | Ohms (Ω) | 1 Ω to 10 kΩ |
| R | Resistance (Real part) | Ohms (Ω) | 1 Ω to 10 kΩ |
| j | Imaginary unit (√-1). Note: TI-36 uses i. | Dimensionless | N/A |
| XL | Inductive Reactance (2πfL) | Ohms (Ω) | 0.1 Ω to 1 kΩ |
| XC | Capacitive Reactance (1 / 2πfC) | Ohms (Ω) | 0.1 Ω to 1 kΩ |
| f | AC Frequency | Hertz (Hz) | 50 Hz, 60 Hz, or kHz ranges |
| L | Inductance | Henries (H) | mH (10-3) to μH (10-6) |
| C | Capacitance | Farads (F) | μF (10-6) to nF (10-9) |
Rearranged Forms & Variable Definitions
On the bench, you rarely just solve for Z. Usually, you know your target impedance or resonant frequency and need to pick a component. Here are the rearranged forms you will plug into your calculator:
- Solving for Inductance (L): L = XL / (2πf) — Use when sizing a choke or filter inductor.
- Solving for Capacitance (C): C = 1 / (2πf × XC) — Use for power factor correction or bypass caps.
- Solving for Resistance (R): R = √(Z2 - (XL - XC)2) — Use when verifying the real power dissipation of a coil.
- Solving for Resonant Frequency (fr): fr = 1 / (2π√(LC)) — Use when tuning an LC tank circuit (where XL = XC).
When This Applies (and When Unit Mistakes Break It)
This formula assumes steady-state sinusoidal AC and linear, lumped-parameter components. It does not apply to transient DC switching, non-sinusoidal waveforms (like square waves from a PWM inverter, which require Fourier analysis), or distributed high-frequency RF systems where trace inductance matters.
- The Micro/Milli Trap: Entering 25 μF as
25 * 10^-3instead of25 * 10^-6. This inflates your capacitance by 1000x, collapsing XC to near zero and ruining your phase angle. - The Radians vs. Degrees Trap: The online TI-36 calculator defaults to Radians in some web emulators. If you calculate arctan(-49.55 / 50) in Radians, you get
-0.78. Your brain expects-44.7°. Always verify the DEG indicator is active on the display before converting rectangular to polar.
Worked Example 1: Finding Total Impedance and Phase Angle
Scenario: You are building a 60 Hz passive audio crossover. The series branch contains a 50 Ω resistor, a 150 mH inductor, and a 25 μF capacitor. Find the total impedance magnitude and phase angle.
Step 1: Calculate Angular Frequency (ω)
ω = 2πf = 2 × π × 60 Hz = 376.99 rad/s
Step 2: Calculate Reactances (with unit tracking)
XL = ωL = 376.99 rad/s × 0.150 H = 56.55 Ω
XC = 1 / (ωC) = 1 / (376.99 rad/s × 0.000025 F) = 106.10 Ω
Step 3: Assemble the Complex Equation
Z = R + j(XL - XC)
Z = 50 + j(56.55 - 106.10)
Z = 50 - j49.55 Ω (Rectangular Form)
Step 4: TI-36 Calculator Keystrokes for Polar Conversion
On the TI-36 interface, switch to CMPLX mode. Enter 50 - 49.55i. Press the conversion key (often labeled >P or accessed via the complex menu) to convert to Polar form.
Magnitude: √(502 + (-49.55)2) = 70.39 Ω
Phase Angle: arctan(-49.55 / 50) = -44.74°
Realistic check: A magnitude of 70 Ω is perfectly normal for an audio crossover, and a negative angle confirms the circuit is currently capacitive (XC > XL).
Worked Example 2: Sizing a Capacitor for Unity Power Factor
Scenario: A 120VAC, 60Hz industrial motor presents an equivalent series impedance of Z = 40 + j30 Ω. The utility company is penalizing you for a lagging power factor. You need to add a parallel capacitor to cancel the inductive reactance (bringing the net reactive current to zero).
Step 1: Identify the Target Reactance
To achieve unity power factor, the capacitive reactance (XC) of your parallel correction bank must exactly equal the inductive reactance (XL) of the motor.
Required XC = 30 Ω
Step 2: Rearrange and Solve for C
C = 1 / (2πf × XC)
C = 1 / (2 × π × 60 Hz × 30 Ω)
C = 1 / 11309.73
C = 0.0000884 F = 88.4 μF
Step 3: Voltage and Safety Sizing
The capacitor will see the full line voltage (120VAC). However, AC motor circuits experience voltage spikes and transients. Standard engineering practice requires a minimum 2x voltage derating for continuous AC duty.
Required Voltage Rating = 120V × 2 = 240VAC minimum.
Decision Path: Picking the Right Calculator Mode & Component
Use this decision tree to configure your online TI-36 calculator and select the physical components for your workbench.
| Condition / Goal | Action / Setting | Concrete Result / Part Pick |
|---|---|---|
| If calculating phase angles or power factor (cos θ) | Set TI-36 angle mode to DEGREES | Yields readable angles (e.g., -44.7° instead of -0.78 rad) |
| If calculating resonant frequency (fr) or angular velocity (ω) | Set TI-36 angle mode to RADIANS | Aligns with standard physics formulas without 2π conversion errors |
| If entering capacitance in μF or inductance in mH | Always use the EE or 10^x key for scientific notation |
Enter 25μF as 25 EE -6, never 0.000025 (prevents typo zeroes) |
| If XC required is ~30Ω at 120VAC 60Hz (from Example 2) | Purchase a standard AC Motor Run Capacitor (CBB60 type) | Buy: Genteq 12110 (100μF, 370VAC) — closest standard value above 88.4μF with safe voltage headroom. |
| If XL required is 56Ω at audio frequencies | Select an air-core or ferrite choke rated for >2x expected RMS current | Buy: Madisound or Dayton Audio 1.5mH to 2.0mH air-core inductor (14 AWG wire to minimize DCR). |
By leveraging the complex number engine of the TI-36 platform, you bypass tedious manual trigonometry and reduce the risk of sign errors in reactive components. Always verify your physical component's voltage and current ratings against the mathematical ideal, and remember that real-world inductors carry parasitic resistance (DCR) that will slightly shift your final phase angle.
References:
1. All About Circuits: Series Resistor-Inductor Circuits
2. Georgia State University HyperPhysics: AC Circuits
3. Texas Instruments Education: TI-36X Pro Scientific Calculator Guide
