How to Use a Texas Instrument Calculator for AC Impedance and Complex Circuit Math

The Core Formula: Series RLC Complex Impedance

When analyzing alternating current (AC) circuits, resistance alone is insufficient to describe opposition to current flow. We must account for phase shifts introduced by inductors and capacitors using complex impedance. The foundational formula for the total complex impedance ($Z$) of a series RLC circuit is:

$Z = R + j(2\pi f L - \frac{1}{2\pi f C})$

Assumptions and Application Boundaries

This formula applies strictly to steady-state, sinusoidal AC circuits with linear, ideal components. It assumes the circuit has reached steady-state (transient DC charging/discharging has settled) and that the source is a pure sine wave. It does not apply to non-linear components like diodes or transistors, nor does it account for the parasitic equivalent series resistance (ESR) of real-world capacitors unless explicitly added to $R$.

Rearranged Forms for Component Sizing

On the bench, you rarely just calculate $Z$ from known components. More often, you have a target impedance or resonant frequency and need to size a specific component. Here are the algebraically rearranged forms solving for each variable, assuming you know the required reactance ($X$) or total impedance magnitude ($|Z|$):

• Solving for Resistance ($R$): $R = \sqrt{|Z|^2 - (X_L - X_C)^2}$
• Solving for Inductance ($L$): $L = \frac{X_L}{2\pi f}$ (where $X_L$ is the required inductive reactance)
• Solving for Capacitance ($C$): $C = \frac{1}{2\pi f X_C}$ (where $X_C$ is the required capacitive reactance)
• Solving for Resonant Frequency ($f_r$): $f_r = \frac{1}{2\pi\sqrt{LC}}$ (The frequency where $X_L = X_C$, making the imaginary part zero)

How to Use a Texas Instrument Calculator for Complex AC Math

While graphing calculators like the TI-84 Plus CE are popular, the TI-36X Pro is the superior tool for electrical engineering math. It handles complex numbers natively in MathPrint mode without the clunky matrix workarounds required on basic scientific calculators. Here is the exact setup sequence for AC impedance calculations:

1. Set Complex Mode: Press MODE, scroll down to CMPLX, and ensure it is set to a+bi (Rectangular) or r\angle\theta (Polar). For initial entry, a+bi is usually easier.
2. Set Angle Mode: In the same MODE menu, ensure DEGREE is selected. Electrical phase angles are universally expressed in degrees (e.g., $-45^\circ$), not radians.
3. Entering the Imaginary Unit: Do not type the letter 'i' using the alpha keyboard. Use the dedicated i key (located above the x^2 key or accessed via 2nd + i depending on your exact hardware revision).
4. Converting Formats: If your calculator is in Rectangular mode but you need Polar (Magnitude and Phase), press 2nd, then CMPLX, and select >r\angle\theta. Paste this conversion function at the end of your equation.

Worked Example 1: Finding Total Impedance and Phase Angle

Scenario: You are building a passive audio crossover. You have a series circuit with a $100 \, \Omega$ resistor, a $50 \text{ mH}$ inductor, and a $10 \, \mu\text{F}$ capacitor. The signal generator is set to $1 \text{ kHz}$. What is the total impedance in Polar form?

Step 1: Convert to Base SI Units

• $R = 100 \, \Omega$
• $L = 50 \text{ mH} = 0.050 \text{ H}$
• $C = 10 \, \mu\text{F} = 0.000010 \text{ F}$
• $f = 1 \text{ kHz} = 1000 \text{ Hz}$

Step 2: Calculate Individual Reactances (Intermediate Tracking)

• $X_L = 2\pi f L = 2 \times \pi \times 1000 \text{ Hz} \times 0.050 \text{ H} = 314.159 \, \Omega$
• $X_C = \frac{1}{2\pi f C} = \frac{1}{2 \times \pi \times 1000 \text{ Hz} \times 0.000010 \text{ F}} = 15.915 \, \Omega$

Step 3: Calculator Entry on TI-36X Pro

Type the following exact sequence into the calculator in a+bi mode:

100 + ( 314.159 - 15.915 ) i [Press ENTER]

Rectangular Result: $100 + 298.24i \, \Omega$

Step 4: Convert to Polar Form

Press 2nd → CMPLX → select >r\angle\theta → ENTER.

Final Polar Result: $314.56 \, \Omega \angle 71.5^\circ$

Sanity Check: The magnitude ($314 \, \Omega$) is larger than the resistance ($100 \, \Omega$), and the phase is highly positive (inductive), which makes sense because $X_L (314 \, \Omega) \gg X_C (15.9 \, \Omega)$. (See All About Circuits: Series RLC for theory verification).

Worked Example 2: Sizing a Capacitor for Target Reactance

Scenario: You need to cancel out the inductive reactance of a $200 \text{ mH}$ relay coil operating at $60 \text{ Hz}$ to achieve unity power factor (resonance). What exact capacitance value is required?

Step 1: Identify Knowns and Target

• $L = 200 \text{ mH} = 0.200 \text{ H}$
• $f = 60 \text{ Hz}$
• Target: $X_C$ must equal $X_L$

Step 2: Calculate $X_L$

• $X_L = 2 \times \pi \times 60 \text{ Hz} \times 0.200 \text{ H} = 75.398 \, \Omega$

Step 3: Rearrange and Solve for $C$

Using the rearranged formula $C = \frac{1}{2\pi f X_C}$:

Enter into TI-36X Pro: 1 / ( 2 * \pi * 60 * 75.398 )

Result: $0.00003517 \text{ F}$

Step 4: Convert to Practical Units

Multiply by $10^6$ to get microfarads: $35.17 \, \mu\text{F}$.
Concrete Pick: Since $35.17 \, \mu\text{F}$ is not a standard E12 capacitor value, select the closest standard non-polarized film capacitor: a $33 \, \mu\text{F}$ or $36 \, \mu\text{F}$ motor run capacitor rated for at least 250VAC.

Calculator Mode Decision Tree for Circuit Analysis

Choosing the wrong calculator mode leads to syntax errors or misinterpreted phase angles. Use this decision path to configure your TI calculator before starting any AC analysis.

By strictly adhering to base SI unit conversions and utilizing the native complex number engine of the TI-36X Pro, you eliminate the algebraic friction of AC circuit analysis. Always verify your final magnitude against the physical reality of your components: if your calculated impedance is lower than your smallest resistor, or your phase angle exceeds $\pm 90^\circ$ in a passive circuit, you have a unit conversion error.