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TEC-ELEC48-02 · Lesson 10 · 12.5 clock hours · lab

Capacitive and Inductive Reactance; Series RLC Circuits

Objective. Students will calculate total impedance and phase angle for a series RLC circuit at a given frequency, then verify results by measuring voltage drops with a True-RMS DMM within ±10%.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by an industry-certified professional or by a certification or approval body. It is a study draft and may contain errors. Do not use it as instructions for work on a job site or on live equipment. Always follow your instructor, your supervising professional, and current official references and regulations.

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Capacitive and Inductive Reactance

Reactance is the opposition that a capacitor or inductor offers to alternating current. Unlike resistance, reactance depends on frequency.

Capacitive reactance (X_C) decreases as frequency rises. The formula is:

X_C = 1 / (2πfC)

where f is frequency in hertz and C is capacitance in farads. A capacitor blocks DC (f = 0 gives infinite reactance) and passes high-frequency AC more easily.

Inductive reactance (X_L) increases as frequency rises. The formula is:

X_L = 2πfL

where L is inductance in henries. An inductor passes DC freely but increasingly opposes higher frequencies.

Both X_C and X_L are measured in ohms, just like resistance.

Impedance in a Series RLC Circuit

When a resistor (R), capacitor (C), and inductor (L) are connected in series, the total opposition to current is called impedance (Z), also in ohms. You cannot simply add R, X_L, and X_C together because they are not in phase with each other.

Because X_L and X_C are exactly 180° apart, they partially cancel. The net reactive component is (X_L − X_C). Impedance is then found with the Pythagorean theorem:

Z = √(R² + (X_L − X_C)²)

This is the impedance triangle: R is the horizontal (adjacent) side, (X_L − X_C) is the vertical (opposite) side, and Z is the hypotenuse.

Phase Angle

The phase angle (θ) tells you how far the total voltage leads or lags the circuit current:

θ = arctan((X_L − X_C) / R)

Phasor Diagram and Impedance Triangle

A phasor is a rotating vector that represents a sinusoidal quantity by its magnitude and phase angle. In the series RLC impedance triangle:

1. Draw R horizontally to the right.

2. Draw (X_L − X_C) vertically — upward if positive (inductive), downward if negative (capacitive).

3. The diagonal from origin to the tip is Z.

4. The angle at the origin is θ.

When you draw this triangle to scale on graph paper and measure Z and θ graphically, you should get results very close to your calculated values. This graphical check is a useful verification skill.

Making Measurements with a True-RMS DMM

A True-RMS DMM accurately measures the root-mean-square value of a non-ideal sine wave. For this lab, use it to measure AC voltage across each component while the function generator drives the circuit.

Key steps:

Verify your measured impedance: Z_measured = V_source / I, where I = V_R / R (since V_R is in phase with current, R is the easiest way to find I).

Safety

Working voltages in this lab are low (typically 5–10 V peak from the function generator) but good habits start here.

Connecting the Oscilloscope

Use the dual-channel oscilloscope to monitor the source waveform (CH1) and the voltage across R (CH2). The CH2 waveform is in phase with the current, making it useful for visualizing the phase relationship between source voltage and circuit current. Adjust the timebase so at least two complete cycles are visible.

Key terms8

Reactance (X)The frequency-dependent opposition to AC current offered by a capacitor or inductor, measured in ohms.
Capacitive reactance (X_C)Opposition offered by a capacitor; calculated as X_C = 1 / (2πfC); decreases as frequency increases.
Inductive reactance (X_L)Opposition offered by an inductor; calculated as X_L = 2πfL; increases as frequency increases.
Impedance (Z)The total opposition to AC current in a circuit combining resistance and reactance, found by the Pythagorean theorem: Z = √(R² + (X_L − X_C)²), measured in ohms.
Phase angle (θ)The angle between total voltage and current in an AC circuit; θ = arctan((X_L − X_C) / R); positive when inductive, negative when capacitive.
PhasorA vector that represents a sinusoidal quantity by its magnitude and phase angle relative to a reference, used to add AC quantities that are out of phase.
True-RMS DMMA digital multimeter that computes the true root-mean-square value of an AC waveform regardless of its shape, giving accurate readings for non-pure sine waves.
ResonanceThe condition in a series RLC circuit where X_L equals X_C, net reactance is zero, impedance equals R, and phase angle is 0°.

Worked examples4

1. A series RLC circuit has R = 100 Ω, L = 50 mH, and C = 10 µF. The source frequency is 500 Hz. Calculate X_L, X_C, Z, and θ.
  1. Calculate X_L: X_L = 2π × 500 × 0.050 = 2 × 3.14159 × 500 × 0.050 = 157.08 Ω
  2. Calculate X_C: X_C = 1 / (2π × 500 × 0.000010) = 1 / (0.031416) = 31.83 Ω
  3. Find net reactance: X_L − X_C = 157.08 − 31.83 = 125.25 Ω (inductive)
  4. Calculate Z: Z = √(100² + 125.25²) = √(10000 + 15688) = √25688 = 160.3 Ω
  5. Calculate θ: θ = arctan(125.25 / 100) = arctan(1.2525) ≈ 51.4° (positive, so voltage leads current)
Answer: X_L = 157.1 Ω, X_C = 31.8 Ω, Z = 160.3 Ω, θ = +51.4°
2. Using the circuit from Example 1 (R = 100 Ω, Z = 160.3 Ω, f = 500 Hz), the source voltage is 10 V RMS. Find the circuit current and the voltage drop across each component.
  1. Find current: I = V_source / Z = 10 / 160.3 = 0.06238 A (62.38 mA)
  2. Voltage across R: V_R = I × R = 0.06238 × 100 = 6.238 V
  3. Voltage across L: V_L = I × X_L = 0.06238 × 157.08 = 9.796 V
  4. Voltage across C: V_C = I × X_C = 0.06238 × 31.83 = 1.986 V
  5. Verify with phasor addition: V_source = √(V_R² + (V_L − V_C)²) = √(6.238² + (9.796 − 1.986)²) = √(38.91 + 61.00) = √99.91 ≈ 9.996 V ≈ 10 V ✓
Answer: I = 62.4 mA; V_R = 6.24 V, V_L = 9.80 V, V_C = 1.99 V (phasor sum ≈ 10 V, confirming results)
3. A series RLC circuit has R = 470 Ω, L = 100 mH, and C = 0.47 µF at a frequency of 1000 Hz. Determine whether the circuit is inductive or capacitive, find Z and θ.
  1. X_L = 2π × 1000 × 0.100 = 628.3 Ω
  2. X_C = 1 / (2π × 1000 × 0.00000047) = 1 / (0.002953) = 338.6 Ω
  3. Net reactance = X_L − X_C = 628.3 − 338.6 = 289.7 Ω (positive → inductive)
  4. Z = √(470² + 289.7²) = √(220900 + 83926) = √304826 = 552.1 Ω
  5. θ = arctan(289.7 / 470) = arctan(0.6163) ≈ 31.6°
Answer: The circuit is inductive. Z = 552.1 Ω, θ = +31.6°
4. In the lab, a student measures V_source = 8.00 V RMS and V_R = 4.80 V RMS across a 220 Ω resistor. Calculate the measured current, the measured impedance, and verify it is within ±10% of the calculated Z = 366 Ω.
  1. Measured current: I = V_R / R = 4.80 / 220 = 0.02182 A (21.82 mA)
  2. Measured impedance: Z_meas = V_source / I = 8.00 / 0.02182 = 366.6 Ω
  3. Percent error: |366.6 − 366| / 366 × 100% = 0.6 / 366 × 100% ≈ 0.16%
  4. 0.16% is well within the ±10% tolerance specified for the lab
Answer: I = 21.8 mA, Z_meas = 366.6 Ω; percent error ≈ 0.2%, which is within ±10% — measurement passes.

The math in this lesson2

Learners compute X_L, X_C, impedance sqrt(100^2 + 125.25^2) = 160.3 ohms and phase angle arctan = 51.4 degrees.
A-SSE.1G-SRT.8
Brush up in the math halls at newSchoolWorks.com:
In class: Problem-set lesson: students solve six series RLC problems calculating XC, XL, Z, and θ using right-triangle phasor method; instructor reviews on board. · Series RLC breadboard lab: build a resistor-capacitor-inductor series circuit driven by a sine-wave function generator; measure voltage across each component with True-RMS DMM at two frequencies; record in a phasor worksheet. · Impedance triangle activity: using measured values, students draw the impedance triangle to scale on graph paper, measure Z and θ graphically, and compare to calculated values; attach drawing to lab report.
Materials: Breadboards, resistors, capacitors, inductors, function generator, True-RMS DMM, dual-channel oscilloscope, graph paper, phasor worksheet, lab report template
Check: All six problem-set answers correct to within rounding. Lab voltage measurements within ±10%; impedance triangle drawn correctly and values consistent with measurements.

Practice check8

Answer each question, then check it. Nothing is recorded; this is your own practice.

AC Circuit Theory
1. A capacitor is connected in series with a 1 kHz AC source. If the frequency is doubled to 2 kHz while all other values remain unchanged, what happens to X_C?
AC Circuit Theory
2. In a series RLC circuit, R = 300 Ω, X_L = 400 Ω, and X_C = 0 Ω. What is the impedance Z?
AC Circuit Theory
3. In a series RLC circuit at resonance, which statement is correct?
AC Circuit Theory
4. A student measures V_R = 3 V, V_L = 7 V, and V_C = 3 V across the components in a series RLC circuit. What is the approximate source voltage?
Test Equipment and Measurement
5. Why must a True-RMS DMM be used instead of an averaging DMM when measuring AC voltages in this lab?
Safety Practices
6. Before removing a capacitor from a powered-down circuit on the breadboard, what safety step must you take first?
AC Circuit Theory
7. On an impedance triangle for a series RLC circuit, the horizontal side represents R = 600 Ω and the hypotenuse (Z) measures 1000 Ω. What is the phase angle θ?
Test Equipment and Measurement
8. When using a dual-channel oscilloscope to observe phase difference between source voltage and circuit current in a series RLC lab, the voltage across which component is most useful as a proxy for current?
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