Impedance of Series RLC Circuit

Core idea

Overview

The impedance (Z) of a series RLC circuit is the total opposition to alternating current (AC) flow, combining the effects of resistance (R), inductive reactance (X_L), and capacitive reactance (X_C). It is a complex quantity, but its magnitude, calculated by this formula, represents the effective resistance of the circuit. This value is crucial for determining current and power in AC circuits, especially when dealing with resonance phenomena.

When to use: Use this equation when analyzing series AC circuits containing resistors, inductors, and capacitors to find the total impedance. It's particularly useful for calculating current (using Ohm's Law, I = V/Z) or understanding the circuit's behavior at different frequencies, especially near resonance.

Why it matters: Understanding impedance is fundamental in electrical engineering for designing and analyzing AC systems, including power distribution, communication circuits, and filter networks. It allows engineers to predict circuit response, optimize performance, and prevent issues like excessive current or voltage drops, ensuring reliable operation of electronic devices.

Symbols

Variables

R = Resistance, $X_{L}$ = Inductive Reactance, $X_{C}$ = Capacitive Reactance, Z = Impedance

R

Resistance

O

X_{L}

Inductive Reactance

O

X_{C}

Capacitive Reactance

O

Z

Impedance

O

Walkthrough

Derivation

Formula: Impedance of Series RLC Circuit

The impedance of a series RLC circuit is the total opposition to AC current, combining resistance and net reactance.

The circuit components (R, L, C) are ideal.
The circuit is a series connection of a resistor, an inductor, and a capacitor.
The AC source is sinusoidal.

1

Representing Components in Phasor Domain:

In AC circuit analysis, components are represented by their impedances in the complex phasor domain. Resistance is purely real, inductive reactance is positive imaginary, and capacitive reactance is negative imaginary.

Z_{R} = R, Z_{L} = j X_{L}, Z_{C} = - j X_{C}

2

Total Impedance in Series:

For components in series, the total impedance is the sum of individual impedances. We combine the real and imaginary parts to get the complex impedance.

Z_{t o t a l} = Z_{R} + Z_{L} + Z_{C} = R + j X_{L} - j X_{C} = R + j (X_{L} - X_{C})

3

Magnitude of Total Impedance:

The formula for the magnitude of a complex number `a + jb` is ` $a^{2} + b^{2}$ `. Applying this to `R + j( $X_{L}$ - $X_{C}$ )` gives the magnitude of the total impedance, which is the scalar value represented by Z.

∣ Z_{t o t a l} ∣ = R^{2} + (X_{L} - X_{C})^{2}

Result

∣ Z_{t o t a l} ∣ = R^{2} + (X_{L} - X_{C})^{2}

Source: Fundamentals of Electric Circuits by C.K. Alexander and M.N.O. Sadiku, Chapter 11: AC Power Analysis

Free formulas

Rearrangements

Solve for $R$

Impedance of Series RLC Circuit: Make R the subject

R = Z^{2} - (X_{L} - X_{C})^{2}

To make R the subject, isolate the $R^{2}$ term by subtracting the squared net reactance from $Z^{2}$ , then take the square root.

Difficulty: 2/5

Solve for $X_{L}$

Impedance of Series RLC Circuit: Make $X_{L}$ the subject

X_{L} = X_{C} \pm Z^{2} - R^{2}

To make $X_{L}$ the subject, isolate the ( $X_{L}$ - $X_{C}$ )^2 term, take the square root, and then add $X_{C}$ . Note that there are two possible solutions for $X_{L}$ - $X_{C}$ .

Difficulty: 3/5

Solve for $X_{C}$

Impedance of Series RLC Circuit: Make $X_{C}$ the subject

X_{C} = X_{L} \mp Z^{2} - R^{2}

To make $X_{C}$ the subject, isolate the ( $X_{L}$ - $X_{C}$ )^2 term, take the square root, and then rearrange. Note that there are two possible solutions for $X_{L}$ - $X_{C}$ .

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a hyperbolic curve where Z increases as R increases, approaching a linear slope at higher values while remaining restricted to a domain where Z is at least the absolute difference of the reactances. For an engineering student, this shape demonstrates that at low resistance, the total impedance is dominated by the circuit reactances, whereas at high resistance, the impedance becomes increasingly dependent on the resistance value itself. The most important feature is that the curve never reaches zero, meaning the total opposition to current flow is always constrained by the inherent reactive components of the circuit.

Graph type: hyperbolic

Why it behaves this way

Intuition

The impedance can be visualized as the hypotenuse of a right triangle in the complex impedance plane, where resistance forms one leg and the net reactance (the difference between inductive and capacitive reactance)

Z

Total opposition to alternating current (AC) flow in a series RLC circuit.

Think of it as the circuit's 'total effective resistance' for AC, combining all forms of opposition. A higher Z means less AC current will flow for a given voltage.

R

Resistance, the opposition to current flow that dissipates energy as heat.

This is the familiar 'friction' for current, always present and converting electrical energy into heat, regardless of whether the current is AC or DC.

X_{L}

Inductive reactance, the opposition to AC current flow specifically due to an inductor.

Inductors resist changes in current. The faster the current tries to change (higher frequency), the more an inductor opposes it, acting like an 'inertial' force against AC.

X_{C}

Capacitive reactance, the opposition to AC current flow specifically due to a capacitor.

Capacitors resist changes in voltage. At higher frequencies, they allow current to flow more easily, so their opposition (reactance) decreases.

Signs and relationships

√(R^2 + (X_L - X_C)^2): This structure represents the magnitude of a vector sum, specifically using the Pythagorean theorem. Resistance (R) is considered 'in-phase' with voltage, while reactances ( $X_{L}$ and $X_{C}$ )
(X_L - X_C): Inductive reactance ( $X_{L}$ ) and capacitive reactance ( $X_{C}$ ) have opposite phase effects on the current relative to the voltage. $X_{L}$ causes current to lag voltage by 90 degrees, while $X_{C}$ causes current to lead voltage by 90

Free study cues

Insight

Canonical usage

All quantities (impedance, resistance, inductive reactance, and capacitive reactance) are consistently expressed in ohms (Ω) within the International System of Units (SI).

Common confusion

A common mistake is forgetting that both inductive reactance ( $X_{L}$ ) and capacitive reactance ( $X_{C}$ ) are measured in ohms (Ω), just like resistance (R) and impedance (Z).

Unit systems

$Z$ Ω - Represents the total opposition to alternating current (AC) flow.

$R$ Ω - Represents the opposition to current flow due to resistive components.

$X_{L}$ Ω - Represents the opposition to AC current flow due to inductance.

$X_{C}$ Ω - Represents the opposition to AC current flow due to capacitance.

One free problem

Practice Problem

A series RLC circuit has a resistance of 30 O, an inductive reactance of 50 O, and a capacitive reactance of 20 O. Calculate the total impedance of the circuit.

Resistance30 O

Inductive Reactance50 O

Capacitive Reactance20 O

Solve for: $Z$

Hint: First, find the net reactance ( $X_{L}$ - $X_{C}$ ), then apply the Pythagorean theorem.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When designing audio crossover networks or tuning radio receivers, Impedance of Series RLC Circuit is used to calculate Impedance from Resistance, Inductive Reactance, and Capacitive Reactance. The result matters because it helps check whether a circuit component is operating within the required voltage, current, power, or resistance range.

Study smarter

Tips

Ensure all reactances ( $X_{L}$ , $X_{C}$ ) and resistance (R) are in Ohms (O).
Remember that $X_{L}$ = 2pfL and $X_{C}$ = 1/(2pfC), where f is frequency, L is inductance, and C is capacitance.
The term ( $X_{L}$ - $X_{C}$ ) represents the net reactance; its sign indicates whether the circuit is inductive or capacitive.
At resonance, $X_{L}$ = $X_{C}$ , making the net reactance zero and impedance equal to resistance (Z=R).

Avoid these traps

Common Mistakes

Incorrectly calculating $X_{L}$ or $X_{C}$ before applying the impedance formula.
Forgetting to square the terms or take the square root at the end.
Confusing impedance with resistance or reactance; impedance is the overall opposition.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The impedance of a series RLC circuit is the total opposition to AC current, combining resistance and net reactance.

Use this equation when analyzing series AC circuits containing resistors, inductors, and capacitors to find the total impedance. It's particularly useful for calculating current (using Ohm's Law, I = V/Z) or understanding the circuit's behavior at different frequencies, especially near resonance.

Understanding impedance is fundamental in electrical engineering for designing and analyzing AC systems, including power distribution, communication circuits, and filter networks. It allows engineers to predict circuit response, optimize performance, and prevent issues like excessive current or voltage drops, ensuring reliable operation of electronic devices.

Incorrectly calculating X_L or X_C before applying the impedance formula. Forgetting to square the terms or take the square root at the end. Confusing impedance with resistance or reactance; impedance is the overall opposition.

When designing audio crossover networks or tuning radio receivers, Impedance of Series RLC Circuit is used to calculate Impedance from Resistance, Inductive Reactance, and Capacitive Reactance. The result matters because it helps check whether a circuit component is operating within the required voltage, current, power, or resistance range.

Ensure all reactances (X_L, X_C) and resistance (R) are in Ohms (O). Remember that X_L = 2pfL and X_C = 1/(2pfC), where f is frequency, L is inductance, and C is capacitance. The term (X_L - X_C) represents the net reactance; its sign indicates whether the circuit is inductive or capacitive. At resonance, X_L = X_C, making the net reactance zero and impedance equal to resistance (Z=R).

References

Sources

Halliday, Resnick, and Walker, Fundamentals of Physics
Alexander and Sadiku, Fundamentals of Electric Circuits
Wikipedia: Electrical impedance
NIST SP 330: The International System of Units (SI)
IUPAC Gold Book
Engineering Circuit Analysis by William H. Hayt Jr., Jack E. Kemmerly, Steven M. Durbin
Fundamentals of Electric Circuits, 7th ed. by Charles K. Alexander and Matthew N.O. Sadiku
Electric Circuits, 11th ed. by James W. Nilsson and Susan A. Riedel

Overview

Variables

Derivation

Representing Components in Phasor Domain:

Total Impedance in Series:

Magnitude of Total Impedance:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Inductive Reactance

Capacitive Reactance

Resonance Frequency of RLC Circuit

Frequently Asked Questions

Sources