Second-Order and RLC Transients

Introduction and Context

A first-order circuit contains one independent energy-storage element and produces a response governed by one time constant. Circuits containing an inductor and a capacitor possess two independent storage mechanisms and are therefore described by a second-order differential equation. Their transient response may decay monotonically, decay at the fastest nonoscillatory rate, or oscillate while its amplitude decreases. These behaviors are central to filters, power converters, motor drives, communication systems, protection circuits, and control interfaces.

This chapter develops the natural-response and complete-response methods for series and parallel RLC circuits. The analysis begins with Kirchhoff’s laws, establishes the characteristic equation, classifies damping, and then applies the results to practical specifications such as settling time, peak time, overshoot, and resonance. Unless stated otherwise, circuits are linear, time-invariant, lumped, and initially relaxed or specified by known initial conditions.

Learning Objectives

By the end of this chapter, you should be able to:

1. Derive the differential equations for series and parallel RLC circuits.
2. Determine natural frequency, damping coefficient, and damping ratio.
3. Classify overdamped, critically damped, and underdamped responses.
4. Calculate complete responses from initial conditions and source changes.
5. Interpret oscillation, stability, resonance, settling time, and overshoot.
6. Select practical component values to meet transient specifications.
7. Recognize modeling assumptions and measurement limitations in real circuits.

1. Energy Storage and Second-Order Behavior

An inductor stores magnetic energy and a capacitor stores electric energy:

Here, L is inductance in henries, i is inductor current in amperes, C is capacitance in farads, and v is capacitor voltage in volts. An inductor current cannot change instantaneously, and a capacitor voltage cannot change instantaneously:

These continuity conditions provide the two initial conditions required by a second-order equation. Resistance dissipates energy according to p_R = Ri² = v²/R, so it determines how rapidly oscillations decay.

For a source-free network, the natural response is the response caused only by energy initially stored in the inductors and capacitors. For a driven network, the complete response is

where x_n(t) is the natural response and x_f(t) is the forced or particular response. In a stable circuit, the natural response tends to zero and the forced response remains as the long-term steady state.

2. Series RLC Circuit

Consider a source-free series circuit containing R, L, and C. Let the loop current be i(t). Kirchhoff’s voltage law gives

Using v_R = Ri, v_L = L di/dt, and i = C dv_C/dt, differentiate the loop equation once to eliminate the capacitor voltage:

Dividing by L gives the standard form

The characteristic equation is

Define the undamped natural angular frequency and damping coefficient as

where ω₀ and α have units of radians per second. The characteristic equation becomes

Its roots are

The damping ratio is

For a series RLC circuit, increasing resistance increases damping; hence

The critical resistance is obtained from α = ω₀:

This expression assumes ideal lumped components and a purely series resistance.

Series-RLC Damping Cases

For the underdamped case,

is the damped angular frequency. The oscillation frequency is lower than the undamped natural frequency because resistance removes energy during each cycle.

3. Parallel RLC Circuit

For a source-free parallel RLC circuit, let v(t) be the common branch voltage. Kirchhoff’s current law gives

Using i_C = C dv/dt, i_R = v/R, and differentiating the inductor relation v = L di_L/dt, one obtains

or

Thus

The critical parallel resistance is

The resistance has the opposite qualitative effect from the series case: in a parallel circuit, a smaller resistance provides stronger damping because it offers a stronger dissipative branch. The classification remains the same, based on the comparison of α and ω₀.

For a general second-order network, it is safer to derive the differential equation or characteristic polynomial first rather than assume that a remembered series formula applies to a parallel circuit.

4. Natural and Complete Responses

The natural response is found by setting independent sources to zero and solving the characteristic equation. For a circuit driven by a constant source, the final value is usually obtained from the dc steady-state circuit: an ideal inductor behaves as a short circuit and an ideal capacitor as an open circuit. The complete response can then be written as

provided the circuit is stable and the input is a constant switched at t = 0. The constants in x_n(t) are determined from the initial value and initial derivative.

For a series circuit with current as the response variable, the initial current is obtained directly from inductor-current continuity. Its initial derivative follows from KVL:

For capacitor voltage, use i_C = C dv_C/dt. These relations prevent a common error: assigning arbitrary initial values to variables whose continuity is physically constrained.

Worked Example 1: Series Underdamped Response

A source-free series circuit has R = 4 Ω, L = 1 H, and C = 0.25 F. At t = 0, the inductor current is i(0) = 2 A and the capacitor voltage is zero. Find the current response.

First,

and

Thus α = ω₀, so the response is critically damped, not underdamped. The repeated-root form is

The initial condition gives A = 2. From KVL at t = 0^+,

Differentiating the response and evaluating at zero gives i'(0)=B−2A. Therefore B−4 = −8, so B = −4. The final response is

The current initially decreases, crosses zero, and then approaches zero without sustained oscillation. The example illustrates that a response can change sign without being underdamped; oscillation requires complex conjugate poles.

Worked Example 2: Underdamped Parameters

Let a series circuit have R = 2 Ω, L = 1 H, and C = 0.25 F. Then

so

The natural response has the form

The exponential envelope decays with time constant 1/α = 1 s, while the oscillation period is approximately 2π/ω_d = 3.63 s.

5. Stability, Oscillation, and Transient Specifications

A continuous-time second-order circuit is asymptotically stable when both characteristic roots have negative real parts. For positive R, L, and C, both series and parallel RLC circuits are stable. If effective resistance becomes negative, as can occur in an active circuit, poles may move into the right half-plane and the response can grow rather than decay.

For an underdamped standard second-order response, the normalized step-response characteristics are commonly expressed in terms of ζ and ω₀. The percent overshoot is

valid for 0 < ζ < 1. The peak time is

and the approximate 2% settling time is

The approximation assumes a conventional dominant second-order step response and a 2% error band. For a 5% band, the corresponding approximation is t_s ≈ 3/(ζω₀). These formulas should not be applied blindly to systems with additional poles, zeros, saturation, or nonlinear loading.

A useful design trade-off follows: increasing damping reduces overshoot but may alter speed and bandwidth. Critical damping gives the fastest response without oscillation only within the ideal second-order model. In a real circuit, parasitic elements and sensor bandwidth can change the observed result.

6. Resonance and Frequency-Domain Interpretation

The series RLC impedance is

At the ideal resonance frequency ω₀ = 1/√(LC), the reactances cancel and the impedance equals R. Consequently, current is maximum for a fixed sinusoidal voltage source. The quality factor for a series RLC circuit is

for the ideal series model. A high Q indicates narrow bandwidth and weak damping, while a low Q indicates broad response and stronger damping.

The time-domain and frequency-domain views describe the same poles. Complex poles close to the imaginary axis produce long-lived oscillations and a sharp resonance. Poles farther into the left half-plane produce faster decay and a broader, less selective response.

7. Practical Design Procedure

A reliable solution procedure is:

1. Identify the response variable and replace switching events with initial and final circuit models.
2. Apply continuity: capacitor voltage and inductor current cannot jump in an ideal circuit.
3. Derive the differential equation using KCL or KVL.
4. Normalize it to s²+2αs+ω₀²=0.
5. Calculate α, ω₀, ζ, and, if necessary, ω_d.
6. Classify the roots and write the corresponding response form.
7. Use the two initial conditions to determine constants.
8. Check units, initial values, final values, and limiting behavior.
9. Compare the predicted transient with component tolerances and parasitic effects.

For example, to design a series RLC circuit with a specified natural frequency, choose

Then select practical L or C, and choose resistance to obtain the desired damping ratio:

The resistance may include a physical resistor, winding resistance, capacitor ESR, switch resistance, and PCB trace resistance. Ignoring these contributions can lead to an incorrect damping prediction.

8. Illustrative Engineering Case: DC-Link Resonance

Background

Consider an illustrative motor-drive dc link represented at a particular operating point by a series inductance of 10 μH and a capacitance of 100 μF. Cable resistance and capacitor ESR provide an effective series resistance of 0.20 Ω. The simplified model is not a complete converter model, but it is useful for estimating ringing after a switching event.

Problem

Determine whether the idealized network is strongly or weakly damped and estimate its oscillation frequency.

Analysis

The undamped natural frequency is

or approximately

The damping coefficient is

Therefore

The response is underdamped, with

The predicted ringing frequency is approximately 4.77 kHz.

Approach and Outcome

A designer could increase damping by adding a properly rated series damping resistor, using a lossy snubber, changing capacitor technology, or modifying layout to reduce parasitic inductance. The choice must account for power loss, voltage stress, switching frequency, thermal limits, and electromagnetic-interference requirements. A resistor selected solely from the transient equation may fail if its pulse-energy rating is inadequate.

Learning Points

The case demonstrates that a short switching transient may be a genuine second-order resonance rather than measurement noise. Oscillation frequency is primarily set by L and C, whereas decay rate depends strongly on effective resistance. Oscilloscope probes also add capacitance and ground-lead inductance, potentially shifting the measured resonance.

9. Limitations, Safety, and Contemporary Practice

Ideal RLC models omit nonlinear magnetic saturation, capacitor voltage dependence, temperature effects, dielectric loss, skin effect, mutual coupling, semiconductor switching behavior, and distributed transmission-line effects. At sufficiently high frequency, a component may no longer behave as a lumped ideal element. Simulation tools should therefore be used alongside datasheets, impedance measurements, and worst-case calculations.

Transient overvoltage and current can damage insulation, switches, capacitors, and human-accessible terminals. Engineers must observe applicable electrical-safety standards, creepage and clearance requirements, fusing, discharge paths, and measurement-category ratings. A capacitor that appears discharged in a steady-state calculation may retain hazardous energy.

Modern design workflows combine symbolic derivation, circuit simulation, automated parameter sweeps, and laboratory validation. AI-assisted tools can help identify likely pole locations or generate simulation scripts, but engineers must verify equations, models, units, and safety assumptions independently. A plausible waveform is not evidence that the underlying model is correct.

Chapter Summary

Second-order RLC circuits contain two independent energy-storage elements and are governed by a second-order differential equation. For the series RLC natural response, α=R/(2L) and ω₀=1/√(LC). For the parallel RLC voltage response, α=1/(2RC) and the same ω₀. The damping ratio ζ=α/ω₀ classifies the response as overdamped, critically damped, or underdamped. Underdamped circuits oscillate at ω_d=ω₀√(1−ζ²) while their envelope decays at rate α.

Natural response analysis requires the characteristic roots and two physically consistent initial conditions. Complete response combines natural and forced components. Pole location determines stability, oscillation, and decay. In practical design, overshoot, settling time, resonance, component tolerances, parasitics, thermal limits, and measurement loading must all be considered.

Key Terms

Second-order circuit: A circuit whose governing differential equation has second derivative terms and generally contains two independent storage states.

Natural response: Response caused by initially stored energy with independent sources set to zero.

Forced response: Response produced by an external source or excitation.

Complete response: Sum of natural and forced responses.

Natural angular frequency: ω₀=1/√(LC) for an ideal RLC network.

Damping coefficient: Parameter α controlling exponential decay.

Damping ratio: Dimensionless ratio ζ=α/ω₀.

Overdamped: Response with two distinct real negative poles.

Critically damped: Response with a repeated real pole and fastest ideal nonoscillatory behavior.

Underdamped: Response with complex conjugate poles and decaying oscillation.

Damped angular frequency: ω_d=√(ω₀²−α²) for an underdamped circuit.

Characteristic equation: Polynomial in s whose roots determine natural modes.

Resonance: Condition at which inductive and capacitive reactances cancel.

Quality factor: Measure of resonance selectivity and damping; for ideal series RLC, Q=1/(2ζ).

Settling time: Time required for a response to enter and remain within a specified error band.

Overshoot: Amount by which a transient exceeds its final value, usually expressed as a percentage.

Continuity condition: Physical constraint that ideal inductor current and capacitor voltage cannot change instantaneously.

Pole: Root of a system denominator or characteristic equation.

Parasitic element: Unintended resistance, inductance, or capacitance associated with real components and interconnections.

Review Questions

Short Answer Questions

1. Why are RLC circuits described by second-order differential equations?
2. Define the damping ratio and explain its role in classification.
3. State the continuity conditions for an ideal inductor and capacitor.
4. What is the difference between natural and forced response?
5. Why does resistance reduce the oscillation amplitude?

Long Answer Questions

1. Derive the source-free series RLC characteristic equation from KVL and identify α and ω₀.
2. Derive the source-free parallel RLC voltage equation from KCL.
3. Explain overdamped, critically damped, and underdamped responses using pole locations and time-domain expressions.
4. Derive the critical resistance for a series RLC circuit and interpret its physical meaning.
5. Explain how time-domain damping and frequency-domain resonance are related.

Analytical / Application Questions

1. A series RLC circuit has L=4 mH, C=25 μF, and R=8 Ω. Calculate α, ω₀, ζ, and classify the response.
2. Design a series RLC network with ω₀=10,000 rad/s and ζ=0.5 if C=10 μF. Determine L and R.
3. A measured ringing waveform has a frequency of 2 kHz and a 2% settling time of approximately 1 ms. Explain what these observations suggest about its poles and identify at least two practical causes for disagreement with an ideal RLC model.

Multiple Choice Questions with Answers

Question 1

For a source-free series RLC circuit, which expression is the damping coefficient?

A. α=R/(2L)

B. α=2L/R

C. α=1/(2RC)

D. α=√(LC)

Correct Answer: A. α=R/(2L)

Question 2

A second-order circuit is critically damped when:

A. α>ω₀

B. α=ω₀

C. α<ω₀

D. α=0

Correct Answer: B. α=ω₀

Question 3

Which quantity cannot change instantaneously in an ideal capacitor?

A. Capacitor current

B. Capacitor power

C. Capacitor voltage

D. Capacitor charge flow rate

Correct Answer: C. Capacitor voltage

Question 4

For an underdamped response, the damped angular frequency is:

A. ω_d=ω₀+α

B. ω_d=α−ω₀

C. ω_d=√(ω₀²−α²)

D. ω_d=ω₀²+α²

Correct Answer: C. ω_d=√(ω₀²−α²)

Question 5

For a parallel RLC circuit, the damping coefficient is:

A. R/(2L)

B. 1/(2RC)

C. 2RC

D. √(LC)

Correct Answer: B. 1/(2RC)

Question 6

A series RLC circuit has L=1 H, C=0.25 F, and R=2 Ω. What is its damping classification?

A. Overdamped

B. Critically damped

C. Underdamped

D. Unstable

Correct Answer: C. Underdamped

Question 7

For a stable underdamped second-order step response, increasing damping ratio generally:

A. Increases overshoot

B. Reduces overshoot

C. Makes the natural frequency zero

D. Creates a right-half-plane pole

Correct Answer: B. Reduces overshoot

Question 8

The ideal series-RLC resonance frequency occurs when:

A. R=0

B. ωL=1/(ωC)

C. ωL=R

D. ωC=1/R

Correct Answer: B. ωL=1/(ωC)

Question 9

For a conventional second-order response, the approximate 2% settling time is:

A. t_s≈4/(ζω₀)

B. t_s≈ζω₀/4

C. t_s≈π/ω₀

D. t_s≈ω₀/ζ

Correct Answer: A. t_s≈4/(ζω₀)

Question 10

For positive R, L, and C in the ideal passive model, the natural-response poles are:

A. Always in the right half-plane

B. On the imaginary axis for every R

C. In the left half-plane

D. Always zero

Correct Answer: C. In the left half-plane