Sinusoidal Steady State, Resonance and Frequency Response

Introduction

Sinusoidal steady-state analysis is fundamental to electrical engineering because power systems, communication circuits, instrumentation networks and filters are commonly excited by periodic signals. In a stable linear time-invariant (LTI) network, the transient response eventually decays, leaving a sinusoid at the excitation frequency. The output generally differs from the input in amplitude and phase.

Phasor analysis replaces differential equations by algebraic equations involving complex impedances. Frequency response then extends the analysis by examining how a network behaves as frequency varies. Resonance is a particularly important frequency-domain phenomenon in which inductive and capacitive effects cancel at one or more frequencies. It can produce useful selectivity, voltage magnification and filtering, but also dangerous overvoltages, overcurrents and sensitivity to component tolerances.

Learning Objectives

After studying this chapter, the reader should be able to:

• derive sinusoidal steady-state responses using phasors;
• calculate impedance, admittance and complex power;
• formulate network functions and interpret poles and zeros;
• distinguish series resonance, parallel resonance and antiresonance;
• derive resonant frequency, half-power frequencies, bandwidth and quality factor;
• apply the correct response definition when specifying bandwidth;
• identify the stability conditions for second-order resonance;
• construct and interpret magnitude, phase and Bode responses;
• assess loading, tolerances, parasitics and measurement limitations; and
• design and evaluate practical resonant networks.

1. Sinusoidal Steady-State Foundations

1.1 Natural and forced responses

The response of an LTI circuit can be written as the sum of its zero-input and zero-state components. Equivalently, it is often described as containing a natural response and a forced response. The natural response is determined by the network poles and initial stored energy. The forced response is produced by the source.

For an asymptotically stable circuit, every natural mode decays because all poles lie strictly in the open left half of the complex plane. A sinusoidal input therefore produces the long-term response

where ω is angular frequency in rad/s, V_m and I_m are peak amplitudes, and φ_v and φ_i are phase angles.

A marginally stable circuit with poles on the imaginary axis requires special care. If an excitation coincides with an undamped natural frequency, an unbounded resonant term can develop; an ordinary bounded sinusoidal steady state then does not exist.

1.2 Phasor representation

Using Euler's identity, a sinusoid is represented as the real part of a complex exponential:

A phasor may use peak or root-mean-square (RMS) magnitude. This chapter uses RMS phasors for power calculations:

Differentiation and integration become algebraic operations:

This correspondence applies to sinusoidal steady state at a single frequency; it does not directly describe transients.

1.3 Impedance and admittance

For ideal lumped elements,

The units are R in ohms, L in henries, C in farads and ω in rad/s. Kirchhoff's laws retain their algebraic form:

Consequently, nodal analysis, mesh analysis, source transformations and Thévenin–Norton equivalents can all be used with complex quantities.

1.4 Complex power

With RMS phasors, complex power is

where S is complex power in volt-amperes, P is average real power in watts, Q is reactive power in vars, and I^* denotes the complex conjugate of current. Apparent power is |S|=|V||I|, and power factor is

Positive Q represents net inductive absorption under the passive sign convention; negative Q represents net capacitive absorption.

Worked Example 1: Phasor current and power

A series circuit has R=20 Ω, L=50 mH and C=100 μF. It is supplied by V=100∠0° V RMS at f=50 Hz.

The angular frequency is ω=2πf=314.159 rad/s. Therefore,

Hence,

The circuit is net capacitive, so current leads voltage. Complex power is

Thus P=302.9 W, Q=-244.1 var and the leading power factor is approximately 0.778.

2. Network Functions and Frequency Response

For a zero-initial-condition LTI network, the network function is

where s=σ+jω, z_k are zeros, p_r are poles and K is a scale factor. Depending on the selected variables, H(s) may be a voltage ratio, current ratio, transfer impedance or transfer admittance.

If the imaginary axis is not a pole location, the frequency response is obtained by setting s=jω:

The steady-state output to x(t)=X_m cos(ωt+φ_x) is

Thus the magnitude determines amplitude scaling and the phase determines phase displacement.

2.1 Decibel magnitude and Bode representation

For voltage or current ratios,

For a power ratio, the corresponding expression is 10log₁₀(P₂/P₁). The factor 20 for an amplitude ratio follows only when power is proportional to the square of amplitude under consistent impedances.

A real pole contributes an asymptotic slope of −20 dB/decade above its break frequency, while a real zero contributes +20 dB/decade. Complex-conjugate pole or zero pairs contribute twice those slopes. Exact values should be used near break frequencies and near lightly damped poles.

2.2 Phase and group delay

The phase response is φ(ω)=arg H(jω). For a narrowband modulated signal, group delay is

where τ_g is measured in seconds. Approximately constant group delay across the signal bandwidth reduces waveform distortion. A circuit may have an acceptable magnitude response but unacceptable phase nonlinearity.

3. Series Resonance

Consider an ideal series RLC network driven by an ideal sinusoidal voltage source. Its impedance is

where R>0, L>0 and C>0. Resonance occurs when the input reactance is zero:

Therefore,

At resonance, Z(ω₀)=R, current is maximal, source voltage and current are in phase, and source power factor is unity. The inductor and capacitor voltages have equal magnitudes and opposite phases. Their individual magnitudes may greatly exceed the source voltage.

3.1 Quality factor

For the ideal series circuit,

This dimensionless quantity measures resonant selectivity and, at resonance, voltage magnification:

This result assumes ideal components and that R represents the total effective series resistance, including source and inductor losses.

3.2 Half-power frequencies and bandwidth

Because the resistor power under fixed RMS source voltage is P_R=|I|²R, half power occurs when

This gives

The positive half-power frequencies are

They satisfy the exact relations

and therefore

The two frequencies are geometrically symmetric about ω₀ on a logarithmic frequency axis, not generally arithmetically symmetric on a linear axis. Only for high Q may one use the approximation ω_1,2≈ω₀∓Δω/2.

Figure: Normalized series-RLC current response for Q = 2. The half-power ordinate is exactly 0.707 of the peak; ω₁/ω₀ ≈ 0.781 and ω₂/ω₀ ≈ 1.281, so ω₁ω₂ = ω₀² without linear-axis symmetry.

Worked Example 2: Exact half-power frequencies

Let R=10 Ω, L=0.10 H and C=100 μF. Then

Thus f₀=50.329 Hz and

The exact half-power frequencies are

Their difference is 100 rad/s, while their product is approximately 100000 rad²/s²=ω₀². They are not equally spaced from ω₀ on a linear axis.

4. Parallel Resonance and Antiresonance

For an ideal parallel RLC network,

Parallel resonance occurs when the susceptance is zero:

At this frequency, input admittance is minimal and purely real, while input impedance reaches its maximum value R. Large, oppositely phased inductor and capacitor branch currents can circulate even when source current is small. This condition is often called antiresonance because the input current has a minimum rather than a maximum under voltage excitation.

4.1 Response-dependent bandwidth

Bandwidth must always be associated with a stated excitation, response and half-power criterion. Suppose the ideal parallel RLC network is driven by an ideal RMS current source I_s. Its voltage response is

The resistor absorbs P_R=|V|²/R. At resonance, |Z|=R and the voltage and resistor power are maximal. The half-power condition is

Equivalently,

For this specifically defined voltage response under ideal current excitation,

In contrast, under ideal voltage excitation the resistor voltage and resistor power are constant with frequency. Consequently, resistor-power half-power frequencies do not exist for that excitation, even though the input source current has a frequency-dependent minimum. Quoting Δω=1/(RC) without defining the current-source voltage response would therefore be misleading.

Practical parallel resonators often contain inductor series resistance rather than an explicit parallel resistor. Converting loss models or deriving the complete admittance is necessary; the ideal parallel formula must not be applied indiscriminately.

5. General Second-Order Frequency Response

A standard low-pass second-order network has

where H₀ is the dc gain, ω_n>0 is the undamped natural angular frequency and ζ is the damping ratio. Its magnitude is

For the standard denominator, asymptotic stability requires ζ>0. A finite resonant magnitude peak above the dc magnitude occurs only when

Under this condition,

and, for positive H₀, the peak magnitude is

If ζ≥1/√2, the low-pass magnitude decreases monotonically and has no resonant peak above its dc value. If ζ=0, poles lie on the imaginary axis and the response is not asymptotically stable; the magnitude becomes unbounded at ω=ω_n in the ideal model. Negative damping produces an unstable network and does not define a finite stable resonance.

For the denominator above, a commonly used pole quality factor is

However, relationships among damping ratio, half-power bandwidth and quality factor depend on transfer-function numerator and response definition. The familiar approximation Q≈ω₀/Δω should not be transferred to arbitrary second-order responses without verification.

Worked Example 3: Stable second-order peaking

For ζ=0.30 and ω_n=1000 rad/s, the system is stable and satisfies 0<ζ<1/√2. Therefore,

For unity dc gain,

The peak is approximately 20log₁₀(1.747)=4.84 dB.

6. Pole–Zero Interpretation

Frequency-response features can be interpreted geometrically in the complex plane. At s=jω,

and the phase is the sum of zero angles minus the sum of pole angles. Poles close to the imaginary axis create sharp magnitude variation and rapid phase change. Zeros suppress transmission near their locations. A zero on the imaginary axis produces a notch at its frequency if it is not cancelled by a pole.

Exact pole-zero cancellation is physically fragile. Component tolerances, parasitic effects and parameter drift leave residual modes. Cancelling an unstable pole with a zero is particularly unacceptable because internal signals can remain unstable despite an apparently benign external transfer function.

The main ideal response classes are:

7. Illustrative Design Case: Series-RLC Band-Pass Sensor Interface

Background

This is an illustrative case. An industrial sensor interface must select a 10 kHz carrier from broadband noise. A passive series-RLC circuit is used, and the output is measured across the resistor, giving a band-pass response.

Problem

The target center frequency is f₀=10 kHz and the desired loaded quality factor is Q=5. An available inductor has L=10 mH. Determine C, the total series resistance R, bandwidth and exact half-power frequencies. Assume initially ideal reactive components and negligible measurement loading.

Analysis and solution

The resonant angular frequency is

From ω₀=1/√(LC),

Using Q=ω₀L/R,

The angular bandwidth is

Hence Δf=Δω/(2π)=2000 Hz. The exact half-power frequencies satisfy f₂-f₁=2000 Hz and f₁f₂=f₀². Therefore,

For a 1 V RMS input, the resonant current is

The resistor power at resonance is I₀²R=7.958 mW, and each half-power point gives approximately 3.979 mW.

Practical refinement

If the inductor has 20 Ω winding resistance and the source has 50 Ω output resistance, only 55.66 Ω of additional series resistance is required to reach 125.66 Ω. Omitting these losses would reduce the actual quality factor and broaden the passband. The measuring instrument must also be included because finite input impedance changes the effective transfer function.

Outcome and learning points

The design meets the nominal selectivity requirement, but exact half-power frequencies are not linearly symmetric around 10 kHz. Realized performance depends on total loaded resistance, component tolerance, parasitic capacitance, inductor self-resonance and temperature. Resonator design must therefore be verified by tolerance analysis and frequency sweep rather than by nominal equations alone.

8. Practical Limitations and Contemporary Practice

8.1 Loading and source resistance

A resonator rarely operates in isolation. Source resistance, load resistance and instrument impedance alter damping and shift the response. Quality factor should be identified as unloaded, external or loaded. The loaded quality factor governs observed bandwidth.

8.2 Nonideal components

Real inductors contain winding resistance, core loss, parasitic capacitance and frequency-dependent permeability. Capacitors have equivalent series resistance, leakage and dielectric absorption. Above an element's self-resonant frequency, its nominal inductive or capacitive behavior may reverse.

8.3 Tolerance and sensitivity

For ω₀=(LC)^-1/2, small relative changes satisfy

Worst-case tolerance analysis is conservative, while Monte Carlo analysis estimates statistical yield when component distributions are known. Automated optimization and machine-learning-assisted surrogate models can accelerate design-space exploration, but they do not replace physical constraints, stability checks or laboratory validation.

8.4 Measurement integrity and safety

Network analyzers and impedance analyzers require calibration at the intended reference plane. Fixture parasitics can dominate high-Q or high-frequency measurements. Excitation amplitude must remain within the linear range; ferromagnetic cores and active devices may otherwise invalidate the LTI model.

High-Q power networks can develop voltages or circulating currents much larger than terminal values. Designs must respect insulation, thermal, electromagnetic-compatibility and component-rating requirements. In regulated products, applicable safety and EMC standards must be considered. Simulation or automated design output should be traceable, independently reviewed and verified by measurement.

8.5 Best-practice workflow

1. Define the excitation, output variable and bandwidth criterion.
2. Establish a complete equivalent circuit including source, load and losses.
3. Derive or compute the exact transfer function.
4. Check pole locations and stability before interpreting resonance.
5. Use exact half-power calculations unless a stated high-Q approximation is justified.
6. Perform tolerance, temperature and parasitic analysis.
7. Verify the design through calibrated frequency-response measurement.

Chapter Summary

Sinusoidal steady-state analysis uses phasors and complex impedances to transform differential circuit equations into algebraic relations. Frequency response describes amplitude and phase as functions of frequency and is determined by the poles and zeros of the network function.

In an ideal series RLC circuit driven by a voltage source, resonance produces maximum current. The exact half-power frequencies satisfy ω₁ω₂=ω₀², and their difference is R/L. In an ideal parallel RLC circuit driven by a current source, the resonant voltage is maximal and the corresponding half-power bandwidth is 1/(RC). This formula is response-dependent: resistor power is constant under ideal voltage excitation.

A standard stable second-order low-pass response exhibits a finite resonant peak only for 0<ζ<1/√2. Practical frequency responses are affected by losses, loading, parasitics, tolerance, nonlinear behavior and measurement fixtures. Reliable design consequently requires explicit response definitions, stability analysis, exact calculations and experimental validation.

Key Terms

Phasor: Complex representation of a sinusoid's magnitude and phase.

Sinusoidal steady state: Long-term sinusoidal response after stable transients decay.

Impedance: Complex voltage-to-current ratio measured in ohms.

Admittance: Reciprocal of impedance, measured in siemens.

Complex power: Quantity S=P+jQ combining real and reactive power.

Network function: Zero-initial-condition ratio of an output transform to an input transform.

Frequency response: Network function evaluated on the imaginary-frequency axis.

Pole: Value of s at which a network function becomes unbounded.

Zero: Value of s at which a network function becomes zero.

Resonance: Frequency condition in which energy-storage effects produce a characteristic response extremum.

Antiresonance: Parallel-resonant condition associated with maximum impedance and minimum source current under voltage excitation.

Half-power frequency: Frequency at which a specified dissipated-power response is half its maximum.

Bandwidth: Difference between specified upper and lower limiting frequencies.

Quality factor: Dimensionless measure of resonant selectivity or stored-to-dissipated energy behavior.

Damping ratio: Dimensionless parameter governing second-order pole damping.

Bode plot: Log-frequency representation of magnitude and phase.

Group delay: Negative derivative of phase with respect to angular frequency.

Loaded quality factor: Quality factor including source, load and internal losses.

Self-resonance: Frequency at which component parasitics change the element's effective behavior.

Review Questions

Short Answer Questions

1. Why does differentiation correspond to multiplication by jω in phasor analysis?
2. Define complex power using RMS phasors.
3. State the exact relationship among ω₁, ω₂ and ω₀ for a series RLC circuit.
4. Why is the parallel-RLC bandwidth formula response-dependent?
5. What pole condition is required for asymptotic stability of a continuous-time rational network?

Long Answer Questions

1. Derive the resonant frequency, half-power frequencies, bandwidth and quality factor of a series RLC circuit.
2. Analyze ideal parallel resonance under current-source excitation and contrast it with voltage-source excitation.
3. Explain how poles and zeros determine magnitude, phase and resonant behavior.
4. Derive the resonant frequency and peak magnitude of a standard second-order low-pass response, stating all stability conditions.
5. Discuss how loading, component nonidealities and measurement fixtures affect a practical resonator.

Analytical / Application Questions

1. A series RLC circuit has R=25 Ω, L=50 mH and C=2 μF. Determine its resonant frequency, exact half-power frequencies, bandwidth and quality factor.
2. Design a series-RLC resistor-output band-pass circuit with f₀=20 kHz, Q=10 and L=2 mH. Calculate R, C and exact half-power frequencies.
3. A second-order low-pass network has ω_n=5000 rad/s and ζ=0.5. Determine whether it has a finite stable resonant peak and calculate the peak frequency and magnitude for unity dc gain.

Multiple Choice Questions with Answers

Question 1

For RMS phasors, which expression gives complex power under the passive sign convention?

A. S=VI

B. S=VI^*

C. S=V^*I^*

D. S=V/I

Correct Answer: B. S=VI^*

Question 2

An ideal series RLC circuit has L=0.20 H and C=50 μF. What is its resonant angular frequency?

A. 100 rad/s

B. 200 rad/s

C. 316.2 rad/s

D. 10,000 rad/s

Correct Answer: C. 316.2 rad/s

Question 3

For a series RLC circuit driven by an ideal voltage source, which relation between the exact half-power frequencies is correct?

A. ω₁+ω₂=ω₀

B. ω₁ω₂=ω₀²

C. ω₂-ω₁=ω₀

D. ω₁=ω₀/2

Correct Answer: B. ω₁ω₂=ω₀²

Question 4

A series RLC circuit has R=12 Ω and L=0.06 H. What is its exact angular half-power bandwidth?

A. 0.72 rad/s

B. 5 rad/s

C. 72 rad/s

D. 200 rad/s

Correct Answer: D. 200 rad/s

Question 5

For an ideal parallel RLC circuit, the formula Δω=1/(RC) applies directly to which specified response?

A. Resistor power under fixed ideal voltage excitation

B. Inductor current under fixed ideal voltage excitation

C. Network voltage and resistor-power response under ideal current excitation

D. Capacitor voltage under fixed ideal voltage excitation

Correct Answer: C. Network voltage and resistor-power response under ideal current excitation

Question 6

At series resonance, an ideal RLC circuit with total series resistance R has an input impedance equal to:

A. Zero

B. R

C. jω₀L

D. 1/(jω₀C)

Correct Answer: B. R

Question 7

A standard second-order denominator is s²+2ζω_ns+ω_n², with ω_n>0. Which condition gives a finite stable resonant peak in its low-pass magnitude response?

A. ζ<0

B. ζ=0

C. 0<ζ<1/√2

D. ζ≥1

Correct Answer: C. 0<ζ<1/√2

Question 8

For a unity-dc-gain second-order low-pass response with ζ=0.5, what is the resonant frequency?

A. 0.5ω_n

B. ω_n/√2

C. ω_n

D. √2ω_n

Correct Answer: B. ω_n/√2

Question 9

A real pole contributes which asymptotic Bode-magnitude slope above its break frequency?

A. −20 dB/decade

B. −10 dB/decade

C. +20 dB/decade

D. +40 dB/decade

Correct Answer: A. −20 dB/decade

Question 10

For a series resonator with ω₀=2000 rad/s and Δω=100 rad/s, what is its quality factor?

A. 2

B. 10

C. 20

D. 200

Correct Answer: C. 20