Inductive Reactance

1. Definition and Mathematical Representation

Inductive Reactance

Definition and Mathematical Representation

Inductive reactance (XL) quantifies the opposition an inductor presents to alternating current (AC) due to Faraday's law of electromagnetic induction. Unlike resistance, which dissipates energy, reactance temporarily stores energy in a magnetic field. The magnitude of XL depends on both the frequency (f) of the AC signal and the inductance (L) of the component.

The fundamental relationship is derived from the inductor's voltage-current phase relationship. For an ideal inductor, voltage leads current by 90°. The time-domain voltage across an inductor is given by:

$$ v(t) = L \frac{di(t)}{dt} $$

For a sinusoidal current i(t) = Ipeak sin(ωt), the voltage becomes:

$$ v(t) = L \frac{d}{dt} \left( I_{\text{peak}} \sin(\omega t) \right) = \omega L I_{\text{peak}} \cos(\omega t) $$

This shows the voltage amplitude scales with ωL. By analogy to Ohm's law (V = IR), the proportionality factor ωL defines the reactance:

$$ X_L = \omega L = 2\pi f L $$

where:

In phasor notation, inductive reactance introduces a +90° phase shift, represented as an imaginary impedance:

$$ \mathbf{Z}_L = jX_L = j\omega L $$

This complex impedance is critical in AC circuit analysis, particularly when combining inductive, capacitive, and resistive elements. For example, in RLC circuits, the total impedance Z becomes:

$$ \mathbf{Z} = R + j\left(\omega L - \frac{1}{\omega C}\right) $$

Practical implications include:

Inductor Voltage-Current Phase Relationship A waveform diagram showing the 90° phase relationship between voltage (cosine) and current (sine) in an inductor. Time (t) Amplitude π/2 v(t) = ωLIₚₑₐₖcos(ωt) i(t) = Iₚₑₐₖsin(ωt) Voltage Current
Diagram Description: The diagram would show the 90° phase relationship between voltage and current waveforms in an inductor, illustrating the time-domain behavior.

Relationship Between Inductance and Frequency

The inductive reactance XL of an inductor is fundamentally tied to both its inductance L and the frequency f of the applied alternating current. This relationship emerges from Faraday's law of induction, where a time-varying current generates a back-emf that opposes the change in current. The faster the current changes (higher frequency), the greater the opposition.

Mathematical Derivation

Starting with the fundamental definition of voltage across an ideal inductor:

$$ v(t) = L \frac{di(t)}{dt} $$

For a sinusoidal current i(t) = Ip sin(ωt), the derivative becomes:

$$ \frac{di(t)}{dt} = I_p \omega \cos(\omega t) $$

Substituting back into the voltage equation:

$$ v(t) = L I_p \omega \cos(\omega t) $$

The peak voltage Vp is therefore:

$$ V_p = L I_p \omega $$

By Ohm's law analogy (V = IX), we identify the inductive reactance:

$$ X_L = \omega L = 2\pi f L $$

Frequency Dependence

The linear proportionality XL ∝ f has critical implications:

Practical Applications

This frequency dependence is exploited in:

Non-Ideal Behavior

Real inductors exhibit:

$$ Q = \frac{X_L}{R_s} = \frac{2\pi f L}{R_s} $$

where Rs is the series resistance. Quality factor Q peaks before self-resonance.

Inductive Reactance vs Frequency A line graph showing the relationship between inductive reactance (X_L) and frequency (f), illustrating linear proportionality, DC behavior, and self-resonance. Frequency (f) Inductive Reactance (Xₗ) 0 DC f₁ fᵣ f_SRF RF Xₗ(max) Xₗ Linear region: Xₗ = 2πfL Resonance (fᵣ) Self-resonance (f_SRF) High frequency region Xₗ = 0 at DC
Diagram Description: The diagram would show the relationship between inductive reactance and frequency, including the linear proportionality and key points like DC behavior and high-frequency dominance.

1.3 Units and Dimensional Analysis

The inductive reactance XL of an inductor is a frequency-dependent opposition to alternating current, quantified in ohms (Ω). Its dimensional analysis reveals fundamental insights into the relationship between inductance, frequency, and reactance.

Derivation of Units

Starting from the defining equation of inductive reactance:

$$ X_L = \omega L = 2\pi f L $$

where:

Breaking down the units:

$$ [X_L] = [\omega][L] = \left(\frac{\text{rad}}{\text{s}}\right) \cdot \left(\frac{\text{V} \cdot \text{s}}{\text{A}}\right) = \frac{\text{V}}{\text{A}} = \Omega $$

The cancellation of seconds (s) confirms that reactance shares the same unit as resistance, despite differing physical origins.

Dimensional Consistency Check

From Faraday's law, the voltage across an inductor is:

$$ V = L \frac{dI}{dt} $$

Rearranged for impedance Z = V/I in the frequency domain:

$$ Z = j\omega L $$

Substituting dimensions:

$$ \left[\frac{V}{I}\right] = \left[\frac{L}{t}\right] \rightarrow \Omega = \frac{H}{s^{-1}} = H \cdot \text{Hz} $$

This aligns with the XL = 2πfL expression, validating consistency.

Practical Implications

In RF circuit design, the reactance unit (Ω) enables direct combination with resistances and capacitive reactances in impedance calculations. For example, a 1 μH inductor at 100 MHz exhibits:

$$ X_L = 2\pi (100 \times 10^6)(1 \times 10^{-6}) = 628\ \Omega $$

This dimensional homogeneity allows vector summation with a capacitor's reactance XC = 1/(2πfC).

Non-Ideal Considerations

Real inductors exhibit parasitic resistance (R) and capacitance (C), making the total impedance:

$$ Z = R + j\omega L + \frac{1}{j\omega C} $$

All terms maintain dimensional consistency in ohms, enabling meaningful phase and magnitude analysis.

2. Derivation from Faraday's Law of Induction

2.1 Derivation from Faraday's Law of Induction

Faraday's Law of Induction states that the electromotive force (EMF) induced in a closed loop is proportional to the rate of change of magnetic flux through the loop. Mathematically:

$$ \mathcal{E} = -\frac{d\Phi_B}{dt} $$

For an inductor with N turns, the total flux linkage is NΦB, and the induced EMF becomes:

$$ \mathcal{E} = -L \frac{dI}{dt} $$

where L is the inductance. In an AC circuit with a sinusoidal current I(t) = I0 sin(ωt), the voltage across the inductor is:

$$ V_L(t) = L \frac{dI}{dt} = L \cdot I_0 \omega \cos(\omega t) $$

This reveals a phase shift where the voltage leads the current by 90°. The amplitude of the voltage is:

$$ V_0 = I_0 \omega L $$

By analogy with Ohm's Law (V = IR), the inductive reactance XL is defined as the ratio of voltage amplitude to current amplitude:

$$ X_L = \frac{V_0}{I_0} = \omega L $$

Expressed in terms of frequency f (since ω = 2πf):

$$ X_L = 2\pi f L $$

Implications in AC Circuits

Inductive reactance increases with frequency, meaning inductors oppose high-frequency currents more than low-frequency ones. This property is exploited in:

Comparison with Capacitive Reactance

Unlike capacitive reactance (XC = 1/(ωC)), which decreases with frequency, inductive reactance exhibits an opposing frequency dependence. This complementary behavior is fundamental to LC resonator design and frequency-selective networks.

$$ Z_L = jX_L = j\omega L $$

where ZL is the complex impedance of the inductor, and j denotes the imaginary unit.

Inductor Voltage-Current Phase Relationship and Reactance A dual-axis diagram showing the 90° phase shift between inductor voltage and current waveforms (top) and the linear relationship between inductive reactance and frequency (bottom). Time (t) Amplitude I(t) V_L(t) 90° phase shift I₀ V₀ Inductor Voltage and Current Waveforms Frequency (f) Reactance (Xₗ) Slope = ωL = 2πfL Xₗ = ωL = 2πfL Inductive Reactance vs. Frequency Inductor Voltage-Current Phase Relationship and Reactance
Diagram Description: The diagram would show the 90° phase relationship between sinusoidal current and voltage waveforms in an inductor, and how inductive reactance scales with frequency.

2.2 Formula: XL = 2πfL

Derivation of Inductive Reactance

The opposition offered by an inductor to alternating current (AC) is termed inductive reactance (XL). Unlike resistance, which dissipates energy, reactance temporarily stores energy in a magnetic field. The fundamental relationship arises from Faraday's law of induction and Lenz's law, which describe how a changing current induces a back-EMF (ε) in the inductor:

$$ \varepsilon = -L \frac{di}{dt} $$

For a sinusoidal current i(t) = Ipeak sin(ωt), the voltage across the inductor becomes:

$$ v(t) = L \frac{di}{dt} = L \cdot \omega I_{peak} \cos(\omega t) $$

The phase shift between voltage and current (voltage leads by 90°) necessitates complex impedance analysis. The magnitude of the reactance is derived from the ratio of peak voltage to peak current:

$$ X_L = \frac{V_{peak}}{I_{peak}} = \omega L = 2\pi f L $$

Key Components of the Formula

Practical Implications

Inductive reactance governs behavior in:

Example Calculation

For a 10 mH inductor at 50 kHz:

$$ X_L = 2\pi \times 50 \times 10^3 \times 10 \times 10^{-3} = 3.14 \ \text{k}\Omega $$

Non-Ideal Considerations

Real inductors exhibit parasitic resistance (RDC) and capacitance, forming a complex impedance Z = R + jXL. At very high frequencies, inter-winding capacitance dominates, causing self-resonance.

Inductor Voltage-Current Phase Relationship A waveform plot showing the 90° phase shift between inductor current (i(t)) and voltage (v(t)) in a sinusoidal AC circuit. Time (t) i(t) = Iₚ sin(ωt) Iₚ -Iₚ v(t) = ωLIₚ cos(ωt) ωLIₚ -ωLIₚ 90° Phase Shift
Diagram Description: The section involves voltage-current phase relationships (90° shift) and time-domain behavior of sinusoidal signals, which are highly visual concepts.

Impact of Core Material on Inductance

The inductance of a coil is profoundly influenced by the magnetic properties of its core material, governed by the relationship:

$$ L = \frac{N^2 \mu A}{l} $$

where L is inductance, N is the number of turns, μ is the permeability of the core material, A is the cross-sectional area, and l is the magnetic path length. The permeability μ is a product of the relative permeability μr and the permeability of free space μ0:

$$ \mu = \mu_r \mu_0 $$

Core Material Classification

Materials are categorized based on their magnetic properties:

Ferromagnetic Core Effects

Ferromagnetic materials drastically increase inductance due to their high μr, but exhibit nonlinearity from hysteresis and saturation. The B-H curve defines their behavior:

$$ B = \mu H $$

where B is magnetic flux density and H is magnetic field intensity. Saturation occurs when further increases in H yield diminishing returns in B, limiting effective permeability.

Practical Core Materials

Engineers select cores based on application-specific trade-offs:

Frequency-Dependent Permeability

Core materials exhibit frequency-dependent losses due to:

$$ \mu(f) = \mu'(f) - j\mu''(f) $$

where μ' is the real part (inductive storage) and μ'' is the imaginary part (loss component). Ferrites, for instance, maintain stable μ' up to a cutoff frequency, beyond which μ'' dominates, increasing core losses.

Core Loss Quantification

Total core losses (Pcore) combine hysteresis and eddy current losses:

$$ P_{core} = k_h f B^\alpha + k_e f^2 B^2 $$

kh and ke are material constants, α is the Steinmetz exponent (~1.6–2.1), and f is frequency. This necessitates careful material selection in high-frequency designs.

B-H Curve for Ferromagnetic Materials A graph showing the B-H curve for ferromagnetic materials, illustrating the relationship between magnetic field intensity (H) and magnetic flux density (B), including the initial magnetization curve, saturation point, and hysteresis loop. Magnetic Field Intensity, H (A/m) Magnetic Flux Density, B (T) 0 H -H B 0 -B Saturation Point μ (slope) Hysteresis Loop
Diagram Description: The B-H curve and saturation behavior of ferromagnetic materials are highly visual concepts that require graphical representation to show nonlinearity and saturation effects.

3. Role in AC Circuits

3.1 Role in AC Circuits

Inductive reactance (XL) governs the opposition an inductor presents to alternating current (AC) due to Faraday's law of induction. Unlike resistance, which dissipates energy as heat, inductive reactance temporarily stores energy in a magnetic field and releases it back into the circuit. The magnitude of XL depends on both the frequency (f) of the AC signal and the inductance (L) of the component:

$$ X_L = 2\pi f L $$

At higher frequencies, the inductor's opposition increases linearly, making it a critical element in frequency-dependent applications such as filters and impedance matching networks. The phase relationship between voltage and current in an ideal inductor is a key distinguishing feature: voltage leads current by 90° (π/2 radians). This phase shift arises from Lenz's law, where the induced EMF opposes changes in current.

Derivation of Inductive Reactance

Starting with Faraday's law, the voltage (V) across an inductor is proportional to the rate of change of current (I):

$$ V(t) = L \frac{dI(t)}{dt} $$

For a sinusoidal current I(t) = I0 sin(ωt), the voltage becomes:

$$ V(t) = L \frac{d}{dt} \left( I_0 \sin(\omega t) \right) = \omega L I_0 \cos(\omega t) $$

The amplitude of the voltage (V0 = ωL I0) reveals the proportionality factor ωL, which defines the reactance XL. Substituting angular frequency ω = 2πf yields the standard formula.

Impedance and Phasor Representation

In complex impedance analysis, an inductor's reactance is expressed as an imaginary component:

$$ Z_L = jX_L = j\omega L $$

This formalism simplifies AC circuit analysis by consolidating magnitude and phase into a single complex number. When combined with resistive (R) and capacitive (XC) elements, the total impedance Z determines the circuit's frequency response.

Practical Implications

Non-Ideal Behavior

Real inductors exhibit parasitic resistance (due to wire windings) and capacitance (inter-turn effects), modeled as a series RLC network. The quality factor (Q) quantifies efficiency:

$$ Q = \frac{X_L}{R} = \frac{\omega L}{R} $$

High-Q inductors minimize energy loss, critical in tuned circuits like oscillators and antenna matching networks.

Inductor Voltage-Current Phase Relationship A waveform diagram showing the 90° phase shift between voltage (V) and current (I) in an inductor, with labeled axes and key phase points. ωt V(t), I(t) π/2 π 3π/2 2π π/2 phase shift V(t) I(t) Voltage (V) Current (I)
Diagram Description: The section discusses the 90° phase shift between voltage and current in an inductor, which is a highly visual concept involving waveform relationships.

3.2 Filtering and Tuning Applications

Inductive reactance (XL) plays a critical role in the design of filters and tuned circuits, where frequency selectivity is paramount. The inherent frequency dependence of XL = 2πfL allows inductors to block high frequencies while permitting lower frequencies to pass, making them indispensable in applications such as RF filters, impedance matching networks, and oscillator circuits.

Low-Pass and High-Pass RL Filters

An RL low-pass filter consists of a resistor (R) and inductor (L) in series, where the output is taken across the resistor. The cutoff frequency (fc) is determined by the inductive reactance equaling the resistance:

$$ f_c = \frac{R}{2\pi L} $$

For frequencies below fc, the inductor’s reactance is negligible, allowing signals to pass. Conversely, a high-pass RL filter outputs across the inductor, attenuating low-frequency signals where XL ≪ R.

Bandpass and Bandstop LC Circuits

When combined with a capacitor, an inductor forms resonant LC circuits. The resonant frequency (fr) is given by:

$$ f_r = \frac{1}{2\pi \sqrt{LC}} $$

At fr, the inductive and capacitive reactances cancel, creating a minimum impedance (bandpass) or maximum impedance (bandstop). The quality factor (Q) quantifies selectivity:

$$ Q = \frac{X_L}{R} = \frac{2\pi f_r L}{R} $$

Higher Q values yield sharper frequency response curves, crucial in radio transceivers and signal processing.

Impedance Matching and RF Applications

Inductive reactance is exploited in impedance matching networks to maximize power transfer. For instance, an L-match network uses a series inductor and shunt capacitor to transform impedance. The Smith chart simplifies these calculations by visualizing complex impedance transformations.

In RF systems, inductors are integral to tank circuits in oscillators, where energy oscillates between L and C at fr. The Colpitts and Hartley oscillators are classic examples leveraging inductive reactance for sustained oscillations.

Practical Considerations

Non-ideal effects, such as parasitic capacitance and core losses, limit real-world inductor performance. Ferrite cores enhance inductance but introduce nonlinearity at high currents. Skin effect and proximity losses become significant at RF frequencies, necessitating litz wire or air-core designs.

Modern applications include EMI filters, where inductors suppress high-frequency noise, and wireless power transfer systems, where resonant inductive coupling enables efficient energy transmission.

RL and LC Filter Circuits with Frequency Response Side-by-side schematics of low-pass RL, high-pass RL, and LC resonant circuits with corresponding frequency response graphs below each. R L Input Output Low-Pass RL Frequency (f) Gain f_c L R Input Output High-Pass RL Frequency (f) Gain f_c L C Input Output LC Resonant Frequency (f) Gain f_r X_L = 2πfL X_C = 1/(2πfC) Q = f_r / Δf
Diagram Description: The section describes RL and LC filter circuits, resonant frequencies, and impedance matching, which are highly visual concepts involving component arrangements and frequency responses.

Inductive Reactance in Transformers

Fundamental Role of Inductive Reactance in Transformer Operation

Inductive reactance (XL) plays a critical role in transformer operation by governing the relationship between primary and secondary currents. In an ideal transformer, the primary winding's inductive reactance determines how much magnetizing current flows to establish the core flux. The reactance is given by:

$$ X_L = 2\pi f L $$

where f is the frequency and L is the inductance of the winding. High XL in the primary winding minimizes the magnetizing current, improving efficiency.

Leakage Reactance and Its Impact

Practical transformers exhibit leakage reactance due to imperfect magnetic coupling between windings. This leakage reactance (Xleak) appears in series with the ideal transformer model:

$$ X_{leak} = 2\pi f L_{leak} $$

where Lleak is the leakage inductance. Leakage reactance causes voltage drops under load and affects the transformer's regulation characteristics.

Mutual Inductance and Coupling

The mutual inductance (M) between primary and secondary windings relates to their self-inductances (L1, L2) and the coupling coefficient (k):

$$ M = k\sqrt{L_1 L_2} $$

For tightly coupled power transformers, k approaches 1, minimizing leakage reactance. The mutual reactance (XM) then dominates:

$$ X_M = 2\pi f M $$

Equivalent Circuit Representation

The complete transformer model includes both magnetizing and leakage reactances:

Frequency Dependence and Saturation Effects

At high frequencies, core losses increase due to eddy currents, while at low frequencies, the magnetizing current rises as XL decreases. The saturation flux density (Bsat) limits the minimum reactance:

$$ X_{L,min} = \frac{V_{rms}}{2\pi f_{min} N A_e B_{sat}} $$

where N is turns count and Ae is core cross-sectional area.

Practical Design Considerations

Transformer designers must balance:

High-frequency transformers often use ferrite cores with distributed air gaps to control inductance while maintaining adequate reactance across the operating band.

4. Laboratory Measurement Techniques

4.1 Laboratory Measurement Techniques

Impedance Bridge Method

Inductive reactance (XL) can be measured precisely using an impedance bridge, such as a Maxwell-Wien or Hay bridge. The bridge balances the unknown inductor against known resistors and capacitors, nullifying the detector signal. For an inductor with series resistance Rs, the balance condition in a Maxwell-Wien bridge is:

$$ Z_x = R_2 R_3 Y_1 $$

where Y1 is the admittance of the parallel RC arm. Solving for XL yields:

$$ X_L = \omega L = \frac{R_2 R_3}{R_1} \left( \frac{\omega^2 C_1^2 R_1^2}{1 + \omega^2 C_1^2 R_1^2} \right) $$

This method achieves accuracies of ±0.1% for frequencies below 10 kHz, but requires careful shielding to minimize stray capacitance.

LCR Meter Measurements

Modern LCR meters apply an AC voltage to the inductor and measure the phase-sensitive current response. The instrument computes XL directly from:

$$ X_L = \frac{V_{\text{rms}}}{I_{\text{rms}}} \sin(\theta) $$

where θ is the phase angle between voltage and current. Auto-balancing bridge LCR meters (e.g., Keysight E4980A) can measure XL from 20 Hz to 2 MHz with ±0.05% basic accuracy. Critical considerations include:

Network Analyzer Techniques

Vector network analyzers (VNAs) provide the most comprehensive characterization by measuring the S-parameters of inductive components. The reactance is derived from the reflection coefficient Γ:

$$ X_L = Z_0 \frac{\text{Im}(1 + \Gamma)}{1 - |\Gamma|^2} $$

where Z0 is the reference impedance. VNAs like the Keysight PNA-L series enable measurements up to 50 GHz, but require:

Practical Error Sources

Common measurement artifacts include:

Impedance Bridge Measurement Setup Lx R3 Detector
Impedance Bridge Measurement Setup A schematic diagram of an impedance bridge setup with inductor (Lx), resistor (R3), and detector connected in a horizontal arrangement. Lx R3 Detector
Diagram Description: The diagram would physically show the circuit layout of an impedance bridge with labeled components (Lx, R3, detector) and their connections.

4.2 Using Oscilloscopes and LCR Meters

Oscilloscope Measurements of Inductive Reactance

When characterizing inductive reactance (XL), an oscilloscope provides real-time visualization of voltage-current phase relationships. For an inductor subjected to a sinusoidal voltage V(t) = V0sin(ωt), the current I(t) lags by 90° due to XL = ωL. To measure this:

  1. Connect the inductor in series with a precision shunt resistor (Rshunt).
  2. Use Channel 1 to measure voltage across the inductor and Channel 2 for the shunt voltage (proportional to current).
  3. Trigger on the input signal and observe the phase shift (Δφ) between waveforms.
$$ X_L = \frac{V_{\text{peak}}}{I_{\text{peak}}} = \frac{V_L}{V_R} \cdot R_{\text{shunt}} $$

The phase difference Δφ should approach π/2 radians for an ideal inductor. Deviations indicate parasitic resistance or core losses.

LCR Meter Techniques

Modern LCR meters apply an AC test signal (typically 1 kHz–1 MHz) and measure complex impedance (Z = R + jXL) directly. Key considerations:

$$ Q = \frac{X_L}{R_s} \quad \text{(Series Model)} \qquad Q = \frac{R_p}{X_L} \quad \text{(Parallel Model)} $$

Error Sources and Mitigation

Oscilloscope limitations: Bandwidth and probe capacitance (typically 10–15 pF) distort high-frequency measurements. Use active probes for ω > 10 MHz.

LCR meter artifacts: Stray inductance in test fixtures becomes significant below 100 nH. Open/short calibration is essential.

Series Parallel

Advanced Techniques

For nonlinear inductors (e.g., ferrite cores), a network analyzer measures S-parameters to derive complex permeability vs. frequency. Time-domain reflectometry (TDR) helps locate distributed capacitance in wound components.

Oscilloscope Waveforms and LCR Meter Models Diagram showing sinusoidal voltage and current waveforms with 90° phase shift, along with series and parallel LCR equivalent circuits. Time Amplitude V_L(t) I_L(t) Δφ=90° R_s X_L Series Equivalent R_p X_L Parallel Equivalent
Diagram Description: The section describes phase relationships between voltage and current waveforms and requires visualization of oscilloscope measurements and LCR meter models.

4.3 Practical Calculation Examples

Deriving Inductive Reactance in AC Circuits

The inductive reactance (XL) of an inductor in an AC circuit is frequency-dependent and given by:

$$ X_L = 2\pi f L $$

where f is the frequency in Hertz (Hz) and L is the inductance in Henrys (H). This relationship shows that reactance increases linearly with both frequency and inductance. For example, a 10 mH inductor at 1 kHz exhibits:

$$ X_L = 2\pi (1000)(0.010) = 62.83 \, \Omega $$

Impedance Calculation in RL Circuits

In a series RL circuit, the total impedance (Z) combines resistive (R) and reactive (XL) components vectorially:

$$ Z = \sqrt{R^2 + X_L^2} $$

Consider a circuit with R = 50 Ω and L = 20 mH operating at 500 Hz. The impedance is calculated as:

$$ X_L = 2\pi (500)(0.020) = 62.83 \, \Omega $$ $$ Z = \sqrt{50^2 + 62.83^2} = 80.3 \, \Omega $$

Phase Angle and Power Factor

The phase angle (θ) between voltage and current in an RL circuit is:

$$ \theta = \arctan\left(\frac{X_L}{R}\right) $$

Using the previous example:

$$ \theta = \arctan\left(\frac{62.83}{50}\right) = 51.5^\circ $$

The power factor (PF) is the cosine of this angle:

$$ PF = \cos(51.5^\circ) = 0.623 \, \text{(lagging)} $$

High-Frequency Applications

At radio frequencies (e.g., 10 MHz), even small inductors (<1 μH) exhibit significant reactance. A 0.5 μH inductor at 10 MHz yields:

$$ X_L = 2\pi (10 \times 10^6)(0.5 \times 10^{-6}) = 31.4 \, \Omega $$

This property is exploited in RF chokes to block high-frequency signals while passing DC.

Real-World Design Considerations

For a 100 μH inductor with 5 Ω DC resistance and 50 pF parasitic capacitance, the self-resonant frequency (fr) is:

$$ f_r = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{(100 \times 10^{-6})(50 \times 10^{-12})}} = 2.25 \, \text{MHz} $$

5. Key Textbooks and Papers

5.1 Key Textbooks and Papers

5.2 Online Resources and Tutorials

5.3 Advanced Topics for Further Study