Impedance Calculator

1. Impedance as a Complex Quantity

Impedance Z extends the concept of resistance to AC circuits. It is a complex number consisting of:

• Real part (R) – resistance, dissipates energy.
• Imaginary part (X) – reactance, stores and releases energy.

Z = R + jX,   where \(j = \sqrt{-1}\) (electrical engineering notation).

The magnitude \(|Z| = \sqrt{R^2 + X^2}\) and phase angle \(\phi = \arctan(X/R)\). For pure inductors, \(X_L = 2\pi f L\) (positive), for capacitors \(X_C = -1/(2\pi f C)\) (negative in series, but often treated as positive magnitude with sign in formula). In series RLC, net reactance \(X = X_L - X_C\).

2. Frequency Dependence

Reactance varies strongly with frequency:

• Inductive reactance \(X_L = 2\pi f L\) ∝ f – increases linearly with frequency.
• Capacitive reactance \(X_C = 1/(2\pi f C)\) ∝ 1/f – decreases as frequency rises.

This leads to three regimes in a series RLC circuit:

• Below resonance (\(f < f_r\)): \(X_C > X_L\) → circuit capacitive (negative phase).
• At resonance (\(f = f_r\)): \(X_L = X_C\) → net reactance zero, impedance purely resistive \(Z = R\).
• Above resonance (\(f > f_r\)): \(X_L > X_C\) → circuit inductive (positive phase).

3. Resonance – Peak or Dip?

Series resonance: Impedance is minimum (\(Z_{\min} = R\)) and current is maximum. Phase crosses zero.

Parallel resonance: Impedance is maximum (\(Z_{\max} \approx Q \cdot X_L\) if \(R\) is small) and current is minimum. Phase also crosses zero but from positive to negative.

The resonant frequency is the same for both: \(f_r = \frac{1}{2\pi\sqrt{LC}}\).

4. Quality Factor (Q) and Bandwidth

Q indicates the sharpness of resonance:

• Series RLC: \(Q = \frac{X_L}{R} = \frac{2\pi f_r L}{R}\).
• Parallel RLC: \(Q = \frac{R}{X_L} = R \cdot 2\pi f_r C\) (since at resonance \(X_L = X_C\)).

Bandwidth (BW) is the range of frequencies where the power drops by half (-3 dB): \(BW = \frac{f_r}{Q}\).

A high-Q circuit is very selective (narrow bandwidth), used in oscillators and filters.

5. Real Components: ESR, ESL, and Parasitics

Actual inductors have winding resistance (DCR) and self-capacitance; capacitors have Equivalent Series Resistance (ESR) and Equivalent Series Inductance (ESL). These parasitics become significant at high frequencies, causing self-resonance and limiting impedance behaviour. For example, a capacitor’s impedance first decreases (capacitive), reaches a minimum at its self-resonant frequency, then increases (inductive).

Always check component datasheets for SRF (Self-Resonant Frequency) and ESR when designing precision circuits.

6. Impedance Matching for Maximum Power Transfer

The maximum power theorem states that maximum power is delivered when the load impedance is the complex conjugate of the source impedance. In RF systems, matching networks (L‑networks, pi‑networks, transformers) transform impedances to 50 Ω. The goal is to cancel reactance and equal resistance.

Common matching elements: series/parallel capacitors and inductors, or transmission line stubs.

7. Measuring Impedance

Impedance can be measured using:

• LCR meters – apply a known AC signal and measure voltage/current ratio and phase (typically at 100 Hz, 1 kHz, 10 kHz, 100 kHz).
• Impedance analyzers – sweep frequency to plot Z vs. f (e.g., Keysight E4990A).
• Network analyzers – measure S‑parameters and convert to impedance (for RF).

8. Detailed Calculation Example

Given: Series RLC with \(R = 100\,\Omega\), \(L = 0.1\,\text{H}\), \(C = 1\,\mu\text{F}\), \(f = 1000\,\text{Hz}\).

1. \(\omega = 2\pi f = 6283.185\,\text{rad/s}\).
2. \(X_L = \omega L = 6283.185 \times 0.1 = 628.32\,\Omega\).
3. \(X_C = 1/(\omega C) = 1/(6283.185 \times 1\times10^{-6}) = 159.15\,\Omega\).
4. Net reactance \(X = X_L - X_C = 469.17\,\Omega\) (inductive).
5. \(Z = R + jX = 100 + j469.17\,\Omega\).
6. \(|Z| = \sqrt{100^2 + 469.17^2} = 479.70\,\Omega\).
7. \(\phi = \arctan(469.17/100) = 77.9^\circ\) (positive, inductive).
8. Resonant frequency \(f_r = 1/(2\pi\sqrt{0.1 \times 1\times10^{-6}}) = 503.3\,\text{Hz}\).
9. Quality factor \(Q = X_L/R = 628.32/100 = 6.28\) (at resonance, but here off-resonance).

This matches the calculator's default output.