Impedance — Revision Notes

⚡ 30-Second Revision

Impedance (Z): — Total opposition to AC current. Unit: Ohm ($\Omega$).\n- Resistance (R): Frequency-independent, dissipates heat. $V_R$ in phase with $I$.\n- **Inductive Reactance ($X_L$):** $X_L = \omega L = 2\pi f L$. Increases with $f$. Current lags voltage by $90^\circ$ (ELI).\n- **Capacitive Reactance ($X_C$):** $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$. Decreases with $f$. Current leads voltage by $90^\circ$ (ICE).\n- Series RLC Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$.\n- **Phase Angle ($\phi$):** $\tan \phi = \frac{X_L - X_C}{R}$. Positive $\phi$ (inductive), Negative $\phi$ (capacitive), $\phi=0$ (resistive/resonance).\n- Resonance: $X_L = X_C \implies Z=R$ (minimum), $I_{max}$, $\phi=0$. Resonant frequency $f_0 = \frac{1}{2\pi\sqrt{LC}}$.\n- Power Factor: $\cos \phi = \frac{R}{Z}$.

2-Minute Revision

Impedance ($Z$) is the AC equivalent of resistance, representing the total opposition to current flow. It's composed of resistance ($R$), inductive reactance ($X_L$), and capacitive reactance ($X_C$).

Resistance is constant, but $X_L$ increases with frequency ($X_L = 2\pi f L$) and $X_C$ decreases with frequency ($X_C = \frac{1}{2\pi f C}$). These components don't add arithmetically due to phase differences: current is in phase with voltage across $R$, lags by $90^\circ$ across $L$, and leads by $90^\circ$ across $C$.

For a series RLC circuit, impedance is calculated as $Z = \sqrt{R^2 + (X_L - X_C)^2}$. The phase angle $\phi = \arctan\left(\frac{X_L - X_C}{R}\right)$ tells us if the circuit is inductive ($X_L > X_C$, voltage leads current) or capacitive ($X_C > X_L$, voltage lags current).

The special condition where $X_L = X_C$ is called resonance, leading to minimum impedance ($Z=R$), maximum current, and zero phase angle. The power factor, $\cos \phi = R/Z$, indicates how effectively power is utilized.

5-Minute Revision

Impedance ($Z$) is the crucial concept for understanding AC circuits, generalizing resistance to include the effects of inductors and capacitors. It's the total opposition to alternating current, measured in Ohms.

Unlike resistance, impedance is frequency-dependent due to its reactive components: inductive reactance ($X_L = 2\pi f L$) and capacitive reactance ($X_C = \frac{1}{2\pi f C}$). $X_L$ increases with frequency, while $X_C$ decreases.

Resistance ($R$) remains constant.\n\nIn a series RLC circuit, these three components combine vectorially. The voltage across the resistor ($V_R$) is in phase with the current ($I$). The voltage across the inductor ($V_L$) leads the current by $90^\circ$, and the voltage across the capacitor ($V_C$) lags the current by $90^\circ$.

This $90^\circ$ phase difference means we use the Pythagorean theorem to find the total impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$. The term $(X_L - X_C)$ is the net reactance, $X$. If $X_L > X_C$, the circuit is inductive, and the total voltage leads the current.

If $X_C > X_L$, it's capacitive, and the total voltage lags the current. The phase angle $\phi$ is given by $\tan \phi = \frac{X_L - X_C}{R}$.\n\nResonance is a critical condition where $X_L = X_C$.

At this specific resonant frequency ($f_0 = \frac{1}{2\pi\sqrt{LC}}$), the net reactance becomes zero, and the impedance is at its minimum value, $Z=R$. This leads to the maximum current in the circuit for a given voltage, and the phase angle $\phi=0$, meaning the circuit behaves purely resistively.

Understanding how to calculate $Z$, $X_L$, $X_C$, and $\phi$ is essential for solving problems related to current, voltage drops across components, and power factor ($\cos \phi = R/Z$) in AC circuits.

Prelims Revision Notes

Impedance (Z) - NEET Quick Notes\n\n1. Definition: Total opposition to AC current flow. Unit: Ohm ($\Omega$).\n\n2. Components of Impedance:\n * Resistance (R):\n * Opposition due to energy dissipation (heat).\n * Independent of frequency.\n * Voltage ($V_R$) and current ($I$) are in phase.\n * Inductive Reactance ($X_L$):\n * Opposition by an inductor (L) to changing current.\n * Formula: $X_L = \omega L = 2\pi f L$.\n * Increases linearly with frequency ($f$).\n * Current lags voltage by $90^\circ$ (ELI: E leads I in L).\n * Capacitive Reactance ($X_C$):\n * Opposition by a capacitor (C) to changing voltage.\n * Formula: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$.\n * Decreases inversely with frequency ($f$).\n * Current leads voltage by $90^\circ$ (ICE: I leads E in C).\n\n3. Series RLC Circuit Impedance:\n * Total impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$.\n * Net Reactance: $X = X_L - X_C$.\n * Impedance Triangle: Right triangle with sides $R$, $X$, and hypotenuse $Z$.\n\n4. Phase Angle ($\phi$):\n * Phase difference between total voltage and total current.\n * Formula: $\tan \phi = \frac{X_L - X_C}{R}$.\n * If $X_L > X_C$: Circuit is inductive, $\phi$ is positive (voltage leads current).\n * If $X_C > X_L$: Circuit is capacitive, $\phi$ is negative (voltage lags current).\n * If $X_L = X_C$: Circuit is purely resistive, $\phi = 0$.\n\n5. Resonance in Series RLC Circuit:\n * Condition: $X_L = X_C$.\n * Resonant Frequency ($f_0$): $f_0 = \frac{1}{2\pi\sqrt{LC}}$.\n * At Resonance: $Z = R$ (minimum impedance), $I_{rms} = V_{rms}/R$ (maximum current), $\phi = 0$.\n\n6. Ohm's Law for AC Circuits:\n * $I_{rms} = V_{rms}/Z$ or $I_{peak} = V_{peak}/Z$.\n\n7. Power Factor (PF):\n * $PF = \cos \phi = \frac{R}{Z}$.\n * Measures how much of the apparent power is real power.\n\nKey Points for NEET:\n* Always convert units (mH to H, $\mu$F to F) before calculation.\n* Remember vectorial addition for $Z$, not arithmetic sum.\n* Master phase relationships (ELI vs ICE).

Vyyuha Quick Recall

To remember the phase relationships in AC circuits: ELI the ICE man\n\n* ELI: In an inductor (L), Voltage (E) leads Current (I).\n* ICE: In a capacitor (C), Current (I) leads Voltage (E).