07Class 12 Physics

Alternating Current formulas

Reactance, impedance, resonance and transformer formulas for Alternating Current. Phase is what separates this from DC circuits.

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AC values

Instantaneous voltage: $\displaystyle v = v_m\sin\omega t$
RMS current: $\displaystyle I = \frac{i_m}{\sqrt2} \approx 0.707\,i_m$
RMS voltage: $\displaystyle V = \frac{v_m}{\sqrt2}$

Reactance and impedance

Inductive reactance: $\displaystyle X_L = \omega L$
Current lags voltage by π/2.
Capacitive reactance: $\displaystyle X_C = \frac{1}{\omega C}$
Current leads voltage by π/2.
Impedance of an LCR circuit: $\displaystyle Z = \sqrt{R^2 + (X_L - X_C)^2}$
Phase angle: $\displaystyle \tan\phi = \frac{X_L - X_C}{R}$

Resonance and power

Resonant frequency: $\displaystyle \omega_0 = \frac{1}{\sqrt{LC}}$
Here X_L = X_C and Z = R.
Quality factor: $\displaystyle Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$
Average power: $\displaystyle P = VI\cos\phi$
Power factor: $\displaystyle \cos\phi = \frac{R}{Z}$

Transformer

Voltage and turns: $\displaystyle \frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$
For an ideal transformer.

Where students slip up

Using peak values where RMS is needed. Meters and power formulas use RMS.
Adding voltages directly in an LCR circuit. They're out of phase, so they add as phasors.
Mixing up lead and lag. A common memory aid is CIVIL: in a C circuit, I leads V, and in an L circuit, V leads I.

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