Question E5A08:
What is the phase relationship between the current through and the voltage across a series resonant circuit at resonance?  

Choices:  
A: The voltage leads the current by 90 degrees  
B: The current leads the voltage by 90 degrees  
C: The voltage and current are in phase  
D: The voltage and current are 180 degrees out of phase    

Correct Answer is  C: The voltage and current are in phase  

At series resonance, the inductor’s reactance cancels the capacitor’s reactance, leaving only resistance. That makes the circuit’s impedance purely real, so voltage and current are in phase.

Why C is correct (voltage and current are in phase):
- Series RLC impedance:  
  $$Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$$
- At resonance:  
  $$\omega_0 = \frac{1}{\sqrt{LC}} \quad \Rightarrow \quad \omega_0 L = \frac{1}{\omega_0 C}$$  
  so the reactive part is zero and  
  $$Z = R \quad \text{(purely real)}$$
- Phase angle:  
  $$\phi = \arctan\left(\frac{X_L - X_C}{R}\right) \to \arctan(0) = 0^\circ$$  
  Therefore, voltage and current are in phase.

Why the other choices are incorrect:
- A: “Voltage leads current by 90°” describes a purely inductive load (current lags by 90°). Not true at resonance where net reactance is zero.
- B: “Current leads voltage by 90°” describes a purely capacitive load. Not true at resonance.
- D: “180° out of phase” would require negative resistance or inversion; a passive series RLC at resonance is purely resistive with 0° phase.

Useful formulas and rules of thumb:
- Resonant frequency:  
  $$f_0 = \frac{1}{2\pi\sqrt{LC}}$$
- Reactances:  
  $$X_L = \omega L, \quad X_C = -\frac{1}{\omega C}$$
- At series resonance: minimum impedance, maximum current, circuit behaves like a resistor, voltage and current in phase.
- At parallel resonance: maximum impedance; source voltage and source current are in phase, while inductor and capacitor branch currents are equal and opposite.