Q.The total impedance of a circuit decreases when a capacitor is added in series with L and R. Explain why.

[!TLDR] Adding C in series introduces a capacitive voltage that is $180^\circ$ out of phase with the inductive voltage, so the net reactance $(X_L-X_C)$ shrinks in magnitude, reducing $Z=\sqrt{R^2+(X_L-X_C)^2}$. [!ANSWER] Impedance decreases because $X_C$ partially cancels the circuit's pre-existing inductive reactance $X_L$, shrinking the net reactive term in $Z=\sqrt{R^2+(X_L-X_C)^2}$.

In a series L-R circuit (before the capacitor is added), the impedance is $Z_1=\sqrt{R^2+X_L^2}$. When a capacitor is now added in series, the voltage phasors across L and across C are exactly $180^\circ$ out of phase with each other (one leads the current by $90^\circ$, the other lags it by $90^\circ$), so instead of adding, their magnitudes SUBTRACT: the new impedance becomes $Z_2=\sqrt{R^2+(X_L-X_C)^2}$. As long as $X_C$ is smaller than $X_L$ (or even if it slightly exceeds it, up to the point where the reduction in the reactive term outweighs any other change), the quantity $(X_L-X_C)^2$ is smaller than $X_L^2$ alone, so $Z_2<Z_1$: the overall impedance decreases. Physically, the capacitor's voltage drop works to CANCEL part of the inductor's voltage drop (since they are exactly out of phase), so less net reactive voltage needs to be supplied by the source for the same current, which is exactly why adding C reduces the impedance. This cancellation is complete (impedance falls all the way to its minimum possible value, Z = R) exactly when $X_L=X_C$, the resonance condition. [!ANSWER] Adding C in series reduces impedance because the capacitor's voltage is $180^\circ$ out of phase with the inductor's voltage, so the net reactance $(X_L-X_C)$ is smaller in magnitude than $X_L$ alone, shrinking $Z=\sqrt{R^2+(X_L-X_C)^2}$.

Compare the impedance formula before ($Z=\sqrt{R^2+X_L^2}$) and after ($Z=\sqrt{R^2+(X_L-X_C)^2}$) adding a series capacitor, and note that $X_L$ and $X_C$ subtract because their phasors are $180^\circ$ apart.

Assuming reactances simply add in magnitude regardless of type, when in fact inductive and capacitive reactance are exactly opposite in phase and so partially or fully cancel when combined in series.