Physics · Ch 7 — Alternating Current

The CR Series Circuit

7.9

The CR Series Circuit

Setting up the circuit. A resistor $R$ and a capacitor $C$ are connected in SERIES to an AC source $v=v_0\sin\omega t$ , again sharing one common current $I$ used as the reference phasor.

The two voltage phasors. As before, $V_R=IR$ is in phase with the current, so its phasor lies along $I$ . The voltage across the capacitor, $V_C=IX_C$ , LAGS the current by $90^\circ$ (Section 7.6 showed the current leads the voltage by $90^\circ$ , which is the same statement read the other way round), so its phasor is drawn at right angles to $I$ , but BEHIND it this time, in the opposite rotational sense to the inductor's $V_L$ phasor.

Adding the two phasors. With $V_R$ and $V_C$ again at right angles, the resultant applied voltage is

V = \sqrt{V_R^2+V_C^2} = \sqrt{(IR)^2+(IX_C)^2} = I\sqrt{R^2+X_C^2}

Reading off the impedance.

Z = \sqrt{R^2+X_C^2}

The phase angle. The angle by which the current LEADS the applied voltage in this circuit is

$\tan\phi = \frac{V_C}{V_R} = \frac{X_C}{R}$ …

Figure 1Impedance-triangle phasor diagram for a series CR circuit

What this figure shows. A right-angled triangle (impedance triangle) is drawn with the horizontal base leg labelled $R$ , the vertical leg labelled $X_C$ (drawn going DOWNWARD from the right end of the base this time, to visually distinguish it from the LR triangle's upward $X_L$ leg), and the hypotenuse labelled $Z$ , with the angle $\phi$ between $R$ and $Z$ marked with a small arc at the origin. Beside it, a small phasor diagram shows the current phasor $I$ along the horizontal reference direction, the resistor's voltage phasor $V_R$ along the same horizontal direction as $I$ , and the capacitor's voltage phasor $V_C$ drawn at $90^\circ$ CLOCKWISE from $V_R$ (behind it, pointing downward); the resultant applied voltage phasor $V$ is the diagonal of the rectangle formed by $V_R$ and $V_C$ , drawn at angle $\phi$ BELOW $V_R$ this time, with a small arc m …

Start your 14-day free trial to unlock the full solution →

← Previous · 7.8The LR Series Circuit Next · 7.10 →The LCR Series Circuit