15.2 Simple ac circuits  (Page 3/7)

where $I_0$ is the peak current in an ac system. The rms voltage    , or the root mean square of the voltage, is

where $V_0$ is the peak voltage in an ac system. The rms current appears because the voltage is continually reversing, charging, and discharging the capacitor. If the frequency goes to zero, which would be a dc voltage, $X_C$ tends to infinity, and the current is zero once the capacitor is charged. At very high frequencies, the capacitor’s reactance tends to zero—it has a negligible reactance and does not impede the current (it acts like a simple wire).

Inductor

Lastly, let’s consider an inductor    connected to an ac voltage source. From Kirchhoff’s loop rule, the voltage across the inductor L of [link] (a) is

The emf across an inductor is equal to $\epsilon =\text{−}L\left(d{i}_L\text{/}dt\right);$ however, the potential difference across the inductor is $v_L\left(t\right)=Ld{i}_L\left(t\right)\text{/}dt$ , because if we consider that the voltage around the loop must equal zero, the voltage gained from the ac source must dissipate through the inductor. Therefore, connecting this with the ac voltage source, we have

The current $i_L\left(t\right)$ is found by integrating this equation. Since the circuit does not contain a source of constant emf, there is no steady current in the circuit. Hence, we can set the constant of integration, which represents the steady current in the circuit, equal to zero, and we have

where $I_0=V_0\text{/}\omega L.$ The relationship between $V_0$ and $I_0$ may also be written in a form analogous to Ohm’s law:

The quantity $X_L$ is known as the inductive reactance    of the inductor, or the opposition of an inductor to a change in current; its unit is also the ohm. Note that $X_L$ varies directly as the frequency of the ac source—high frequency causes high inductive reactance.

A phase difference of $\pi \text{/}2$ rad occurs between the current through and the voltage across the inductor. From [link] and [link] , the current through an inductor lags the potential difference across an inductor by $\pi \text{/}2\text{rad}$ , or a quarter of a cycle. The phasor diagram for this case is shown in [link] .

Summary

• For resistors, the current through and the voltage across are in phase.
• For capacitors, we find that when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle. Since a capacitor can stop current when fully charged, it limits current and offers another form of ac resistance, called capacitive reactance, which has units of ohms.
• For inductors in ac circuits, we find that when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle.
• The opposition of an inductor to a change in current is expressed as a type of ac reactance. This inductive reactance, which has units of ohms, varies with the frequency of the ac source.

Conceptual questions

Problems