15.2 Simple AC Circuits

Resistor

First, consider a resistor connected across an ac voltage source. From Kirchhoff’s loop rule, the instantaneous voltage across the resistor of Figure 15.5(a) is

and the instantaneous current through the resistor is

Figure 15.5 (a) A resistor connected across an ac voltage source. (b) The current $i_R\left(t\right)$ through the resistor and the voltage $v_R\left(t\right)$ across the resistor. The two quantities are in phase.

Here, $I_0=V_0\text{/}R$ is the amplitude of the time-varying current. Plots of $i_R\left(t\right)$ and $v_R\left(t\right)$ are shown in Figure 15.5(b). Both curves reach their maxima and minima at the same times, that is, the current through and the voltage across the resistor are in phase.

Graphical representations of the phase relationships between current and voltage are often useful in the analysis of ac circuits. Such representations are called phasor diagrams. The phasor diagram for $i_R\left(t\right)$ is shown in Figure 15.6(a), with the current on the vertical axis. The arrow (or phasor) is rotating counterclockwise at a constant angular frequency $\omega ,$ so we are viewing it at one instant in time. If the length of the arrow corresponds to the current amplitude $I_0,$ the projection of the rotating arrow onto the vertical axis is $i_R(t)=I_0\text{sin}\omega t,$ which is the instantaneous current.

Figure 15.6 (a) The phasor diagram representing the current through the resistor of Figure 15.5. (b) The phasor diagram representing both $i_R\left(t\right)$ and $v_R\left(t\right)$.

The vertical axis on a phasor diagram could be either the voltage or the current, depending on the phasor that is being examined. In addition, several quantities can be depicted on the same phasor diagram. For example, both the current $i_R\left(t\right)$ and the voltage $v_R\left(t\right)$ are shown in the diagram of Figure 15.6(b). Since they have the same frequency and are in phase, their phasors point in the same direction and rotate together. The relative lengths of the two phasors are arbitrary because they represent different quantities; however, the ratio of the lengths of the two phasors can be represented by the resistance, since one is a voltage phasor and the other is a current phasor.

Capacitor

Now let’s consider a capacitor connected across an ac voltage source. From Kirchhoff’s loop rule, the instantaneous voltage across the capacitor of Figure 15.7(a) is

Recall that the charge in a capacitor is given by $Q=CV.$ This is true at any time measured in the ac cycle of voltage. Consequently, the instantaneous charge on the capacitor is

Since the current in the circuit is the rate at which charge enters (or leaves) the capacitor,

where $I_0=\omega C{V}_0$ is the current amplitude. Using the trigonometric relationship $\text{cos}\omega t=\text{sin}\left(\omega t+\pi \text{/}2\right),$ we may express the instantaneous current as

Dividing $V_0$ by $I_0$, we obtain an equation that looks similar to Ohm’s law:

The quantity $X_C$ is analogous to resistance in a dc circuit in the sense that both quantities are a ratio of a voltage to a current. As a result, they have the same unit, the ohm. Keep in mind, however, that a capacitor stores and discharges electric energy, whereas a resistor dissipates it. The quantity $X_C$ is known as the capacitive reactance of the capacitor, or the opposition of a capacitor to a change in current. It depends inversely on the frequency of the ac source—high frequency leads to low capacitive reactance.

Figure 15.7 (a) A capacitor connected across an ac generator. (b) The current $i_C\left(t\right)$ through the capacitor and the voltage $v_C\left(t\right)$ across the capacitor. Notice that $i_C\left(t\right)$ leads $v_C\left(t\right)$ by $\pi \text{/}2$ rad.

A comparison of the expressions for $v_C\left(t\right)$ and $i_C\left(t\right)$ shows that there is a phase difference of $\pi \text{/}2\text{rad}$ between them. When these two quantities are plotted together, the current peaks a quarter cycle (or $\pi \text{/}2\text{rad}$) ahead of the voltage, as illustrated in Figure 15.7(b). The current through a capacitor leads the voltage across a capacitor by $\pi \text{/}2\text{rad},$ or a quarter of a cycle.

The corresponding phasor diagram is shown in Figure 15.8. Here, the relationship between $i_C\left(t\right)$ and $v_C\left(t\right)$ is represented by having their phasors rotate at the same angular frequency, with the current phasor leading by $\pi \text{/}2\text{rad}.$

Figure 15.8 The phasor diagram for the capacitor of Figure 15.7. The current phasor leads the voltage phasor by $\pi \text{/}2$ rad as they both rotate with the same angular frequency.

To this point, we have exclusively been using peak values of the current or voltage in our discussion, namely, $I_0$ and $V_0.$ However, if we average out the values of current or voltage, these values are zero. Therefore, we often use a second convention called the root mean square value, or rms value, in discussions of current and voltage. The rms operates in reverse of the terminology. First, you square the function, next, you take the mean, and then, you find the square root. As a result, the rms values of current and voltage are not zero. Appliances and devices are commonly quoted with rms values for their operations, rather than peak values. We indicate rms values with a subscript attached to a capital letter (such as $I_{\text{rms}}$).

Although a capacitor is basically an open circuit, an rms current, or the root mean square of the current, appears in a circuit with an ac voltage applied to a capacitor. Consider that

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 Figure 15.9(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

Figure 15.9 (a) An inductor connected across an ac generator. (b) The current $i_L\left(t\right)$ through the inductor and the voltage $v_L\left(t\right)$ across the inductor. Here $i_L\left(t\right)$ lags $v_L\left(t\right)$ by $\pi \text{/}2$ rad.

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 Equation 15.6 and Equation 15.7, 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 Figure 15.10.

Figure 15.10 The phasor diagram for the inductor of Figure 15.9. The current phasor lags the voltage phasor by $\pi \text{/}2$ rad as they both rotate with the same angular frequency.