2.3: Alternating Current Circuits

A direct current has a fixed value that is independent of time. An alternating current, on the other hand, has a value that changes with time. This change in current follows a pattern that we can characterize by it period—the time, \(t_p\), for one complete cycle—or by its frequency, \(f\), which is the reciprocal of its period

\[f = \frac{1}{t_p} \label{sine1} \]

Frequency is reported in hertz (Hz), which is equivalent to one cycle per second.

Sinusoidal Currents and Voltages

Although we can draw many periodic signals—and will do so in later chapters—the simplest periodic signal is a sine wave: as shown on the right side of Figure \(\PageIndex{1}\), the sine is a propagating wave whose amplitude, \(A\), is a function of time, \(t\), which we write as \(a(t)\).

Figure \(\PageIndex{1}\). Illustration showing how a sine wave can be explained by a rotating vector. See text for an explanation of the symbols.

The left side of Figure \(\PageIndex{1}\) provides a rotating vector representation of the sine wave (a representation we will encounter again in Chapter 19 on NMR spectroscopy). The vector is the arrow that extends from the center of the circle to the circle's edge. It is rotating to the left with an angular velocity given by \(\omega\) and that is expressed in radians per the sine wave's period, \(t_p\); thus

\[\omega = \frac{2 \pi}{t_p} = 2 \pi f \label{sine2} \]

where \(f\) is the frequency. The amplitude of the sine wave as a function of time, \(a(t)\), is equivalent to the projection of the rotating vector onto the x-axis; thus

\[a(t) = A\sin{\omega t} = A\sin{2 \pi f t} \label{sine3} \]

In the context of this chapter, the amplitude is either a current, \(i\), or a voltage, \(v\).

\[i(t) = I\sin{\omega t} = I\sin{2 \pi f t} \label{sine4} \]

\[v(t) = V\sin{\omega t} = V\sin{2 \pi f t} \label{sine5} \]

where \(I\) is the maximum, or peak current, and \(V\) is the maximum, or peak voltage.

Equations \ref{sine4} and \ref{sine5} require that a sine wave's time-dependent amplitude, \(a(t)\), has a value of zero when \(t = n \pi\), where \(n\) is an integer. There is no reason to insist on this and two sine waves can be separated from each other in time, as shown in Figure \(\PageIndex{2}\), by a phase angle, \(\Phi\). The equation for the sine wave when \(\Phi \ne 0\) becomes

\[a(t) = A\sin{(\omega t + \Phi)} = A\sin{(2 \pi f t + \Phi)} \label{sine6} \]

Figure \(\PageIndex{2}\). Two sine waves that are out-of-phase with each other. The angle, \(\Phi\), between the two rotating vectors is the phase angle.

One complication of an alternating current is that the net current over the course of a single cycle is zero. This is a problem for us because the equation for power in a resistor is

\[P = \frac{I^2}{R} \ne 0 \label{sine7} \]

Figure \(\PageIndex{3}\) shows several ways to report current in AC circuits.

Figure \(\PageIndex{3}\). Different ways to report current: I_p is the maximum absolute current, or peak current; I_pp is the peak-to-peak current; I_rms is the root-mean-square current; and I_avg is the numerical average current.

The root-mean-square current, \(I_{rms}\), is defined as

\[I_{rms} = \sqrt{\frac{I_p^2}{2}} = \sqrt{2} \times \frac{I_p}{2} = 0.707 \times I_p \label{sine8} \]

and yields the same power in an AC circuit as a direct current of equal value in a DC circuit. The average current, \(I_{avg}\), is

\[I_{avg} = \frac{1}{\pi} \int_{0}^{\pi}I_p \sin{\omega t}\, dt = \frac{2 I_p}{\pi} = 0.6371 \times I_p \label{sine9} \]

Capacitors

A capacitor is a component of circuits that is capable of storing charge. Figure \(\PageIndex{4}\) shows the design of a typical capacitor and its symbol when constructing an electrical circuit. The capacitor consists of two conducting plates separated by a thin layer of an insulating, or dielectric material. The plates have areas of \(A\) and are separated by a distance, \(d\). The dielectric material has a dielectric constant, \(\epsilon\). A simple capacitor might consist of two pieces of a metal foil separated by air, which serves as as the dielectric material. A capacitor's ability to store charge, \(Q\), is given by

\[Q = C \times V \label{cap1} \]

where \(V\) is the voltage applied across the two plates and where \(C\) is the capacitor's capacitance, which, in turn, is defined as

\[C = \frac{\epsilon A}{d} \label{cap2} \]

Capacitance is measured in units of farads, where one farad is equal to one coulomb per volt.

Figure \(\PageIndex{4}\). Schematic diagram of a capacitor. In an electrical circuit, a capacitor is represented by the symbol at the lower right.

Resistor and Capacitor in Series

Figure \(\PageIndex{5}\) shows a resistor, with a resistance of \(R\), and a capacitor, with a capacitance of \(C\), in series with a voltage source, with a voltage of \(V\).

Figure \(\PageIndex{5}\). A simple circuit that consists of a resistor and capacitor in series.

When the switch it closed, current flows as the capacitor builds up a charge. From Kirchoff's laws, we know that

\[V = v_R + v_C = iR + \frac{Q}{C} \label{cap3} \]

where \(v_R\) and \(v_C\) are, respectively, the time-dependent voltages across the resistor and the capacitor. Because \(V\) has a fixed value, any increase in \(v_C\) as the capacitor is charged is offset by a decrease in \(V_r\). Given that the values of \(v_C\) and \(v_R\)—and the associated currents—are time-dependent, we can differentiate Equation \ref{cap3} with respect to time

\[\frac{dV}{dt} = 0 = \left( R \times \frac{di}{dt} \right) + \left( \frac{1}{C} \times \frac{dq}{dt} \right) = \left(R \times \frac{di}{dt}\right) + \frac{i}{C} \label{cap4} \]

Rearranging Equation \ref{cap4} gives

\[\frac{di}{i} = - \frac{1}{RC}dt \label{cap5} \]

Integrating both sides of this equation

\[ \int_{I_{0}}^{i} \frac{1}{i} di = -\frac{1}{RC} \int_{0}^{t} dt \label{cap6} \]

leads to the following relationship between the current at time \(t\) and the initial current, \(I_0\)

\[i_t = I_0 \times e^{-t/RC} \label{cap7} \]

which tells us that the current decreases exponentially as the capacitor becomes fully charged. Replacing the current in equation \ref{cap7} with \(\frac{V}{R}\) and substituting back into Equation \ref{cap3}

\[v_C = V_0 \left( 1 - e^{-t/RC} \right) \label{cap8} \]

shows us that during the time the capacitor is being charged, the current flowing through the capacitor is decreasing exponentially to its limit of zero, and the voltage across the capacitor is increasing exponentially to its limit of the applied voltage.

Time Constant

The value \(RC\) in Equation \ref{cap7} and in Equation \ref{cap8} is the circuit's time constant. It takes approximately five time constants for the capacitor to fully charge or fully discharge. Figure \(\PageIndex{6}\) shows the voltage across the capacitor, \(v_C\), as it is allowed to charge and to discharge. Time is shown in terms of the number of elapsed time constants, and voltage is expressed as a fraction of the maximum voltage. The dashed line shows that the time constant, \(RC\), is equivalent to \(0.63 \times\) the maximum voltage.

Figure \(\PageIndex{6}\). Voltage across a capacitor as it is charged and then discharged. The scale on the x-axis is time in increments of the time constant, \(RC\), and the scale on the y-axis is the fraction of the applied voltage. The dashed line shows that the time constant is equivalent to \(0.63 \times\) the applied voltage.

Response of a Series RC Circuit to a Sinusoidal Input

If we replace the DC voltage source in Figure \(\PageIndex{5}\) with an AC source, then the capacitor will undergo a continuous fluctuation in its voltage and current as a function of time. We know, form Equation \ref{cap1} that charge, \(Q\), is the product of capacitance, \(C\), and voltage,\(V\), which we can write as a derivative with respect to time.

\[\frac{dq}{dt} = C \times \frac{dv}{dt} \label{ac1} \]

Phase Shift in an AC Circuit

In an AC circuit, as we learned earlier in Equation \ref{sine4}, the current, which is equivalent to \(dq/dt\) is

\[i = I_p \sin{2 \pi f t} \label{ac2} \]

where \(I_p\) is the peak current. Substituting into Equation \ref{ac1} gives

\[i = I_p \sin{2 \pi f t} = C \times \frac{dv}{dt} \label{ac3} \]

Rearranging this equation and integrating over time gives the time-dependent voltage across the capacitor, \(v_C\), as

\[v_C = \frac{I_p}{2 \pi f C} \left( -\cos{2 \pi f t} \right) \label{ac4} \]

We can rewrite this equation in terms of a sine function instead of a cosine function by recognizing that the two are 90° out of phase with each other; thus

\[v_C = \frac{I_p}{2 \pi f C} \left( \sin{2 \pi f t -90} \right) = V_p \left( \sin{2 \pi f t -90} \right) \label{ac5} \]

Comparing Equation \ref{ac2} and Equation \ref{ac5}, we see that the current and the voltage are 90° out-of-phase with each other; Figure \(\PageIndex{7}\) shows this visually.

Figure \(\PageIndex{7}\). Response of a capacitor's voltage to a sinusoidal input current showing that they are 90° (\(\pi/2\)) out-of-phase.

Capacitive Reactance, Resistance, and Impedence

From Equation \ref{ac5} we see that

\[V_p = \frac{I_p}{2 \pi f t} \label{ac6} \]

Dividing both sides by \(I_p\) gives

\[\frac{V_p}{I_p} = X_C = \frac{1}{2 \pi f t} \label{ac7} \]

where \(X_C\) is the capacitor's reactance, which, like a resistor's resistance, has units of ohms. Unlike a resistor, however, a capacitor's reactance is frequency dependent and, given the reciprocal relationship between \(X_C\) and \(f\), it becomes smaller at higher frequencies.

In a RC circuit, both the resistor and the capacitor contribute to the circuit's impedence of the alternating current. Because the contribution of the capacitor is 90° out-of-phase to the contribution from the resistor, the net impedence, \(Z\), is

\[Z = \sqrt{R^2 + X_C^2} \label{ac8} \]

as shown in Figure \(\PageIndex{8}\) where the vector that represents \(Z\) is the hypotonus of a right triangle defined by the resistor's resistance and the capacitor's reactance.

Figure \(\PageIndex{8}\): The relationship between a resistor's resistance, a capacitor's reactance, and the impedence of the RC circuit.

Substituting in Equation \ref{ac7} shows the effect of frequency on impedence.

\[Z = \sqrt{R^2 + \left( \frac{1}{2 \pi f t} \right)^2} \label{ac9} \]

Writing Ohm's law in terms of impedence, \(V_p = I_p \times Z\), and substituting it into Equation \ref{ac9}, defines \(I_p\) and \(V_p\) in terms of impedence.

\[V_p = I_p \times \sqrt{R^2 + \left( \frac{1}{2 \pi f t} \right)^2} \label{ac10} \]

\[I_p = \frac{V_p}{\sqrt{R^2 + \left( \frac{1}{2 \pi f t} \right)^2}} \label{ac11} \]

Filters Based on RC Circuits

The frequency dependence of an RC circuit provides us with the ability to attenuate some frequencies and to pass other frequencies. This allows for the selective filtering of an input signal. Here we consider the design of a low-pass filter that removes higher frequency signals, and the design of a high-pass filter that removes lower frequency signals. Figure \(\PageIndex{9}\) shows that (a) a low-pass filter places the resistor before the capacitor and measures the output voltage, \(V_{out}\), across the capacitor, and that (b) a high-pass filter places the capacitor before the resistor and measures the output voltage, \(V_{out}\), across the resistor.

Figure \(\PageIndex{9}\). Circuits for (a) a low-pass filter and (b) a high-pass filter.

Low-Pass Filter

For the low-pass filter in Figure \(\PageIndex{9}a\), the ratio of the voltage across the capacitor, \((V_p)_{out}\), to the peak input voltage, \((V_p)_{in}\), is equal to the fraction of the circuit's impedence, \(Z\), attributed to the capacitor's reactance, \(X_C\), as expected for a voltage divider that consist of elements in series.

\[\frac{(V_p)_{out}}{(V_p)_{in}} = \frac{X_C}{Z} = \frac{(2 \pi f C)^{-1}}{\sqrt{R^2 + \left( \frac{1}{2 \pi f C}\right)^2}} \label{lowpass1} \]

Figure \(\PageIndex{10}a\) shows the frequency response for a low-pass filter with a \(1 \times 10^6 \text{ Hz}\) resistor and a \(1 \times 10^{-6} \text{ F}\) capacitor, removing all frequencies greater than approximately \(10^1\) Hz.

Figure \(\PageIndex{10}\). The output of (a) a low-pass filter and (b) a high-pass filter.

High-Pass Filter

For the high-pass filter in Figure \(\PageIndex{9}b\), the ratio of the voltage across the resistor, \((V_p)_{out}\), to the peak input voltage, \((V_p)_{in}\), is equal to the fraction of the circuit's impedence, \(Z\), attributed to the resistor's resistance, \(R\), as expected for a voltage divider that consist of elements in series.

\[\frac{(V_p)_{out}}{(V_p)_{in}} = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + \left( \frac{1}{2 \pi f C}\right)^2}} \label{lowpass2} \]

Figure \(\PageIndex{10}b\) shows the frequency response for a low-pass filter with a \(1 \times 10^5 \text{ Hz}\) resistor and a \(1 \times 10^{-7} \text{ F}\) capacitor, removing all frequencies less than approximately \(10^{-1}\).