7.5. AC VOLTAGE APPLIED TO A CAPACITOR

Figure 7.7 shows an ac source e generating ac voltage v = v sin wt connected to a capacitor only, a purely capacitive ac circuit. When a capacitor is connected to a voltage source in a dc circuit, current will flow for the short time required to charge the capacitor. As charge accumulates on the capacitor plates, the voltage across them increases, opposing the current.

In this chunk, we discuss how a capacitor interacts in an AC circuit. When connected to an alternating current supply, the capacitor does not behave like it does in a direct current (DC) scenario. In DC, it initially allows current to flow while charging up to a certain voltage, after which the current stops. In AC, however, the current continuously reverses direction, causing the capacitor to charge and discharge periodically.

Think of a water tank that fills up when water flows in and empties out when water flows back. In a DC circuit, the tank fills until it reaches a certain level and then the flow stops. In an AC circuit, the water continually fills and drains as the pressure changes direction, similar to how a capacitor charges and discharges in response to AC voltage.

Let q be the charge on the capacitor at any time t. The instantaneous voltage v across the capacitor is q / C. From the Kirchhoff’s loop rule, the voltage across the source and the capacitor are equal, q / C = v sin(wt).

Here, we establish a mathematical relationship between the charge (q) on the capacitor and the voltage (v) across it. The voltage across a capacitor is given by the formula v = q / C, where C is the capacitance. In an AC circuit, as the voltage from the source varies sinusoidally, the expression q / C = v sin(wt) shows that the voltage across the capacitor is induced by the changing charge that is affected by the alternating current.

Imagine that the water tank mentioned earlier is also connected to a pressure gauge. As water flows in and out (in rhythm with the changing voltage), the pressure in the tank (analogous to the voltage across the capacitor) changes according to how full the tank is. The gauge fluctuates up and down even though the total volume of water (representing the charge) may remain constant over a given period.

To find the current, we use the relation i = dq/dt. By deriving this, we have i = vC sin(wt) = wC v cos(wt).

This section focuses on determining the current flowing through the capacitor as it relates to the capacitor’s charge. The equation i = dq/dt highlights that the current is the rate of change of charge over time. When applying this to an AC voltage, the current is expressed as a function of time, showing that it leads the voltage by a phase of π/2 (90 degrees). This means that at the moment when the voltage is at its peak, the current reaches its peak value earlier.

Using the tank example again, imagine that the water flows into and out of the tank at different times. The rate of filling (current) reacts faster than the overall height of water (voltage). When you turn off the water supply and then turn it back on quickly, the gauge (voltage) hits a peak after the inflow has already peaked, illustrating how the current can change faster than the voltage.

Comparing it to i = v / R for a purely resistive circuit, we find that (1/wC) plays the role of resistance. It is called capacitive reactance and is denoted by X_c, where X_c = 1/wC.

In this chunk, we define capacitive reactance (X_c), which acts similarly to resistance in a resistive circuit, but inversely. This means that as the frequency of the AC voltage increases, the capacitive reactance decreases, allowing more current to flow through the capacitor. This indicates that capacitors impede AC signals differently than DC signals.

Consider a narrow river where water flows quickly (high frequency), allowing for little blockage (low reactance). If you widen the river (decrease frequency), the water encounters resistance (high reactance) and doesn't flow as freely. This analogy helps visualize how capacitors behave under varying frequencies of AC.

A comparison of Eq. (7.16) with the equation of source voltage shows that the current is π/2 ahead of voltage.

This chunk emphasizes the lead of current over voltage in a capacitive circuit, measured as π/2 radians or 90 degrees. This lead demonstrates a crucial characteristic of capacitors in AC circuits. Understanding this time relationship is essential for analyzing AC circuits' behavior.

If you think of the current and voltage as two people dancing, the person representing current would take a step forward (peak) before the person representing voltage does. This portrays their relationship during the alternating cycle, where the current moves ahead of the voltage.

The instantaneous power supplied to the capacitor is p = i v = (i v cos(ωt) sin(ωt)). The average power P equals 0 since the average of sin(2ωt) over a complete cycle is zero.

In this section, we see how to compute instantaneous power in an AC circuit involving a capacitor. It essentially combines the current and voltage expressions and reflects the fluctuating nature of power. Importantly, while current flows, the average power consumed by the capacitor over time is confirmed to be zero, indicating that capacitors do not dissipate energy.

If you were to continuously fill and drain the water tank (like the current in an AC circuit), for every cycle of filling and draining, the overall energy used averages out to zero because it effectively returns to the original state - no work is done at the end of the process.

Thus, we see that in the case of an inductor, the current lags the voltage by π/2 and in the case of a capacitor, the current leads the voltage by π/2.

Finally, we summarize the contrasting behaviors of inductors and capacitors in AC circuits regarding their voltage and current relationships. This highlights an essential principle in AC circuit analysis that informs subsequent studies of complex circuits.

Think of inductors and capacitors as two people in a three-legged race – one consistently lags behind (the inductor), while the other always manages to get ahead (the capacitor). This analogy encapsulates how they interact with alternating voltage and current.

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