In an AC circuit, with only resistors, the voltage and current are in phase. What that means is that the voltage and current change in lock step. When the voltage increases, the current increases. When the voltage decreases, the current decreases.
When there are capacitors and inductors in an AC circuit, however, the phase relationship between the voltage and current changes. Specifically, the relationship between the current through a capacitor and the voltage across a capacitor is that the current leads voltage by 90 degrees. (E5B09) We could also say that the voltage lags the current by 90 degrees. See figure below.
What that means is that the current through a capacitor increases and decreases before the voltage across a capacitor increases and decreases. We say that the current leads the voltage by 90 degrees because it starts increasing one-quarter of a cycle before the voltage starts increasing.
The relationship between the current through an inductor and the voltage across an inductor is that the voltage leads current by 90 degrees. (E5B10) We could also say that the current lags the voltage. See figure below.
What that means is that the voltage across an inductor increases and decreases before the current through the inductor increases and decreases. We say that the voltage leads the voltage by 90 degrees because it starts increasing one-quarter of a cycle before the current starts increasing.
When there are resistors as well as a capacitor or inductor or both in a circuit, the relationship is a little more complicated. Let’s look at what happens in the series RLC circuit shown below.
In this circuit, there is resistance, capacitive reactance, and inductive reactance. The reactances subtract from one another. If the capacitive reactance is greater than the inductive reactance, the net reactance will be capacitive. If the inductive reactance is greater than the capacitive reactance, the net reactance will be inductive.
The resistance and the reactance add to one another, but they add vectorially. The reason for this is that the reactance will be 90 degrees out of phase with the resistance. This is shown in the figure below.
The magnitude of the impedance, Z, will be equal to sqrt(R2 + X2) and the sine of the phase angle will be equal to X/R. Let’s see how this works in several examples.
If XC is 500 ohms, R is 1 kilohm, and XL is 250 ohms, the phase angle between the voltage across and the current through the series RLC circuit is 14.0 degrees with the voltage lagging the current. (E5B07) Here’s how to calculate that:
X = XC – XL = 250 ohms (capacitive)
phase angle = tan-1 (250/1000) = 14 degrees.
and because the reactance is capacitive, the voltage will lag the current.
If XC is 100 ohms, R is 100 ohms, and XL is 75 ohms, the phase angle between the voltage across and the current through the series RLC circuit is 14 degrees with the voltage lagging the current. (E5B08) Here’s the calculation:
X = XC – XL = 25 ohms (capacitive)
phase angle = tan-1 (25/100) = 14 degrees.
and because the reactance is capacitive, the voltage lags the current.
If XC is 25 ohms, R is 100 ohms, and XL is 50 ohms, the phase angle between the voltage across and the current through the series RLC circuit is 14 degrees with the voltage leading the current. (E5B11) Here’s the calculation:
X = XL – XC = 25 ohms (inductive)
phase angle = tan-1 (25/100) = 14 degrees.
and because the reactance is inductive, the voltage leads the current.
If XC is 75 ohms, R is 100 ohms, and XL is 50 ohms, the phase angle between the voltage across and the current through the series RLC circuit is 14 degrees with the voltage lagging the current. (E5B12) Here’s the calculation:
X = XC – XL = 25 ohms (capacitive)
phase angle = tan-1 (25/100) = 14 degrees.
and because the reactance is capacitive, the voltage lags the current.
If XC is 250 ohms, R is 1 kiloohm, and XL is 500 ohms, the phase angle between the voltage across and the current through the series RLC circuit is 14 degrees with the voltage lagging the current. (E5B13) Here’s the calculation:
X = XL – XC = 250 ohms (inductive)
phase angle = tan-1 (250/1000) = 14 degrees.
and because the reactance is capacitive, the voltage leads the current.
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