# Why is the capacitive reactance negative

## AC capacitor circuits

### AC capacitor circuits

#### Chapter 4 - Reactance and Impedance - Capacitive

### Capacitors Vs. Resistors

Capacitors don't behave like resistors. While resistors allow a flow of electrons through them in direct proportion to the voltage drop, capacitors counter voltage changes by drawing or supplying current while they are being charged or discharged to the new voltage level. The flow of electrons through a capacitor is directly proportional to the rate of change in the voltage across the capacitor. This contrast to the change in voltage is a different form of reactance, but it is exactly the opposite of what inductors exhibit.

### Capacitor circuit properties

In mathematical terms, the relationship between the current through the capacitor and the rate of voltage change across the capacitor is as such:

The expression de / dt is a value from the calculus, which means the rate of change of the instantaneous voltage (e) over time in volts per second. The capacitance (C) is in farads and the instantaneous current (i) is of course in amperes. Sometimes you will find the rate of instantaneous voltage change over time as dv / dt instead of de / dt: use the lowercase "v" instead of or "e" to represent voltage, but it means exactly the same thing. To show what happens to alternating current, let's analyze a simple capacitor circuit: (Figure below)

Pure capacitive circuit: The capacitor voltage delays the capacitor current by 90^{°}

If we were to draw the current and voltage for this very simple circuit it would look something like this: (image below)

Pure capacitive waveforms.Remember that the current through a capacitor is a response to the change in voltage. Therefore, the instantaneous current is always zero when the instantaneous voltage is at a peak (zero change or level slope on the voltage sine wave), and the instantaneous current is at a peak wherever the instantaneous voltage changes maximally (the points of steepest slope on the voltage wave where it crosses the zero line). This results in a stress wave that is 90º from the ^{Current wave} is out of phase. Looking at the graph, the current wave appears to have a "lead" over the stress wave; the current "carries" the voltage, and the voltage "leaves" behind the current. (Picture below)

As you may have guessed, the same unusual wave of energy that we saw with the simple inductor circuit is also present in the simple capacitor circuit: (Figure below)

In a purely capacitive circuit, the instantaneous power can be positive or negative.As with the simple inductor circuit, the 90 degree phase shift between voltage and current results in a power wave that changes equally between positive and negative. This means that a capacitor does not dissipate power as it reacts to changes in voltage. it alternately absorbs and emits force.

### The reactance of a capacitor

The resistance of a capacitor to a change in voltage leads to an opposition to alternating voltage in general, which by definition always changes in magnitude and direction. For any given amount of AC voltage at a given frequency, a capacitor of the given size "will" conduct a certain amount of AC current. Just as the current through a resistor is a function of the voltage across the resistor and the resistance offered by the resistor, the alternating current through a capacitor is a function of the alternating voltage across it and the reactance offered by the capacitor. As with inductors, the reactance of a capacitor is expressed in ohms and is represented by the letter X (or X _{C.}to be more precise).

Since the capacitors "conduct" the current in proportion to the rate of change in voltage, they will pass more current for faster changing voltages (when charged and discharged to the same voltage spikes in less time) and less current for slower changing voltages. This means that the reactance in ohms for each capacitor is inversely proportional to the frequency of the alternating current. (Table below)

Reactance of a 100 μF capacitor:Frequency (Hertz) | Reactance (ohms) |
---|---|

60 | 26.5258 |

120 | 13.2629 |

2500 | 0.6366 |

Please note that the ratio of capacitive reactance to frequency is exactly the opposite of that of inductive reactance. The capacitive reactance (in ohms) decreases with increasing AC frequency. Conversely, the inductive reactance (in ohms) increases with increasing AC frequency. Inductors are against faster changing currents by creating larger voltage drops; Capacitors counteract rapidly changing voltage drops by allowing larger currents.

As with inductors, the 2πf term of the reactance equation can be replaced with the lowercase Greek letter omega (ω), which is referred to as the angular velocity of the AC circuit. Thus, the equation X _{C.} = 1 / (2? FC) can also be written as X _{C.} = 1 / (? _{C.} ), where & ohgr; is poured in units of radians per second.

The alternating current in a simple capacitive circuit is equal to the voltage (in volts) divided by the capacitive reactance (in ohms), just as either the alternating or direct current in a simple resistor circuit is equal to the voltage (in volts) divided by the resistance (in Ohm). The following circuit illustrates this mathematical relationship using an example: (Figure below)

Capacitive reactance.However, we must note that the voltage and current are not in phase here. As previously shown, the current has a phase shift of + 90 ° with respect to the voltage. If we mathematically represent these phase angles of voltage and current, we can calculate the phase angle of the reactive opposition of the capacitor to the current.

The voltage in a capacitor is around 90^{°}lagging behind.

Mathematically we say that the phase angle of the resistance of a capacitor to the current is -90 °, which means that the resistance of a capacitor to the current is a negative imaginary quantity. (Figure above) This phase angle of reactive opposition to current becomes critical in circuit analysis, especially for complex AC circuits where reactance and resistance work together. It will prove advantageous to represent the resistance of a component to the current in the form of complex numbers rather than just scalar quantities of resistance and reactance.

**REVIEW:**- • Capacitive reactance is the resistance that a capacitor offers to the alternating current due to its phase-shifted storage and release of energy in its electrical field. The reactance is symbolized by the capital letter "X" and, like the resistance (R), is measured in ohms.
- • The capacitive reactance can be calculated using the following formula: X
_{C.}= 1 / (2πfC) - • The capacitive reactance decreases with increasing frequency. In other words, the higher the frequency, the less it opposes the alternating current flow of electrons (the more it "conducts").

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