Home > Textbooks > Lessons In Electric Circuits > Vol. II - AC > Reactance and Impedance - Capacitive > AC Capacitor Circuits

Chapter 4: REACTANCE AND IMPEDANCE - CAPACITIVE

# AC Capacitor Circuits

Capacitors do not behave the same as resistors. Whereas resistors allow
a flow of electrons through them directly proportional to the voltage
drop, capacitors oppose *changes* in voltage by drawing or
supplying current as they charge or discharge to the new voltage level.
The flow of electrons "through" a capacitor is directly proportional to
the *rate of change* of voltage across the capacitor. This opposition to voltage change is another form of *reactance*, but one that is precisely opposite to the kind exhibited by inductors.

Expressed mathematically, the relationship between the current "through" the capacitor and rate of voltage change across the capacitor is as such:

The expression *de/dt* is one from calculus, meaning the rate of
change of instantaneous voltage (e) over time, in volts per second. The
capacitance (C) is in Farads, and the instantaneous current (i), of
course, is in amps. Sometimes you will find the rate of instantaneous
voltage change over time expressed as dv/dt instead of de/dt: using the
lower-case letter "v" instead or "e to represent voltage, but it means
the exact same thing. To show what happens with alternating current,
let's analyze a simple capacitor circuit: (Figure below)

*Pure capacitive circuit: capacitor voltage lags capacitor current by 90 ^{o}*

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

*Pure capacitive circuit waveforms.*

Remember, the current through a capacitor is a reaction against the *change*
in voltage across it. Therefore, the instantaneous current is zero
whenever 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 is at maximum change (the points
of steepest slope on the voltage wave, where it crosses the zero line).
This results in a voltage wave that is -90^{o} out of phase
with the current wave. Looking at the graph, the current wave seems to
have a "head start" on the voltage wave; the current "leads" the
voltage, and the voltage "lags" behind the current. (Figure below)

*Voltage lags current by 90 ^{o} in a pure capacitive circuit.*

As you might have guessed, the same unusual power wave that we saw with the simple inductor circuit is present in the simple capacitor circuit, too: (Figure below)

*In a pure capacitive circuit, the instantaneous power may 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 alternates equally between positive and negative. This means that a capacitor does not dissipate power as it reacts against changes in voltage; it merely absorbs and releases power, alternately.

A capacitor's opposition to change in voltage translates to an
opposition to alternating voltage in general, which is by definition
always changing in instantaneous magnitude and direction. For any given
magnitude of AC voltage at a given frequency, a capacitor of given size
will "conduct" a certain magnitude 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 AC current through a
capacitor is a function of the AC voltage across it, and the *reactance*
offered by the capacitor. As with inductors, the reactance of a
capacitor is expressed in ohms and symbolized by the letter X (or X_{C} to be more specific).

Since capacitors "conduct" current in proportion to the rate of voltage
change, they will pass more current for faster-changing voltages (as
they charge and discharge to the same voltage peaks in less time), and
less current for slower-changing voltages. What this means is that
reactance in ohms for any capacitor is *inversely* proportional to the frequency of the alternating current. (Table below)

*Reactance of a 100 uF capacitor:*

Frequency (Hertz) | Reactance (Ohms) |
---|---|

60 | 26.5258 |

120 | 13.2629 |

2500 | 0.6366 |

Please note that the relationship of capacitive reactance to frequency is exactly opposite from that of inductive reactance. Capacitive reactance (in ohms) decreases with increasing AC frequency. Conversely, inductive reactance (in ohms) increases with increasing AC frequency. Inductors oppose faster changing currents by producing greater voltage drops; capacitors oppose faster changing voltage drops by allowing greater currents.

As with inductors, the reactance equation's 2πf term may be replaced by
the lower-case Greek letter Omega (ω), which is referred to as the *angular velocity* of the AC circuit. Thus, the equation X_{C} = 1/(2πfC) could also be written as X_{C} = 1/(ωC), with ω cast in units of *radians per second*.

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

*Capacitive reactance.*

However, we need to keep in mind that voltage and current are not in
phase here. As was shown earlier, the current has a phase shift of +90^{o}
with respect to the voltage. If we represent these phase angles of
voltage and current mathematically, we can calculate the phase angle of
the capacitor's reactive opposition to current.

*Voltage lags current by 90 ^{o} in a capacitor.*

Mathematically, we say that the phase angle of a capacitor's opposition to current is -90^{o}, meaning that a capacitor's opposition to current is a negative imaginary quantity. (Figure above)
This phase angle of reactive opposition to current becomes critically
important in circuit analysis, especially for complex AC circuits where
reactance and resistance interact. It will prove beneficial to
represent *any* component's opposition to current in terms of complex numbers, and not just scalar quantities of resistance and reactance.

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

## Related Content:

Video lecture: Capacitive Reactance, Impedance, Power Factor, AC Circuits, Physics