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 90o If we were to plot the current and voltage for this very simple circuit, it would look something like this: (Figure below)air filters in hvac-systems Pure capacitive circuit waveforms.how to defrost frozen ac unit Remember, the current through a capacitor is a reaction against the change in voltage across it. ice on air conditioner inside unitTherefore, 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 -90o 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. Voltage lags current by 90o 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 XC 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. Reactance of a 100 uF capacitor: 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 XC = 1/(2πfC) could also be written as XC = 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) 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 +90o 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 90o in a capacitor. Mathematically, we say that the phase angle of a capacitor’s opposition to current is -90o, meaning that a capacitor’s opposition to current is a negative imaginary quantity. 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.Likewise, when the supply voltage is reduced the charge stored in the capacitor also reduces and the capacitor discharges. But in an AC circuit in which the applied voltage signal is continually changing from a positive to a negative polarity at a rate determined by the frequency of the supply, as in the case of a sine wave voltage, for example, the capacitor is either being charged or discharged on a continuous basis at a rate determined by the supply frequency.

As the capacitor charges or discharges, a current flows through it which is restricted by the internal resistance of the capacitor. This internal resistance is commonly known as Capacitive Reactance and is given the symbol XC in Ohms. Unlike resistance which has a fixed value, for example, 100Ωs, 1kΩ, 10kΩ etc, (this is because resistance obeys Ohms Law), Capacitive Reactance varies with the applied frequency so any variation in supply frequency will have a big effect on the capacitors, “capacitive reactance” value. As the frequency applied to the capacitor increases, its effect is to decrease its reactance (measured in ohms). Likewise as the frequency across the capacitor decreases its reactance value increases. This variation is called the capacitors complex impedance. Complex impedance exists because the electrons in the form of an electrical charge on the capacitor plates, pass from one plate to the other more rapidly with respect to the varying frequency.

As the frequency increases, the capacitor passes more charge across the plates in a given time resulting in a greater current flow through the capacitor appearing as if the internal resistance of the capacitor has decreased. Therefore, a capacitor connected to a circuit that changes over a given range of frequencies can be said to be “Frequency Dependant”. Capacitive Reactance has the electrical symbol “Xc” and has units measured in Ohms the same as resistance, ( R ). It is calculated using the following formula: Calculate the capacitive reactance of a 220nF capacitor at a frequency of 1kHz and again at 20kHz. At a frequency of 1kHz, Again at a frequency of 20kHz, where: ƒ = frequency in Hertz and C = capacitance in Farads Therefore, it can be seen from above that as the frequency applied across the 220nF capacitor increases, from 1kHz to 20kHz, its reactance value decreases, from approx 723Ωs to just 36Ωs and this is always true as capacitive reactance, Xc is inversely proportional to frequency with the current passed by the capacitor for a given voltage being proportional to the frequency.

For any given value of capacitance, the reactance of a capacitor, Xc expressed in ohms can be plotted against the frequency as shown below. By re-arranging the reactance formula above, we can also find at what frequency a capacitor will have a particular capacitive reactance ( XC ) value. At which frequency would a 2.2uF Capacitor have a reactance value of 200Ωs? Or we can find the value of the capacitor in Farads by knowing the applied frequency and its reactance value at that frequency. What will be the value of a capacitor in farads when it has a capacitive reactance of 200Ω and is connected to a 50Hz supply. We can see from the above examples that a capacitor when connected to a variable frequency supply, acts a bit like a “frequency controlled variable resistor”. At very low frequencies, such as 1Hz our 220nF capacitor has a high capacitive reactance value of approx 723.3KΩs (giving the effect of an open circuit). At very high frequencies such as 1Mhz the capacitor has a low capacitive reactance value of just 0.72Ω (giving the effect of a short circuit).

So at zero frequency or steady state DC our 220nF capacitor has infinite reactance looking more like an “open-circuit” between the plates and blocking any flow of current through it. We remember from our tutorial about Resistors in Series that different voltages can appear across each resistor depending upon the value of the resistance and that a voltage divider circuit has the ability to divide its supply voltage by the ratio of R2/(R1+R2). Therefore, when R1 = R2 the output voltage will be half the value of the input voltage. Likewise, any value of R2 greater or less than R1 will result in a proportional change to the output voltage. Consider the circuit below. We now know that a capacitors reactance, Xc (its complex impedance) value changes with respect to the applied frequency. If we now changed resistor R2 above for a capacitor, the voltage drop across the two components would change as the frequency changed because the reactance of the capacitor affects its impedance.

The impedance of resistor R1 does not change with frequency. Resistors are of fixed values and are unaffected by frequency change. Then the voltage across resistor R1 and therefore the output voltage is determined by the capacitive reactance of the capacitor at a given frequency. This then results in a frequency-dependent RC voltage divider circuit. With this idea in mind, passive Low Pass Filters and High Pass Filters can be constructed by replacing one of the voltage divider resistors with a suitable capacitor as shown. The property of Capacitive Reactance, makes capacitors ideal for use in AC filter circuits or in DC power supply smoothing circuits to reduce the effects of any unwanted Ripple Voltage as the capacitor applies an short circuit signal path to any unwanted frequency signals on the output terminals. So, we can summarize the behaviour of a capacitor in a variable frequency circuit as being a sort of frequency controlled resistor that has a high capacitive reactance value (open circuit condition) at very low frequencies and low capacitive reactance value (short circuit condition) at very high frequencies as shown in the graph above.