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Taking the derivative and multiplying by C , gives a first-order differential equation :.


With this assumption, solving the differential equation yields. As the capacitor reaches equilibrium with the source voltage, the voltages across the resistor and the current through the entire circuit decay exponentially.

In the case of a discharging capacitor, the capacitor's initial voltage V Ci replaces V 0. The equations become.


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Impedance , the vector sum of reactance and resistance , describes the phase difference and the ratio of amplitudes between sinusoidally varying voltage and sinusoidally varying current at a given frequency. Fourier analysis allows any signal to be constructed from a spectrum of frequencies, whence the circuit's reaction to the various frequencies may be found. The reactance and impedance of a capacitor are respectively. Impedance decreases with increasing capacitance and increasing frequency. Conversely, for very low frequencies, the reactance is high, so that a capacitor is nearly an open circuit in AC analysis—those frequencies have been "filtered out".

Capacitors are different from resistors and inductors in that the impedance is inversely proportional to the defining characteristic; i. A capacitor connected to a sinusoidal voltage source causes a displacement current to flow through it. The ratio of peak voltage to peak current is due to capacitive reactance denoted X C.

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If X C approaches 0, the capacitor resembles a short wire that strongly passes current at high frequencies. If X C approaches infinity, the capacitor resembles an open circuit that poorly passes low frequencies. The current of the capacitor may be expressed in the form of cosines to better compare with the voltage of the source:. When using the Laplace transform in circuit analysis, the impedance of an ideal capacitor with no initial charge is represented in the s domain by:.

Capacitors deviate from the ideal capacitor equation in a number of ways. Some of these, such as leakage current and parasitic effects are linear, or can be analyzed as nearly linear, and can be dealt with by adding virtual components to the equivalent circuit of an ideal capacitor. The usual methods of network analysis can then be applied. There is yet another group, which may be linear but invalidate the assumption in the analysis that capacitance is a constant. Such an example is temperature dependence.

Finally, combined parasitic effects such as inherent inductance, resistance, or dielectric losses can exhibit non-uniform behavior at variable frequencies of operation. Above a particular electric field, known as the dielectric strength E ds , the dielectric in a capacitor becomes conductive.

The voltage at which this occurs is called the breakdown voltage of the device, and is given by the product of the dielectric strength and the separation between the conductors, [29]. The maximum energy that can be stored safely in a capacitor is limited by the breakdown voltage.

Due to the scaling of capacitance and breakdown voltage with dielectric thickness, all capacitors made with a particular dielectric have approximately equal maximum energy density , to the extent that the dielectric dominates their volume. As the voltage increases, the dielectric must be thicker, making high-voltage capacitors larger per capacitance than those rated for lower voltages.

The breakdown voltage is critically affected by factors such as the geometry of the capacitor conductive parts; sharp edges or points increase the electric field strength at that point and can lead to a local breakdown. Once this starts to happen, the breakdown quickly tracks through the dielectric until it reaches the opposite plate, leaving carbon behind and causing a short or relatively low resistance circuit. The results can be explosive, as the short in the capacitor draws current from the surrounding circuitry and dissipates the energy.

It happens because a metal melts or evaporates in a breakdown vicinity, isolating it from the rest of the capacitor. The usual breakdown route is that the field strength becomes large enough to pull electrons in the dielectric from their atoms thus causing conduction.

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Other scenarios are possible, such as impurities in the dielectric, and, if the dielectric is of a crystalline nature, imperfections in the crystal structure can result in an avalanche breakdown as seen in semi-conductor devices. Breakdown voltage is also affected by pressure, humidity and temperature. An ideal capacitor only stores and releases electrical energy, without dissipating any. In reality, all capacitors have imperfections within the capacitor's material that create resistance. This is specified as the equivalent series resistance or ESR of a component. This adds a real component to the impedance:.

As frequency approaches infinity, the capacitive impedance or reactance approaches zero and the ESR becomes significant. This is usually significant only at relatively high frequencies. As inductive reactance is positive and increases with frequency, above a certain frequency capacitance is canceled by inductance.

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High-frequency engineering involves accounting for the inductance of all connections and components. If the conductors are separated by a material with a small conductivity rather than a perfect dielectric, then a small leakage current flows directly between them. The capacitor therefore has a finite parallel resistance, [38] and slowly discharges over time time may vary greatly depending on the capacitor material and quality.

The quality factor or Q of a capacitor is the ratio of its reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the capacitor, the closer it approaches the behavior of an ideal capacitor. Ripple current is the AC component of an applied source often a switched-mode power supply whose frequency may be constant or varying. Ripple current causes heat to be generated within the capacitor due to the dielectric losses caused by the changing field strength together with the current flow across the slightly resistive supply lines or the electrolyte in the capacitor.

The equivalent series resistance ESR is the amount of internal series resistance one would add to a perfect capacitor to model this. Some types of capacitors , primarily tantalum and aluminum electrolytic capacitors , as well as some film capacitors have a specified rating value for maximum ripple current. The capacitance of certain capacitors decreases as the component ages. In ceramic capacitors , this is caused by degradation of the dielectric. The type of dielectric, ambient operating and storage temperatures are the most significant aging factors, while the operating voltage usually has a smaller effect, i.

The aging process may be reversed by heating the component above the Curie point. Aging is fastest near the beginning of life of the component, and the device stabilizes over time.

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In contrast with ceramic capacitors, this occurs towards the end of life of the component. It can usually be taken as a broadly linear function but can be noticeably non-linear at the temperature extremes.

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The temperature coefficient can be either positive or negative, sometimes even amongst different samples of the same type. In other words, the spread in the range of temperature coefficients can encompass zero. Capacitors, especially ceramic capacitors, and older designs such as paper capacitors, can absorb sound waves resulting in a microphonic effect. Vibration moves the plates, causing the capacitance to vary, in turn inducing AC current.

Some dielectrics also generate piezoelectricity. The resulting interference is especially problematic in audio applications, potentially causing feedback or unintended recording. In the reverse microphonic effect, the varying electric field between the capacitor plates exerts a physical force, moving them as a speaker. This can generate audible sound, but drains energy and stresses the dielectric and the electrolyte, if any.

Current reversal occurs when the current changes direction.

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Voltage reversal is the change of polarity in a circuit. Reversal is generally described as the percentage of the maximum rated voltage that reverses polarity. In DC circuits and pulsed circuits, current and voltage reversal are affected by the damping of the system. Voltage reversal is encountered in RLC circuits that are underdamped.

The current and voltage reverse direction, forming a harmonic oscillator between the inductance and capacitance. The current and voltage tends to oscillate and may reverse direction several times, with each peak being lower than the previous, until the system reaches an equilibrium.