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This chapter is from the book

This chapter is from the book

3.5 Impedance of an Ideal Capacitor in the Time Domain

In an ideal capacitor, there is a relationship between the charge stored between the two leads and the voltage across the leads. The capacitance of an ideal capacitor is defined as:

Equation 3-4

03equ04.gif


where:

  • C = the capacitance, in Farads

  • V = the voltage across the leads, in volts

  • Q = the charge stored between the leads, in Coulombs

The value of the capacitance of a capacitor describes its capacity to store charge at the expense of voltage. A large capacitance means the ability to store a lot of charge at a low voltage across the terminals.

The impedance of a capacitor can only be calculated based on the current through it and the voltage across its terminals. In order to relate the voltage across the terminals with the current through it, we need to know how the current flows through a capacitor. A real capacitor is made from two conductors separated by a dielectric. How does current get from one conductor to the other, when it has an insulating dielectric between them? This is a fundamental question and will pop up over and over in signal integrity applications.

The answer is that real current probably doesn't really flow through a capacitor, it just acts as though it does when the voltage across the capacitor changes. Suppose the voltage across a capacitor were to increase. This means that some positive charge had to be added to the top conductor and some negative charge had to be added to the bottom conductor. Adding negative charge to the bottom conductor is the same as pushing positive charge out; it is as though positive charges were added to the top terminal and positive charges were pushed out of the bottom terminal. This is illustrated in Figure 3-5. The capacitor behaves as though current flows through it, but only when the voltage across it changes.

03fig05.jpgFigure 3-5. Increasing the voltage across a capacitor adds positive charge to one conductor and negative charge to the other. Adding negative charge to one conductor is the same as taking positive charges from it. It looks like positive charge enters one terminal and comes out of the other.

By taking derivatives of both sides of the previous equation, a new definition of the I-V behavior of a capacitor can be developed:

Equation 3-5

03equ05.gif


where:

  • I = the current through the capacitor

  • Q = the charge on one conductor of the capacitor

  • C = the capacitance of the capacitor

  • V = the voltage across the capacitor

This relationship points out, as we saw previously, that the only way current flows through a capacitor is when the voltage across it changes. If the voltage is constant, the current through a capacitor is zero. We also saw that for a resistor, the current through it doubled if the current doubled. However, in the case of a capacitor, the current through it doubles if the rate of change of the voltage across it doubles.

This definition is consistent with our intuition. If the voltage changes rapidly, the current through a capacitor is large. If the voltage is nearly constant, the current through a capacitor is near zero. Using this relationship, we can calculate the impedance of an ideal capacitor in the time domain:

Equation 3-6

03equ06.gif


where:

  • V = the voltage across the capacitor

  • C = the capacitance of the capacitor

  • I = the current through the capacitor

This is a complicated expression. It says that the impedance of a capacitor depends on the precise shape of the voltage waveform across it. If the slope of the waveform is large (i.e., if the voltage changes very fast), the current through it is high and the impedance is small. It also says that a large capacitor will have a lower impedance than a small capacitor for the same rate of change of the voltage signal.

However, the precise value of the impedance of a capacitor is more complicated. It is hard to generalize what the impedance of a capacitor is other than it depends on the shape of the voltage waveform. The impedance of a capacitor is not an easy term to use in the time domain.

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