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## Capacitor and Inductor

Through out these notes, we use upper-case letters and to represent DC (direct current) current and voltage that are constants in time, and lower-case letters and (sometimes simply and ) to represent time-varying such as AC (alternating current) current and voltage.

Capacitor

Conceptually a capacitor is composed of a pair of plates of conductor which can be charged by a voltage source. The voltage between the two plates is proportional to the charge , but inversely proportional to the capacity of the capacitor:

The current through C is:

If the voltage is DC which does not change over time, then and the current through C is zero. If the voltage is AC which changes polarity over time, then and the current through C is not zero.

Inductor

• Electromagnetic Interaction: Electricity to Magnetism

Magnetic field (flux) is generated in the space around a current flowing through a piece of conductor:

The magnetic field around a coil is the superposition of the magnetic flux generated by each section of the coil:

• Electromagnetic Interaction: Magnetism to Electricity

Electric current is induced in a conductor when there is changing magnetic flux in the surrounding space.

• Self and Mutual Induction

A time-varying electric current in a coil will cause a time-varying magnetic field in the surrounding space, which in turn will induce electric voltage and then current in the same coil (self-induction) or a different coil in the neighborhood (mutual-induction).

The self-induced voltage across the coil due to a current is proportional to the rate of change over time of the total magnetic flux caused by the current:

Here is the total magnetic flux in the coil, and is the inductance of the coil that represents how much flux can be generated by the coil per unit current.

• Lenz's Law:

The polarity of the self-induced voltage in a coil is such that it tends to produce a current which induces a magnetic flux to oppose the change of the magnetic field that induced the voltage, thereby opposing any change in current that is causing the magnetic flux.

When current increases, the induced voltage tends to resist it, when current decreases, the induced voltage tends to sustain it.

Ideal Transformer

The ratio between the primary voltage and the secondary voltage of a transformer is proportional to the ratio between the numbers of turns:

If there is no power loss by the transformer, then the transformer is ideal and we have

The ratio between the primary current and the secondary current of a transformer is inversely proportional to the ratio between the numbers of turns. Also note the reference directions of the currents and and the reference polarities of the voltages and , reflecting the fact that the secondary current is caused by the induced voltage (consistent polarity), while is the induced voltage opposing the current .

We can also find the ratio between the primary and secondary impedances based on the assumption that there is no power loss in the transformer, i.e.,

Next: Linear Elements Up: Chapter 1: Basic Quantities Previous: Basic Quantities
Ruye Wang 2012-07-02