Capacitance and RC Circuits Figure: 03-00CO Caption: Circuit and waveforms for Example 3.1.

Capacitors are constructed by separating 2 sheets of conductor, which is usually metallic, by a thin layer of insulating material. In parallel-plate capacitor, the sheets are flat and parallel. The insulating material between the plates, called a dielectric, can be air, Mylar, polyester, polypropylene, mica, etc. Figure: 03-01 Caption: A parallel-plate capacitor consists of two conductive plates separated by a dielectric layer.

Figure: 03-02ab Caption: A capacitor and its fluid-flow analogy.

Stored charge in terms of voltage The constant of proportionality is the capacitance C, which has units of farads (F). Farads are equivalent to coulombs per volt. Current in terms of voltage Note: the current reference direction points into the positive reference polarity.

Voltage in terms of current Suppose that we know the current i(t) flowing through a capacitance C and we want to compute charge and voltage. Figure: 03-03 Caption: The circuit symbol for capacitance, including references for the current i(t) and voltage v(t).

Stored Energy Energy stored in the capacitance is given by

Figure: 03-04a-dEXM Caption: Circuit and waveforms for Example 3.1.

Figure: 03-09 Caption: Three capacitances in parallel.

Figure: 03-10 Caption: Three capacitances in series.

Capacitance of the parallel-plate capacitor If the distance d between the plates is much smaller than both the width and the length of the plates, the capacitance is approximately In which e is the dielectric constant of the material between the plates. For vacuum, the dielectric constant is e = e0 = 8.85x10-12 F/m Figure: 03-11 Caption: A parallel-plate capacitor, including dimensions.

Figure: 03-12 Caption: Practical capacitors can be constructed by interleaving the plates with two dielectric layers and rolling them up. By staggering the plates, connection can be made to one plate at each end of the roll.

Figure: 03-13 Caption: The circuit model for a capacitor, including the parasitic elements R_s, L_s, and R_p.

Figure: 03-14EXM Caption: See Example 3.5.

An inductor is constructed by coiling a wire around some type of form. Inductance An inductor is constructed by coiling a wire around some type of form. Current flowing through the coil creates a magnetic field or flux that links the coil. Frequently the coil form is composed of a magnetic material such as iron or iron oxide that increases the magnetic flux for a given current. Figure: 03-15a-c Caption: An inductor is constructed by coiling a wire around some type of form.

When the current changes in value, the resulting magnetic flux changes according to Faraday’s law of electromagnetic induction, time-varying magnetic flux linking a coil induces voltage across the coil. For an ideal inductor, the voltage is proportional to the time rate of change of the current. Furthermore, the polarity of the voltage is such as to oppose the change in current. The constant of proportionality is called inductance, L.

Current in terms of voltage Figure: 03-16 Caption: Circuit symbol and the v-i relationship for inductance. Stored Energy

Figure: 03-17a-dEXM Caption: Waveforms for Example 3.6.

Figure: 03-18a-cEXM Caption: Circuit and waveforms for Example 3.7.

Figure: 03-19abEXM Caption: See Exercise 3.7.

Figure: 03-20ab Caption: Inductances in series and parallel are combined in the same manner as resistances.

Figure: 03-21abEXM Caption: See Exercise 3.10.

Figure: 03-22 Caption: Circuit model for real inductors including several parasitic elements.

Figure: 03-22-01PA3-1 Caption:

Figure: 03-22-02PA3-2 Caption:

Figure: 03-23ab Caption: Circuit symbols and v-i relationships for mutually coupled inductances.

Figure: 03-24 Caption: A linear variable differential transformer used as a position transducer.

Figure: 03-24-01P3-11ab Caption:

Figure: 03-24-02P3-12 Caption:

Figure: 03-24-03P3-14ab Caption:

Figure: 03-24-04P3-24ab Caption:

Figure: 03-24-05P3-25ab Caption:

Figure: 03-24-06P3-28 Caption:

Figure: 03-24-07P3-35 Caption:

Figure: 03-24-08P3-36 Caption:

Figure: 03-24-09P3-39 Caption:

Figure: 03-24-10P3-45 Caption:

Figure: 03-24-11P3-48 Caption:

Figure: 03-24-12P3-51 Caption:

Figure: 03-24-13P3-62ab Caption:

Figure: 03-24-14P3-63ab Caption:

Figure: 03-24-15P3-67 Caption:

Figure: 03-24-16P3-68 Caption:

Figure: 03-24-17P3-71 Caption:

Figure: 03-24-18P3-72 Caption:

Figure: 03-24-19P3-74 Caption:

Figure: 03-24-20P3-76 Caption: