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EE1000, Lecture 4 Capacitance, Inductance, Resonance Capacitance The symbol of a capacitor: Examples: Capacitors in parallel add. Q: What about capacitors in series? A: Variable capacitor (increase or decrease capacitance): overlapping plates Inductance The symbol of an inductor: Examples: The cylindrical shape in the middle is called a solenoid. Inductors in series add. Q: What about inductors in parallel? A: Variable Inductor (increase or decrease inductance): ferrite slug moves into and out of the coil. Capacitors A capacitor's energy exists in its surrounding electric fields. It is proportional to the square of the field strength, which is proportional to the charges on the plates. If we assume the plates carry charges that are the same in magnitude, +q and -q, then the energy stored in the capacitor must be proportional to q2. Let’s also assume that the voltage across the capacitor = V. For historical reasons, we write the constant of proportionality as ½ C, 1 q2 1 EC CV 2 …This gives us an important relationship: q CV 2C 2 The constant C is a geometrical property of the capacitor, called its capacitance: C r 0 A d , where εr is the relative permittivity of the dielectric and ε0 = 8.854x10-12 F/m. Inductors Any current will create a magnetic field, so in fact every current-carrying wire in a circuit acts as an inductor! All the loops' contribution to the magnetic field add together to make a stronger field. Unlike capacitors and resistors, practical inductors are easy to make by hand. One can for instance spool some wire around a short wooden dowel, put the spool inside a plastic aspirin bottle with the leads hanging out, and fill the bottle with epoxy to make the whole thing rugged. The energy density is proportional to the square of the magnetic field strength, which is in turn proportional to the current flowing through the coiled wire, so the energy stored in the inductor must be proportional to I2. EL 1 2 LI 2 The quantity L is called the inductance of the inductor: L 0 N 2 A l , Where, A=cross-sectional of the coil/solenoid, N =the number of turns (or loops of wire) in the coil/solenoid and µ0 = 4π×10-7 Henrys per meter= permeability of free space. Oscillations and RLC Circuit The simplest type of resonator (the device that resonates or oscillates) is an RLC circuit. A series RLC circuit: A parallel RLC circuit: Resonance effect The resonance effect occurs when inductive and capacitive reactances are equal in magnitude. The frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit is o 1 LC where L is the inductance in henries (H), and C is the capacitance in farads (F). The angular frequency has units of radians per second (rad/s). The resonant frequency in units of Hertz (Hz) is fo o 1 2 2 LC LC circuits are often used as filters; the L/C ratio determines their "Q" and so selectivity. For a series resonant circuit with a given resistance, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit, the opposite applies. At resonant frequency: series = short; parallel = open. The most common application of an RLC circuit is tuning radio transmitters and receivers. For example, when we tune a radio to a particular station, the RLC circuits are set at resonance for that particular carrier frequency. The Home-Made Inductor (Coil) and Home-made Variable Capacitor Inductor [HW on the last page…] Variable Capacitor ****************************** Homework Assignment ********************************** ************************************************************************************* EE1000 HW #4 The capacitance of a parallel plate capacitor with area A and distance d between the plates is C Name:________________________________ r 0 A d , where εr is the relative permittivity of the dielectric and ε0 = 8.854x10-12 Farads per meter. The inductance of an air-core solenoid of length l and cross sectional area A is given by L 0 N 2 A l , where N is the number of turns (or loops of wire) in the solenoid and µ0 = 4π×10-7 Henrys per meter is the permeability of free space. The resonant frequency is given by fo 1 2 LC [Hz]. Problems: All units must be converted to m, m^2, F, H, etc. and then modified later for the final answer. 1. Find the area (in square centimeters) of a 1.5x10-9 F parallel plate capacitor with a distance between the plates of 0.002” (=5.08x10^-5 m). Assume a relative permittivity, εr = 2.5. 2. Find the inductance of a 5 cm (centi meter) diameter solenoid that is 10 cm (centi meter) in length and has 156 turns. Assume an air core. 3. What is the resonant frequency of a 25 pF (pico Farads) capacitor connected to a 500 μH (micro Henry) inductor?