BG_4991_chapters_4_and_5

BG_4991_chapters_4_and_5 - Grossman/ Melkonian 1 Chapter 4...

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Unformatted text preview: Grossman/ Melkonian 1 Chapter 4 and 5 Time Varying Circuits Capacitors Inductors Time Constant T C Waveforms RC Circuits Step Input RL Circuits Step Input Grossman/ Melkonian 2 CAPACITORS: Where: C is the capacitance in Farads. The farad is a large unit. Typical values of capacitance are in F or pF. is the permittivity of the dielectric medium (8.854x10-12 F/m for air). A is the cross-sectional area of the two parallel conducting plates. d is the distance between the two plates. C = A d Dielectric Insulator = permittivity A d Metal plates of area A Section 4.1 Grossman/ Melkonian 3 + When a voltage is applied to the plates, an electric field is produced that causes an electric charge q(t) to be produced on each plate. +--Dielectric Insulator = permittivity Metal plates of area A Standard Notations (passive sign convention) +-+-v C v C V s i C (t) i C (t) CAPACITORS: Grossman/ Melkonian 4 since: i(t) = dq(t) / dt and: q(t) = Cv C (t) Differentiating: dq(t) = C dv C (t) i C (t) = dq(t) / dt = C(dv C (t) / dt ) q (t) = Cv C (t) i C (t) = C[dv C (t) / dt] CAPACITORS: Charge of a capacitor i-v relationship for a capacitor Grossman/ Melkonian 5 Integrating the i-v relationship of the capacitor: CAPACITORS: dv C (t) = 1 / C i C (t)dt = v C (t) t t Letting V C (0) = V represent the initial voltage across the capacitor at some time t = t : v C (t) = V + 1 / C i C (t)dt t 0 t t i C (t) = C[dv C (t) / dt] Voltage across a capacitor Current through a capacitor Grossman/ Melkonian 6 CAPACITORS: Capacitor Power: p C (t) = i C (t) v C (t) = C[dv C (t) / dt]v C (t) p C (t) = d / dt[Cv C 2 (t)] Power associated with a capacitor Capacitor Energy: Energy is the integral of power, therefore, w C (t) = Cv C 2 (t) Energy associated with a capacitor Grossman/ Melkonian 7 CAPACITORS: Example 1: Given i C (t) = I o [e-t/Tc ] for t 0 and v C (0) = 0V, f ind the capacitors power and energy. v c (t) = [(I o T c / C)(1-e-t/Tc )]V Power: p c (t) = i C (t) v C (t) v C (t) = V + 1 / C i C (x)dx = 0V + 1 / C I o [e-x/Tc ]dx t t p C (t) = [I o e-t/Tc ][(I o T C / C)(1-e-t/Tc )]W p c (t) = I o 2 T c / C (e-t/Tc-e-2t/Tc ) i C (t) v C (t) Power can be positive or negative Grossman/ Melkonian 8 CAPACITORS: Example 1 cont.: Energy: w C (t) = Cv C 2 (t) w C (t) = [(I o T C ) 2 / 2C](1-e-t/Tc ) 2 t I o i C (t) Waveforms: i C (t) = I o [e-t/Tc ] Energy is always positive Grossman/ Melkonian 9 t v C (t) I o T c / C CAPACITORS: Example 1 cont.: v c (t) = [(I o T c / C)(1-e-t/Tc )]V t p C (t) I o 2 T c / 4C T c ln2 p c (t) = I o 2 T c / C (e-t/Tc-e-2t/Tc ) Grossman/ Melkonian 10 CAPACITORS: Example 1 cont.: t w c (t) I o 2 T c 2 /2C w C (t) = [(I o T C ) 2 / 2C](1-e-t/Tc ) 2 Grossman/ Melkonian 11 CAPACITORS: Series and Parallel Capacitors: C 1 Series: Capacitors connected in series combine like resistors connected in parallel....
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This note was uploaded on 02/11/2012 for the course ECE 4991 taught by Professor Crandel during the Fall '07 term at FIT.

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BG_4991_chapters_4_and_5 - Grossman/ Melkonian 1 Chapter 4...

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