(RLC circuit) The circuit in the figure consists of a resistor (R ohms), an inductor (L henrys), a capacitor (C farads), and an initial voltage source. Let h = R/(2L), and suppose R, L, and C have been selected so that b also equals 1//LC. (This is done, for instance, when the circuit is used in a voltmeter.) Let v(t) be the voltage (in volts) at time t, measured across the capacitor. It can be shown that v is in the null space H of the linear transformation that maps v(t) into Lv"(t) + Rv'(t) + (1/C)v(t), and H consists of all functions of the form v(t) = e¬b' (c1 + c2t). Find a basis for H. Voltage source
(RLC circuit) The circuit in the figure consists of a resistor (R ohms), an inductor (L henrys), a capacitor (C farads), and an initial voltage source. Let h = R/(2L), and suppose R, L, and C have been selected so that b also equals 1//LC. (This is done, for instance, when the circuit is used in a voltmeter.) Let v(t) be the voltage (in volts) at time t, measured across the capacitor. It can be shown that v is in the null space H of the linear transformation that maps v(t) into Lv"(t) + Rv'(t) + (1/C)v(t), and H consists of all functions of the form v(t) = e¬b' (c1 + c2t). Find a basis for H. Voltage source
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(RLC circuit) The circuit in the figure consists of a resistor (R ohms), an inductor (L henrys), a capacitor (C farads), and an
initial voltage source. Let h = R/(2L), and suppose R, L, and C have been selected so that b also equals
1//LC. (This is done, for instance, when the circuit is used in a voltmeter.) Let v(t) be the voltage (in volts) at time t,
measured across the capacitor. It can be shown that v is in the null space H of the linear transformation that maps
v(t) into Lv"(t) + Rv'(t) + (1/C)v(t), and H consists of all functions of the form
v(t) = e¬b' (c1 + c2t). Find a basis for H.
Voltage
source
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