Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 7: Q4MP (page 387)
The diagram shows a 鈥渞elaxation鈥� oscillator. The chargeqon the capacitor builds up until the neon tube fires and discharges the capacitor (we assume instantaneously). Then the cycle repeats itself over and over.
(a) The charge q on the capacitor satisfies the differential equation
, here R is the Resistance, C is the capacitance and Vis the
Constant d-c voltage, as shown in the diagram. Show that if q=0 when
t=0 then at any later time t (during one cycle, that is, before the neon
Tube fires),
(b) Suppose the neon tube fires at. Sketch q as a function of t for
several cycles.
(b) Expand the periodic q in part (b) in an appropriate Fourier series.
Short Answer
Answer
Use the initial condition q-0 when t=0 ,to prove that
.
The sketch of q as a function of t is as shown in figure below.
The periodic q expanded in Fourier series is
.
Step by step solution
Given information
The given diagram of a 鈥渞elaxation鈥� oscillator is shown below.
The differential equation for the chargeis
. It is to be proved that
if
,when
.
The sketch ofas a function of
is to be drawn if the neon tube is fired at
and the periodic
is to be expanded in Fourier series.
Write the complex series formulas
The following are the formulas for the complex Fourier series.
Here, the period ofis
and the frequencies of the terms in the series are
.
Prove for q
The given differential equation is.
Convert the equation in separable form.
Integrate both the sides as shown below.
Here,is the integral constant.
Use the initial value conditionwhen
.
Therefore,.
Sketch q
The sketch of q as a function of t, if the neon tube is fired at, is shown in figure below.
Expand q in Fourier series
Evaluate coefficients for the series.
Ifthen
.
Substitute the value of u and du to calculate.
Here,
And
Substitute the values ofand
.
Solve further for.
Thus, the function is.
Over 30 million students worldwide already upgrade their learning with 91影视!
