The RC Circuit: An Approach with Fourier Transforms In this article PDF

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The RC Circuit: An Approach with Fourier Transforms In this article

equations that come up in network analysis. The use of FT comes smoothly from first principles – precisely what we set out to demonstrate here. We start with

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CLASSROOM

Mitrajyoti Ghosh

83, Mitrapara 2nd Lane, Harinavi,

Kolkata 700148, West Bengal,

India.

Email:

mijospeakingnow@gmail.com The RC Circuit: An Approach with Fourier Transforms In this article we shall mathematically analyse the Resistor-

Capacitor (RC) circuit with the help of Fourier transforms(FT). This very general technique gives us a lot of insight into

solvingfirst order differential equations with source terms de- pending on time. In itself, the RC circuit is by far the most commonplace entity in modern electronics. But the method of FT is not the accepted custom for an electronic engineer, who

isprobablymorecomfortableworkingwithcompleximpedan-ces and phasors while solving problems in network analysis.

In fact, what is used much more extensively is the Laplace transform. Butalotofthings, (includingthecompleximpedance itself, and some insight into complex analysis) can be under- stood better if we use the FT approach to solve the differential

equations that come up in network analysis. The use of FTcomes smoothly fromfirst principles - precisely what we set

out to demonstrate here. We start with the circuit shown inFigure1, with the initial con- ditions that att=0 , chargeqacross the capacitor is 0, and the currenti= dq dt =0. We shall also impose the very important condition thatV in 0?t<

0The importance of this condition shall be clear in due course as

we look at the methods employed to solve this differential equa- tion for different forms of the input voltage applied.

1. The RC Circuit and its Differential Equation

For the circuit shown inFigure1, the differential equation for

Keywords

Fourier transforms, contour

integration, circuit theory. chargeqon the capacitor is given by, dq dt qRC V in R .(1)

RESONANCE|November 20161029

CLASSROOM

Figure 1.The RC Cir-

cuit with time dependent in- put voltageV in Note that this is a problem involving aninhomogeneous differ- ential equation with homogeneous boundary conditions.

Let ˜qbe the FT ofq.

Then we have,

q= 1 2π

˜qe

iωt dω,(2)

˜q=

1 2π qe -iωt dt.(3) Thus, dq dt 1 2π

˜qiωe

iωt dω.(4)

Andalso,wehavetheFTofV

in and the inverse transform, V in 1 2π V in e -iωt dt,(5) V in 1 2π V in e iωt dω.(6)

Solvingqfor different

forms ofV in (t) can reveal many aspects of RC circuits and aid learning some subteleties of contour integration in physics along the way. Using (2-6) and substituting them back into (1) wefinally get, 1 2π

˜qiω+

˜q RC V in R e iωt dω=0,(7)

Hence,

˜qiω+

˜q RC V in R =0,(8)

1030RESONANCE|November 2016

CLASSROOM

Where (8) is basically an analog ofKirchhoff's Law in the fre- quency domain, which gives us,

˜q=

V in C

1+iωRC

.(9) Therefore, by inverting the FT, the charge on the capacitorqcan be written as, q= 1 2π V in C

1+iωRC

e iωt dω.(10) In the upcoming sections, we shall solveqfor different forms of V in (t) and thereby explore the RC circuit in greater detail. This will reveal certain aspects of RC circuits such as their functioning as differentiators, integrators andfilters.

2. Some Simple Forms ofV

in to Illustrate Contour

Integration

2.1The Voltage Spike

Suppose there is a voltage spike att=t

0 Mathematically, we can model this using the Dirac delta function as, V in =A 0

δ(t-t

0 ),(11) whereA 0 has suitable dimensions. Replicating the strategy as shown in the previous section, we should obtain: V in 1 2π A 0 e -iωt 0 ,(12) q= A 0 C 2π e iω(t-t 0 dω

1+iωRC

A 0

2πiR

e iω(t-t 0 dω -i/RC .(13) This integral is clearly a complex integral and we need to move over to the complexω-plane to solve it. However,ωis entirely real. Therefore we can best do by using the semi-circular contour (of radiusRas shown below), for integration.

RESONANCE|November 20161031

CLASSROOM

Figure 2.Closing the con-

tour in the upper half plane fort>t 0

Jordan's Lemmaforfinding integrals of the form

f(z)e ikz dz says that fork>0,the contour must be closed in the upper half plane (Figure2) for the integration around the semicircular part of the contour to vanish asR→∞. This ensures that the only quantity we shall be left with, whenR→∞is the integral along the realωline from-∞to∞. Fork<0, the contour should be closed in the lower half plane for the integral on the semicircular contour to vanish.

In this casek≡(t-t

Fort>t

quotesdbs_dbs8.pdfusesText_14

[PDF] The RC Circuit: An Approach with Fourier Transforms In this article

CLASSROOM

Mitrajyoti Ghosh

83, Mitrapara 2nd Lane, Harinavi,

Kolkata 700148, West Bengal,

India.

Email:

0The importance of this condition shall be clear in due course as

1. The RC Circuit and its Differential Equation

Keywords

Fourier transforms, contour

RESONANCE|November 20161029

CLASSROOM

Figure 1.The RC Cir-

Let ˜qbe the FT ofq.

Then we have,

˜qe

˜q=

˜qiωe

Andalso,wehavetheFTofV

Solvingqfor different

˜qiω+

Hence,

˜qiω+

1030RESONANCE|November 2016

CLASSROOM

˜q=

1+iωRC

1+iωRC

2. Some Simple Forms ofV

Integration

2.1The Voltage Spike

Suppose there is a voltage spike att=t

δ(t-t

1+iωRC

2πiR

RESONANCE|November 20161031

CLASSROOM

Figure 2.Closing the con-

Jordan's Lemmaforfinding integrals of the form

In this casek≡(t-t

Fort>t