2) The circuit shown at right has an AC generator of peak EMF max and frequency

Question

2) The circuit shown at right has an AC generator of peak EMF max and frequency w connected in series with a resistor R and a capacitor C. a) Use Kirchoff's loop rule to write an equation using dQ/dt, Q,R, and C. b) Given that we have an ac source so E = Emax cos (wt), and knowing that the phase of Q may be different than the phase of E, try Q(t) = Q1 cos(wt) + Q2 sin(wt), as a solution to get an algebraic equation from the differential equation in part a). c) Equate the coefficients of the sin(wt) function and the coefficients of the cos(wt) function to find two linear equations for your unknowns Qi and Q2. d) Using the total Q(t) you can write the function Vcapacitor(t) and show that the magnitude of this voltage Is when cu Is small and negligible when w is large. e) A common application of this circuit is in audio amplifiers, which process music signals containing many different frequencies. In this application, the generator EMF is produced by something like a CD player, or an old-fashioned record player. This input signal Vm is sent through the resistor and capacitor, and Q p an output signal is produced by connecting an output device (like a speaker) in parallel across C. Suppose C our music input comes from an old record-player, and is subject to hiss-and-pop scratch noise. These distortions of the music are very short-lived, and so are of very high frequency. Thus, we would like our amplifier to remove signals of very high frequency. This is called a low-pass filter (or ''scratch filter'') when the input signal is of low frequency, the input signal V is passed through to V01 without much change ... but high-frequency Input signals are blocked.

Explanation / Answer

a)

according to kvl law write the equation

E=IR+(integration of current over time interval)/C

where I is current flowing through circuit

Q=CV

rewrite the equation involving charge Q

E=IR+Q/C

b)

E=Emaxcos(wt)

Q=Q1cos(wt)+Q2sin(wt)

substitute these equations in part a equation involving E,Q,I,

c)

Emaxcos(wt)=IR+(Q1cos(wt)+Q2sin(wt))/C

equating coefficients of cos (wt)

Emax=Q1/C

IR+Q2sin(wt)/C =0

d)

E(t)=IR+V(t)

e)

main application of RC circuit is we is in low pass circuits

we have good advantage of this circuit

it allows only low frequency signals and eliminates high frequency components

even though spped of response is less

it eliminates distortion in the signals like audio speaker

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