Re: [Phys-L] weighting in the wings ... damped harmonic oscillator ... bandwidth ... algebra ... bug hunting

["Donald Polvani" <dgpolvani@verizon.net>]

On Thursday, November 24, 2016 12:53 PM, John Denker wrote:

> Different questions lead to different answers:
>  -- Equation [1] describes the current in the resistor, whereas
>  -- I was looking at the voltage across the capacitor.

I was looking at the current in the resistor and the voltage across the
resistor.  I've now had a chance to look at the voltages across all three
circuit elements R, L, and C.  I assume an AC source voltage:

E = E_m*cos(w*t + phi_v)

With common circuit current:

i = i_m*cos(w*t + phi_i) 

With i_m = E_m/Z and Z = the magnitude of the complex impedance = sqrt(R^2 +
(L*w - 1/(w*C))^2)

1) For the resistor (R):

The magnitude of the voltage across the resistor is:

V_mR = i_m*R = E_m*R/Z

R/Z = R/sqrt(R^2 + (L*w - 1/(w*C))^2) = 1/sqrt(1 + Q^2*(w/w_0 - w_0/w)^2) =
x/sqrt(x^2 + Q^2*(x^2 - 1)^2)

Where w_0 = resonance frequency = 1/sqrt(L*C), Q = w_0*L/R, and x = w/w_0 =
f/f_0

The frequency dependence of V_mR is given by:

G_R(x) = R/Z = x/sqrt(x^2 + Q^2*(x^2 - 1)^2)

The frequency dependence of V_mR^2 is given by:

G_R(x)^2 = x^2/(x^2 + Q^2*(x^2 - 1)^2)

The maximum value of G_R(x)^2 is 1, and occurs at x = x_mR = 1, so that the
maximum value occurs at resonance (w_0).

The FWHM is given by:

X_UR = (1 + sqrt(1 + 4*Q^2))/(2*Q)
X_LR = (-1 + sqrt(1 + 4*Q^2))/(2*Q)

FWHM = X_UR - x_LR = 1/Q

2) For the inductor (L)

The magnitude of the voltage across the inductor is:

V_mL = i_m*w*L = (E_m*w_0*L/R)*(w/w_0)*R/Z = x*G_R(x)

Therefore:

G_L(x)  = x^2/sqrt(x^2 + Q^2*(x^2 - 1)^2)

G_L(x)^2 = x^4/(x^2 + Q^2*(x^2 - 1)^2)

The maximum value of G_L(x)^2 is:

G_Lm^2 = 4*Q^2/(4*Q^2 - 1)

Which occurs at x_mL = sqrt(2)*Q/sqrt(2*Q^2 - 1)

Note G_Lm^2 is > 1, but approaches 1 as Q approaches infinity, and x_mL is >
1 but approaches 1 as Q approaches infinity. We also need Q > 1/2 to keep
G_Lm^2 > 0, but Q must be even larger than this as specified below for the
FWHM.

The FWHM is given by:

x_UL = sqrt(((2*Q^2 - 1) + sqrt(1 + 4*Q^2))/(2*(Q^2 - 2)))

x_LL =  sqrt(((2*Q^2 - 1)  - sqrt(1 + 4*Q^2))/(2*(Q^2 - 2)))

x_UL - x_LL = sqrt(((2*Q^2 - 1) + sqrt(1 + 4*Q^2))/(2*(Q^2 - 2))) -
sqrt(((2*Q^2 - 1) - sqrt(1 + 4*Q^2))/(2*(Q^2 - 2)))

To keep x_UL real, we require Q > sqrt(2).  For values of Q < sqrt(2),
G_L(x)^2 stays above 1/2 to the right of the maximum, so a FWHM doesn't
exist.

3) For the capacitor (C)

The magnitude of the voltage across the capacitor is:

V_mC = i_m/(w*C) = (E_m/(R*C*w_0))*(w_0/w)*R/Z = G_R(x)/x

Therefore:

G_C(x)  = 1/sqrt(x^2 + Q^2*(x^2 - 1)^2)

G_C(x)^2 = 1/(x^2 + Q^2*(x^2 -1)^2)

The maximum value of G_C(x)^2 is:

G_Cm^2 = 4*Q^2/(4*Q^2 - 1)

Which is the same as G_Lm^2 but occurs at x_mC = sqrt(4*Q^2 - 2)/(2*Q)

Note G_Cm^2 is > 1, but approaches 1 as Q approaches infinity, and x_mC is <
1 but approaches 1 as Q approaches infinity.

Again we require Q > 1/2 to keep G_Cm^2 > 0, but this requirement is again
superseded by the even larger value specified below for the FMHM.

The FWHM is given by:

x_UC = sqrt((2*Q^2 - 1) + sqrt(1 + 4*Q^2))/(sqrt(2)*Q)

x_LC =  sqrt((2*Q^2 - 1)  - sqrt(1 + 4*Q^2))/(sqrt(2)*Q)

x_UC - x_LC = sqrt((2*Q^2 - 1) + sqrt(1 + 4*Q^2))/(sqrt(2)*Q) - sqrt((2*Q^2
- 1) - sqrt(1 + 4*Q^2))/(sqrt(2)*Q)

To keep x_LC real, we again require Q > sqrt(2).  For values of Q < sqrt(2),
G_C(x)^2 stays above 1/2 to the left of the maximum, so a FWHM doesn't
exist.

I've computed and plotted these results and found agreement with the
theoretical predictions.

Don Polvani
Adjunct Faculty, Physics, Retired
Anne Arundel Community College
Arnold, MD 21012