Parallel resonance LC circuit question?

[uzair_ha91]

"At resonance frequency, the circuit impedance is maximum. It is resistive and its value is given L/Cr..."
The book doesn't explain how impedance equals L/Cr, so I'm confused here. How is it derived?

 

[uzair_ha91]

I googled it and found the closest result here, but again not explained where the equation was derived from: http://www.hobbyprojects.com/phasors_and_resonance/parallel_resonance_in_AC_circuits.html

 

[Bob S]

Here are the basic equations for a parallel L and C, and no resistance. (w = 2 pi f)

1/Z = 1/jwL + jwC = jwC - j/wL = j(w²LC-1)/wL

So Z = jwL/(1-w²LC)

When LC=1/w², Z= infinity

[Edit] If you also have a parallel resistor such that

1/Z = 1/jwL + jwC +1/R. then

Z =~ R at resonance.

 

[uzair_ha91]

But what about Z=L/C*R where R is the resistance of the coil if we don't neglect it?

 

[uzair_ha91]

How do we get that?

 

[Bob S]

But what about Z=L/C*R where R is the resistance of the coil if we don't neglect it?

Exactly right.
Using the same equations I posted earlier, , and the inductance has a small series resistance R, then

1/Z = 1/(jwL + R) + jwC

et cetera (like in prev. post), leading to (at LC resonance)

Z= (jwL + R)/jwCR.

If R<< jwL, then Z = jwL/jwCR = L/CR

(note: L/C has units ohms². so L/CR has units ohms)

 

[uzair_ha91]

:-) Thanks!

 

[uzair_ha91]

uh, can you tell me why/how j²=-1?

Actually I'm studying high school physics so I'm studying the equations without the "j" operator or whatever it is..

 

[negitron]

By definition, j = sqrt(-1). So, obviously, j² = -1 since that's the inverse operation. It's just the imaginary number; in mathematics sqrt(-1) = i. EEs like to use j because i already means current.

 

[Bob S]

uh, can you tell me why/how j²=-1?

Actually I'm studying high school physics so I'm studying the equations without the "j" operator or whatever it is..

Hi uzair
There is a real physical basis for j² = -1. As I showed in an earlier post, the time derivitaves of a circular function V(ωt) = V₀ sin(ωt) are

d V(ωt)/dt = ω V₀ cos(ωt) = jω V(ωt)

d² V(ωt)/dt² = -ω² V₀ sin(ωt) = (jω)² V(ωt) = j² ω² V(ωt) = -ω² V(ωt)

So you can see in red the shorthand notation for a 90 degree phase shift caused by taking the time derivative, and why j² = -1.
Bob S

 

[Schecter5150]

Sorry for the bump, but I'm working on the problem including the series R with L, and can't seem to figure out how to derive the resonance frequency w_o in that case. Wikipedia lists it as w_0 = sqrt((1/LC) - (R/L)^2), where the impedance is only real.

I have the equation Zin = (R + jwL)/(jwRC-w^2*LC+1), which I have rearranged as

Zin = ((L/C) - jR(w/C))/(R + j(wL - 1/wc))

When I split up the fractions in the rearranged form, I can easily see where the 1/LC term comes from in w_0, but I'm having difficulty seeing how the (R/L)^2 term is derived. Any advice would be appreciated.

 

[Bob S]

Use Z_in = -(R + jωL)/(1 + ω²LC -jωRC)

and multiply both numerator and denominator by 1 + ω²LC +jωRC

This will get the j out of the denominator. The denominator will be (1 + ω²LC)² + (ωRC)².

Bob S