Where is the RLC circuit time constant"2L/R" coming from?

In summary, the time constant for reaching the steady state here is undefined and it depends on the circuit arrangement.
  • #1
garylau
70
3
Sorry
i want to ask a question about RLC circuit~
i don't understand where is this "2L/R"coming from
and also i don't know what is the function of this" time constant"(it is meaningless in physical sens?e)

in the note it states"that the time constant for reaching the steady state here..."

how did he knows this is the time constant that lead to the steady state?

thank you​
 

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  • #2
This is a resonant circuit consisting of C and L together with some losses represented by R.
The oscillations gradually diminish, in an exponential manner, similar to the discharge of a capacitor. Eventually the amplitude falls almost to zero. The time constant is the period of time after which the amplitude has fallen to 1/e.
 
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  • #3
Unfortunately, you didn`t show us neither the circuit diagram nor the corresponding transfer function.
Nevertheless, from the transient response one can deduce how the circuit will look like.
It is a simple RLC- lowpass.

The transfer function is H(s)=(1/sC)/(R+sL+1/sC)=1/(1+sRC+s²LC)
The poles of this function can be found by settimg the denominator to zero: 1+sRC+s²LC=0.
Result (two real poles):
s1,2=-R/2L(+-)SQRT(R²/4L²-1/LC).

Knowing that
(1) the denominator of the transfer function (frequency domain) is identical to the characteristic polynominal (time domain) and
(2) that, therefore, the the real part of the pole (-R/2L) appears in the solution of the differential equation as exp(-R/2L)t,

we can derive that the damping coefficient for the decaying oscillation is σ=R/2L.
If we compare this result with the general expression exp(-t/τ) we get τ=2L/R.
 
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  • #4
LvW said:
Unfortunately, you didn`t show us neither the circuit diagram nor the corresponding transfer function.
Nevertheless, from the transient response one can deduce how the circuit will look like.
It is a simple RLC- lowpass.

The transfer function is H(s)=(1/sC)/(R+sL+1/sC)=1/(1+sRC+s²LC)
The poles of this function can be found by settimg the denominator to zero: 1+sRC+s²LC=0.
Result (two real poles):
s1,2=-R/2L(+-)SQRT(R²/4L²-1/LC).

Knowing that
(1) the denominator of the transfer function (frequency domain) is identical to the characteristic polynominal (time domain) and
(2) that, therefore, the the real part of the pole (-R/2L) appears in the solution of the differential equation as exp(-R/2L)t,

we can derive that the damping coefficient for the decaying oscillation is σ=R/2L.
If we compare this result with the general expression exp(-t/τ) we get τ=2L/R.
Oh Sorry
 

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  • #5
Also for your circuit arrangement, the result of my calculation example still applies.
The reason is as follows: The current through the whole circuit (your task) has the the same form (in the time domain) as the voltage across the capacitor (my example). That means, the damping properties are identical - and hence the time constant.
 
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  • #6
LvW said:
Also for your circuit arrangement, the result of my calculation example still applies.
The reason is as follows: The current through the whole circuit (your task) has the the same form (in the time domain) as the voltage across the capacitor (my example). That means, the damping properties are identical - and hence the time constant.
Oh i See

but when i try to put "2L/R" into the equation
i didn't see anything special?
there is no well-defined "amplitude" in this case?
because there are two terms
De^(Rt/2L) * Sin(omega*t)

if i try to put t=2L/R then it will become (63.4%)*D*sin(omega* 2L/R)

it seems (63.4%)* D = Amplitude of that D which the power is dissipated 50%?
 

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  • #7
tech99 said:
This is a resonant circuit consisting of C and L together with some losses represented by R.
The oscillations gradually diminish, in an exponential manner, similar to the discharge of a capacitor. Eventually the amplitude falls almost to zero. The time constant is the period of time after which the amplitude has fallen to 1/e.
However i got confused here
there is no well-defined definition of amplitude in this case as the sin function terms never allow the function reaches "the new amplitude" value?
 
  • #8
garylau - what now is your problem? Please, try a clear question or problem description.
 
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  • #9
LvW said:
garylau - what now is your problem? Please, try a clear question or problem description.

Does RLC circuit has well-defined time constant?
In the case of RL and Rc circuit,it is clearly that there is time constant as there is only "e" terms in the equation.
but in the case of RLC circuit,the whole equation include both "e" and "sin" terms and it behaves strange.
the sin terms never allow the the function reaches its "new Amplitude" value except time=0
So it looks different from the case of LC and RC cases,is it meaningless to understand what is "time constant in RLC circuit"?
from a website
somebody said "I think that in a RLC circuit one doesn't define time constant. There has two expressions with dimensional unit: time, but that is not time constant.
T=1/((-R/2L)±SQRT((R/2L)^2-1/(LC)))"

http://forum.allaboutcircuits.com/threads/time-constant-for-rlc-circuit.877/
 
  • #10
garylau said:
Does RLC circuit has well-defined time constant?
In the case of RL and Rc circuit,it is clearly that there is time constant as there is only "e" terms in the equation.
but in the case of RLC circuit,the whole equation include both "e" and "sin" terms and it behaves strange.
the sin terms never allow the the function reaches its "new Amplitude" value except time=0
So it looks different from the case of LC and RC cases,is it meaningless to understand what is "time constant in RLC circuit"?
from a website
somebody said "I think that in a RLC circuit one doesn't define time constant. There has two expressions with dimensional unit: time, but that is not time constant.
T=1/((-R/2L)±SQRT((R/2L)^2-1/(LC)))"

http://forum.allaboutcircuits.com/threads/time-constant-for-rlc-circuit.877/
RC and RL circuits are called 'first order circuits' and their behaviour is analysed using a defined 'time constant'.
An RLC circuit is a second order circuit and its behaviour can be analysed using parameters like rise time, peak time, delay time, damping factor, natural frequency of oscillation etc. (look up step response of a second order circuit). I have not seen the concept of time constant being used directly in case of 2nd order circuits.
images (2).jpg
 
Last edited:
  • #11
cnh1995 said:
I have not seen the concept of time constant being used directly in case of 2nd order circuits.
Cnh1995 - didn`t you read my contribution in post#3 ?
In a 2nd-order sysytem the time constant defines the slope of the envelope of the decaying oscillation.
Hence, it has a direct relation to the settling time.
 

1. Where does the formula "2L/R" for the time constant of an RLC circuit come from?

The formula for the time constant of an RLC circuit, "2L/R," comes from the mathematical analysis of the circuit's behavior using Kirchhoff's laws and Ohm's law. It is derived from the differential equation that describes the voltage across the capacitor in an RLC circuit.

2. Why is the time constant of an RLC circuit important?

The time constant of an RLC circuit is important because it represents the rate at which the energy stored in the circuit's components (resistor, inductor, and capacitor) decays. It is a critical parameter in determining the response of the circuit to changes in input voltage or current.

3. How does the value of the inductor and resistor affect the time constant of an RLC circuit?

The value of the inductor and resistor directly affect the time constant of an RLC circuit. A larger inductance or resistance will result in a longer time constant, meaning it will take longer for the energy in the circuit to decay. Conversely, a smaller inductance or resistance will result in a shorter time constant.

4. Can the time constant of an RLC circuit be changed?

Yes, the time constant of an RLC circuit can be changed by adjusting the values of the components in the circuit. In particular, changing the inductance or resistance will have a direct impact on the time constant. Additionally, the time constant can also be changed by changing the input voltage or current to the circuit.

5. Are there any other factors that can affect the time constant of an RLC circuit?

In addition to the values of the components and the input voltage or current, there are other factors that can affect the time constant of an RLC circuit. These include temperature, the presence of nearby magnetic fields, and the quality of the components used. These factors can introduce additional resistance or inductance, which can impact the overall time constant of the circuit.

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