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Differential Equation of Series RL Circuit

Hi!

Could you possibly give me a physical interpretation of the homogeneous
solution of the differential equation describing a series RL circuit
powered by a sinusoidal source?

The equation that describes the circuit is:

Vo*sin(wt)-L*d/dt(i(t)) = i(t)*R

The solution (using i(0)=0 as initial condition):

i(t)= transient_response + steady_state_response

(or homogeneous + (particular or inhomogeneous))

where

transient_response = [Vo * wL / (R^2+(wL)^2)] * exp(-R/L*t)
steady_state_response = Vo / (sqrt(R^2+(wL)^2)) * sin(w*t-arctan(wL/R))

using also:

a * sin(wt) + b * cos (wt) = 1/sqrt(a*a+b*b) * sin(wt + arctan (b/a))

What is the physical interpretation of the transient response?

Thank you in advance,

Hugo.
 
N

Noway2

Jan 1, 1970
0
Could you possibly give me a physical interpretation of the homogeneous
solution of the differential equation describing a series RL circuit
powered by a sinusoidal source?
transient_response = [Vo * wL / (R^2+(wL)^2)] * exp(-R/L*t)
steady_state_response = Vo / (sqrt(R^2+(wL)^2)) * sin(w*t-arctan(wL/R))

What is the physical interpretation of the transient response?
One of the properties of any LTI system, such as the RL circuit you
describe, is that the application of a sinusoidal source will give a
sinusoidal result that is altered in magnitude and phase. This portion
is apparent in your steady state response where the output is a
sinusoid of magnitude and phase determined by the system paramaters
(W,L, and R).

The transient response, in your example, is an applied voltage, that is
super imposed (added to) the steady state response, but its effects
decay with time, exponentially according to the time constant (-R/L*t).
From a physical perspective, the transient response represents the
initial charge up or inrush into the storage elements, or the
transition from the devices from the at rest condition. A simpler
example, is to think of the step response of the system. If you apply
a constant voltage, at t0, the current will exponentially ramp up to
the steady state of V/R. The relative amounts of inductance and
resistance will determine the rate (time constant) at which the current
ramps (the storage elements charge).
 
J

John Larkin

Jan 1, 1970
0
Hi!

Could you possibly give me a physical interpretation of the homogeneous
solution of the differential equation describing a series RL circuit
powered by a sinusoidal source?

The equation that describes the circuit is:

Vo*sin(wt)-L*d/dt(i(t)) = i(t)*R

The solution (using i(0)=0 as initial condition):

i(t)= transient_response + steady_state_response

(or homogeneous + (particular or inhomogeneous))

where

transient_response = [Vo * wL / (R^2+(wL)^2)] * exp(-R/L*t)
steady_state_response = Vo / (sqrt(R^2+(wL)^2)) * sin(w*t-arctan(wL/R))

using also:

a * sin(wt) + b * cos (wt) = 1/sqrt(a*a+b*b) * sin(wt + arctan (b/a))

What is the physical interpretation of the transient response?

Look at it this way: the first half-cycle of the applied sine wave is
all positive, so it has a DC component. After that, all the successive
full cycles average zero. So there's an initial asymmetric, DC kick
shot in at startup. After many cycles, the circuit sort of forgets
about that initial insult and settles down.

If r=0, meaning you apply a sine wave to a pure inductor starting at
t=0, the dc component never dies away.

Something like that.

John
 
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