RL circuit with polynomial forcing function solution (units)

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Hello.
This is the differential equation.

$$
i \cdot \dfrac{R}{L} + \dfrac{di}{dt} = t
$$
My solution path:
Homogenous solution:
$$
r + \dfrac{R}{L} = 0 \\ \\

r = -\dfrac{R}{L} \\ \\

i_{h}(t) = C_{0} e^{ -\dfrac{R}{L} t} \\ \\ $$

Particular solution:

Try $$ y_{p} = at + b \\ \\

y_{p} \dfrac{R }{L } + y_{p}' = t \\ \\

(at + b) \dfrac{R}{L} + a = t \\ \\

at\dfrac{R}{L} = t \\ \\

b \dfrac{R}{L} + a = 0 \\ \\

a = \dfrac{L}{R} \\ \\

b = - \dfrac{ L^2}{R^2} \\ \\ $$

Verifying particular solution:

$$ y_{p} = \dfrac{L}{R} \cdot ( t - \dfrac{L}{R} ) \\ \\

y_{p}' = \dfrac{L}{R} \\ \\

y_{p} \dfrac{R}{L} + y_{p}= t \\ \\

\dfrac{L}{R} \cdot ( t - \dfrac{L}{R} ) \cdot \dfrac{R}{L} + \dfrac{L}{R} = t \\ \\

t - \dfrac{L}{R} + \dfrac{L}{R} = t $$
Boundary conditions given:
$$
\boxed{I(0) = 0 } \text{inital conditions} \\ \\

0 = C_{0} - \dfrac{L^2}{R^2} \\ \\

C_{0} = \dfrac{L^2}{R^2} \\ \\

I(t) = \dfrac{L^2}{R^2} e^{ -\frac{R}{L} t} + \dfrac{ L}{ R} ( t - \dfrac{ L}{ R}) \\ \\

\boxed{i(t) = \dfrac{L^2}{R^2} \cdot e^{( -\frac{R}{L} t)} + \left( \dfrac{ L}{ R}( t - \dfrac{ L}{ R}) \right) \,\,\,\, (\text{A} ) } \,\,\,\, \text{Complete solution} $$
My question is, despite the complete solution fitting the boundary conditions, the units must be amperes, and I dont think the units are amperes. How can i get a solution with the correct units? My solution path seems correct to me as t is a first order polynomial.
 
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Answers and Replies

  • #2
Delta2
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The problem with the units is right from the start of the ODE

##I\frac{R}{L}+\frac{dI}{dt}=t## (1)

because the left hand side of (1) has units of ##\frac{A}{s}## while the right hand side has units of just ##s## (seconds). So to fix this you must multiply only the right hand side by a constant C which will be just 1 but will have units of ##\frac{A}{s^2}## so ##C=1\frac{A}{s^2}## so the differential equation will be

##I\frac{R}{L}+\frac{dI}{dt}=Ct##(2) and now the units in both sides of (2) are ##\frac{A}{s}##.

(or alternatively you can multiply only the left hand side of (1) by a constant D such that ##D=1\frac{s^2}{A}##.

Your final complete solution will not change at all as number, because it will be just multiplied by C which is just 1, but it will change as units and it will be amperes. (##\frac{A}{s^2}s^2=A## now as it is, the units are ##s^2##).
 
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  • #3
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The problem with the units is right from the start of the ODE

##I\frac{R}{L}+\frac{dI}{dt}=t## (1)

because the left hand side of (1) has units of ##\frac{A}{s}## while the right hand side has units of just ##s## (seconds). So to fix this you must multiply only the right hand side by a constant C which will be just 1 but will have units of ##\frac{A}{s^2}## so ##C=1\frac{A}{s^2}## so the differential equation will be

##I\frac{R}{L}+\frac{dI}{dt}=Ct##(2) and now the units in both sides of (2) are ##\frac{A}{s}##.

(or alternatively you can multiply only the left hand side of (1) by a constant D such that ##D=1\frac{s^2}{A}##.

Your final complete solution will not change at all as number, because it will be just multiplied by C which is just 1, but it will change as units and it will be amperes. (##\frac{A}{s^2}s^2=A## now as it is, the units are ##s^2##).
I made a mistake in my solution.
The initial DE ACTUALLY given was:
$$
iR + L\dfrac{di}{dt} = t
$$
Multiplying the DE by $$ \dfrac{1}{L}$$
Gives:
$$ i \dfrac{R}{L} + \dfrac{di}{dt} = \dfrac{t}{L} $$
I hurried the solution and forgot to multiply the source term by $$ \dfrac{1}{L} $$
But even without multiplying the entire equation, the initial units are:
$$ V = s $$
Which is wrong. So I think this question means that there is a voltage source which is outputs the voltage equal to the duration of time.
My apologies, the person who gave it to me to solve was not clear about it.
 
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  • #4
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i will assume $$ t \,\, (V) $$ and solve it again for the source term as $$ \dfrac{t}{L} \,\, \frac{V}{H} $$
 
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  • #5
Delta2
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Yes the initial DE has to be ##iR+L\frac{di}{dt}=V_0t## where ##V_0=1\frac{V}{s}##
 
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  • #6
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Yes the initial DE has to be ##iR+L\frac{di}{dt}=V_0t## where ##V_0=1\frac{V}{s}##
Yes. Here is the solution that was realised through your insight:
$$
V_{0} = \frac{{V} }{ s} \\ \\
i\dfrac{R}{L} + \dfrac{di}{dt} = \dfrac{ V_{0} t}{L} \\
\dfrac{R}{L} + r = 0 \\
r = - \dfrac{R}{L} $$

$$ i_{h}(t) = C_{0} e^{(- \dfrac{R}{L} t )} \\

\text{Particular solution:}$$

$$ i_{p}(t) = at + b $$

$$ (at + b ) \dfrac{R}{L} + a = \dfrac{V_{0}t}{L} $$

$$ at \frac{R}{L} + (a + b) = \dfrac{V_{0}t}{L} $$

$$ at \frac{R}{L} = \dfrac{V_{0}t}{L} $$

$$ \boxed{ a = \dfrac{ V_{0} }{ {R} } } \tag{a}$$

$$ \dfrac{V_{0} }{ {R} } + b \frac{R}{L} = 0 $$

$$ \boxed{ b = - \dfrac{ V_{0} L}{ {R^2} } } \tag{b} $$

$$ i(t) = C_{0} e^{(- \frac{R}{L} t )} + (\dfrac{ V_{0} }{ {R} } t + - \dfrac{ V_{0} L}{ {R^2} } ) \tag{} $$

$$ \text{Initial values to solve for constants}$$

$$ i(0) = 0 \\ \\

0 = C_{0} - \dfrac{ V_{0} L}{ {R^2} } \\ \\

C_{0} = \dfrac{ V_{0} L}{ {R^2} } \\ \\

{ i(t) = \frac{ V_{0} L }{ {R^2} }e^{(- \frac{R}{L} t )} +\dfrac{ V_{0} }{ R } t - \dfrac{ V_{0} L}{ {R^2} } } \\ \\ $$

$$ \boxed{ i(t) = \dfrac{ V_{0} L}{ {R^2} } (e^{(- \frac{R}{L} t )} - 1 ) +\dfrac{ V_{0} }{ R } t } \tag{Complete}
$$
The units work out now:
$$ \frac{ \frac{V}{s} \cdot H }{\Omega^2} + \frac{ \frac{V}{s} \cdot s}{\Omega} = A \\
\frac{V F }{ s} + \frac{V}{\Omega} = A \\
(\frac{ V}{ s} \frac{Q}{V} = \frac{Q}{s}=A) + A = A
$$
Thank you.
 
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  • #7
Delta2
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I just checked the units of the complete solution,the first term is ##\frac{V}{s Ohm}\frac{H}{Ohm}=\frac{A}{s}s=A## and the second term
##\frac{V}{s}\frac{s}{Ohm}=\frac{V}{Ohm}=A## so everything looks fine.
 
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