# LC Impedance Conventions

It is fairly easy to derive AC impedance conventions for inductors and capacitors, starting with time-domain equations.

For the inductor:

$v(t) = L \frac{d}{dt} i(t)$

Take the Laplace transform:

$V(s) = L s I(s)$

Substituting in $s = j \omega$.

$\frac{V(j \omega)}{I(j \omega)} = Z_L = j \omega L$

For the capacitor:

$i(t) = C \frac{d}{dt} v(t)$

Take the Laplace transform:

$I(s) = C s V(s)$

Substituting in $s = j \omega$.

$\frac{V(j \omega)}{I(j \omega)} = Z_C = \frac{1}{j \omega C} = - \frac{j}{\omega C}$

This shows why inductors have $+ j$ impedance and capacitors have $- j$ impedance

Another way to show it is to examine the test signal $\sin(\omega t)$. Suppose that this is the expression for the inductor current:

$i(t) = \sin(\omega t) \\ v(t) = L \frac{d}{dt} i(t) = \omega L \cos (\omega t)$

We can see that for the inductor, the voltage is greater than the current by a factor $\omega L$, and it leads by 90 degrees (agreeing with the $+ j$) term derived earlier.

Now examine the capacitor in the time domain:

$v(t) = \sin(\omega t) \\ i(t) = C \frac{d}{dt} v(t) = \omega C \cos (\omega t)$

We see that the voltage is smaller than the current by a factor $\omega C$, and that the voltage lags the current by 90 degrees, agreeing with the transfer function derived earlier.

Note by Steven Chase
9 months, 3 weeks ago

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@Steven Chase Thanks.

- 9 months, 3 weeks ago

@Steven Chase is there any other way to prove it, other than laplace transform.

- 9 months, 3 weeks ago

I have added a bit to the end

- 9 months, 3 weeks ago

@Steven Chase Yeah Thanks.

- 9 months, 3 weeks ago

Your knowledge of Alternating current is next level.
I feel very unconfident while solving Alternating current problems.

- 9 months, 3 weeks ago

@Steven Chase Therefore i have started series of Alternating current problems.
Make sure to drop a solution.
Hope I am not disturbing you.

- 9 months, 3 weeks ago

@Steven Chase https://brilliant.org/problems/alternating-current-series-4/

- 9 months, 3 weeks ago

I'm not sure what the physical mechanism for that one is supposed to be. Do you have an idea?

- 9 months, 3 weeks ago

@Steven Chase Yeah right now I got a YouTube channel where this problem is discussed.
Here is the video.

- 9 months, 2 weeks ago

@Steven Chase Hello, Good Morning.
I have posted a new problem. Please check my answer is correct or not?

- 9 months, 3 weeks ago

@Steven Chase https://brilliant.org/problems/electromagnetic-induction-4/

- 9 months, 3 weeks ago

@Steven Chase Good. Evening,Help me in this

- 9 months, 3 weeks ago

@Steven Chase hello Good Afternoon,Try this problem

- 9 months, 2 weeks ago

Initially, the left capacitor has voltage V and the right capacitor has no voltage. Then after the switch opens, the system is subject to the following constraints.

1) The sum of the two capacitor voltages must be V
2) Since the capacitors are in series, and they have the same capacitance, the change in one capacitor's voltage must exactly match the change in the other capacitor's voltage.

There are three scenarios to consider then:

1) The capacitor voltages are exactly the same after switch opening as before switch opening
2) Both capacitors increase their voltages relative to before switch opening
3) Both capacitors decrease their voltages relative to before switch opening

Scenarios 2 and 3 can't happen without violating constraint #1. Therefore nothing happens, and no heat is dissipated.

- 9 months, 2 weeks ago

@Steven Chase Your answer is correct but one thing which I have not understand properly is
“The capacitor voltages are exactly the same after switch opening as before switch opening”

- 9 months, 2 weeks ago

In other words, they don't change at all

- 9 months, 2 weeks ago

@Steven Chase do you have edited the note?

- 9 months ago

It is the same as it was earlier

- 9 months ago

@Steven Chase You have very much knowledge of electrical engineering.
Nowadays I am feeling jealous with you. :)

- 9 months ago

No worries. I have been doing this a long time

- 9 months ago

@Steven Chase I was joking :). Just to make you feel better :)

- 9 months ago

@Steven Chase Do you mean that, you are making people feel jealous from long time ??

- 9 months ago

No, I've been studying electrical engineering for a long time

- 9 months ago

@Steven Chase I know that, I was kidding you.

- 9 months ago