B The total voltage drop in a series RLC circuit

[mymodded]

TL;DR

is the total voltage across all components in a series rlc circuit really equal to $v_{L} + v_{C} + v_{R}$ ?

my textbook said this

which I don't think is true, tho the textbook still said this (this equation is the true one)

so is the textbook right about the first equation?

 

[berkeman]

TL;DR Summary: is the total voltage across all components in a series rlc circuit really equal to $v_{L} + v_{C} + v_{R}$ ?

which I don't think is true

Why not? The loop voltage is the sum of the individual voltage drops (and any voltage gains). How can it be otherwise? Are you trying to ask about scalar versus complex voltages somehow?

BTW, your posting of math as images is making it hard for me to quote parts of your post that I want to focus on. I will send you a PM with hints on using LaTeX at PF to post math equations. Please start using LaTeX instead of images as soon as you can. Thank you.

 

[DaveE]

is the total voltage across all components in a series rlc circuit really equal to $v_{L} + v_{C} + v_{R}$ ?

yes. Look into Kirchhoff's Voltage Law.

my textbook said this

which I don't think is true

Why? Please explain what is confusing you. Otherwise we don't really know how to give useful answers.

Imagine you have a voltmeter and you fix the - probe at a circuit node. Then you start with the + probe at that node (where is should say zero) and sequentially move around the loop. When you get back to the starting point would you expect the voltage to be zero? doesn't that mean that the voltages all sum to zero? If not where would the extra voltage be?

Don't be confused by time varying signals. The Kirchhoff's Voltage Law applies at each instant in time.

Khan Academy has some really good electronics tutorials. I would check those out if I was you.

 

[mymodded]

Why? Please explain what is confusing you. Otherwise we don't really know how to give useful answers.

there is a question where we had to solve for the inductor's voltage which turned out to be about 500V, even though the AC source was only 12V and the only reason it made sense is because $V_{m}^{2} =V_{R}^2 + (V_{L} - V_{C})^{2}$ and $V_{C}$ was equal to $V_{L}$ (at resonance), so they cancel out, I don't know how the first equation holds true when the second equation holds true as well. Or it could be that it has something to do with one of them being a voltage drop while the other is a voltage rise. Also if the first equation $V = v_{C} + v_{L} + v_{R}$ is true, why do we have the second equation?

edit: I've got a question, in the equation $V_{m}^{2} =V_{R}^2 + (V_{L} - V_{C})^{2}$, $V_{m}, V_{R}, V_{L}, V_{C},$ correspond to the maximum voltages, but does this equation hold true when the actual voltages (I mean when they are not at their maximum) are used instead of the maximum voltages? I know this equation is derived from adding the phasor vectors but I'm just wondering.

 

[DaveE]

there is a question where we had to solve for the inductor's voltage which turned out to be about 500V, even though the AC source was only 12V and the only reason it made sense is because $V_{m}^{2} =V_{R}^2 + (V_{L} - V_{C})^{2}$ and $V_{C}$ was equal to $V_{L}$ (at resonance), so they cancel out, I don't know how the first equation holds true when the second equation holds true as well. Or it could be that it has something to do with one of them being a voltage drop while the other is a voltage rise. Also if the first equation $V = v_{C} + v_{L} + v_{R}$ is true, why do we have the second equation?

edit: I've got a question, in the equation $V_{m}^{2} =V_{R}^2 + (V_{L} - V_{C})^{2}$, $V_{m}, V_{R}, V_{L}, V_{C},$ correspond to the maximum voltages, but does this equation hold true when the actual voltages (I mean when they are not at their maximum) are used instead of the maximum voltages? I know this equation is derived from adding the phasor vectors but I'm just wondering.

The KVL equation $v_{C}(t) + v_{L}(t) + v_{R}(t) = v_{S}(t)$ has instantaneous voltage terms. So, in your example $v_{S}(t) = 12 \sqrt{2} sin(\omega t)$, etc. The $\sqrt{2}$ factor is because it's common to use RMS voltages when describing AC sources, but you could leave it out and just refer to peak magnitude.

The other equation $V_{R}^2 + (V_{L} - V_{C})^{2}$ is a calculation of the magnitude squared of the voltage sum. So, for example $|v_{S}(t)| = |12 \sqrt{2} sin(\omega t)| = 12 \sqrt{2} = V_{S}$ or 12V_rms. Then $V_{R}^2 + (V_{L} - V_{C})^{2} = |v_{R}(t) + v_{L}(t) + v_{C}(t)|^2$. Of course the phase of each voltage term matters when you add them up.

 

[mymodded]

The KVL equation $v_{C}(t) + v_{L}(t) + v_{R}(t) = v_{S}(t)$ has instantaneous voltage terms. So, in your example $v_{S}(t) = 12 \sqrt{2} sin(\omega t)$, etc. The $\sqrt{2}$ factor is because it's common to use RMS voltages when describing AC sources, but you could leave it out and just refer to peak magnitude.

The other equation $V_{R}^2 + (V_{L} - V_{C})^{2}$ is a calculation of the magnitude squared of the voltage sum. So, for example $|v_{S}(t)| = |12 \sqrt{2} sin(\omega t)| = 12 \sqrt{2} = V_{S}$ or 12V_rms. Then $V_{R}^2 + (V_{L} - V_{C})^{2} = |v_{R}(t) + v_{L}(t) + v_{C}(t)|^2$. Of course the phase of each voltage term matters when you add them up.

Thank you so much appreciate it a lot

But wait, $|v_{R}(t) + v_{L}(t) + v_{C}(t)|^2$ is not necessarily always equal to the peak squared while $V_{R}^2 + (V_{L} - V_{C})^{2}$ appears to always equal the peak squared (as it uses the maximum voltage for each component) so how is that so?

 

[DaveE]

Sorry, I'm working and traveling for a few days. Plus I find your last question a bit confusing, which makes it hard to answer. I suggest you review voltage dividers, complex impedance, and phasor representation of AC circuits. Khan Academy has some really good material on this. Maybe others will step in to help?

The solution process will go like this:
Find the loop current by adding the impedances (including their phase) across the source.
Then find the inductor voltage by multiplying that current by the inductor impedance.
So, the voltage gain $\frac{V_L}{V_S} = |\frac{j \omega L}{R+j \omega L+\frac{1}{j \omega C}}|$
Also note that $V_L = | j \omega L⋅ I|$, etc.

 

[Sagittarius A-Star]

Maybe others will step in to help?

Additional explanation: It is useful to calculate RLC circuits, connected to a sinusoidal voltage, with complex quantities.

Source:
W. Ameling: German book with translated title "basics of electrical engineering I"
https://www.amazon.com/Grundlagen-E...chaft-Technik/dp/3528491493?tag=pfamazon01-20

The discontinued electronic calculator Casio fx-61f had buttons for such calculations.

 

[Sagittarius A-Star]

But wait, $|v_{R}(t) + v_{L}(t) + v_{C}(t)|^2$ is not necessarily always equal to the peak squared while $V_{R}^2 + (V_{L} - V_{C})^{2}$ appears to always equal the peak squared (as it uses the maximum voltage for each component) so how is that so?

Which textbook do you refer to?

The term $|v_{R}(t) + v_{L}(t) + v_{C}(t)|^2$ is the square of the instantaneous total voltage at time $t$.

The overall complex impedance can be written as
$\underline {Z} = R +jX$
with $R$ = resistance and $X$ = reactance.
It follows the absolute value of the complex impedance:
$\left | \underline Z \right | = \sqrt{R^2 + X^2}$.

 

[tech99]

For a series LCR circuit a good way of analysing it is to first assume a current of 1 Amp. This is the same everywhere in the circuit. Then find the reactances, XL and XC. If 1 amp is flowing in a reactance of say, 10 Ohms, then the voltage across it is 10 volts. You can draw a phasor diagram where I is the reference and VX is then drawn leading it by 90 degrees and VC is lagging it by 90 degrees. VR is in phase with the current. It is then a simple matter to add the three voltages vectorially and find the voltage across LCR.
For a parallel LCR circuit, we assume a 1 volt generator, so we have 1 volt across each element, and find the currents in each element in a similar way as I have described.

 

[mymodded]

Which textbook do you refer to?

I refer to this textbook [PDF link redacted by the Mentors due to Copyright violation] (you can find this lesson on page 274 in the pdf or 267 in the actual book), they don't usually do this, but I feel like they missed out on many pieces of information in this unit.

The term $|v_{R(t)}+v_{L(t)}+v_{C(t)}|^{2}$ is the square of the instantaneous total voltage at time t.

thanks a lot!