Charging of Inductors: Explained Without the Duality

[flexifirm]

Most people understand (gut understanding) the charging of capacitors more than that of inductors. They just apply the duality to inductors and move on (without seeking the same understanding of inductors).

Teachers of my past never could explain exactly why the charging of a capacitor slowed down with time, but I figured it out by thinking it over. It's because the more electrons that move from one plate to the other, the harder it is for the ones next in line to do so. Why harder? Because there are now more -ve charges repelling them at the destination plate, and more +ve charges attracting them at their originating plate.

Thus, the charging of capacitors slows down with time (universal time constant curve).

But what about inductors? They're current graph (charging of flux) is exactly the same shape as a capacitor's voltage graph (charging by electrons). WHY?

Without using differential equations and any sort of method that just points at the duality with caps, can someone explain why the initial FLUX into a inductor has a much easier time building up, than the later FLUX (FLUX is proportional to current through).

 

[flexifirm]

Hmmm...

It's true that the voltage across the inductor sinks as more current is supplied through it (because that current will also drop across the R in the circuit.. and to obey KVL, inductor voltage sinks!)

But why does the current's rate of change slow down? (without calculus or anything)

And why is the charge rate of an inductor increased when the R in the circuit is increased? Intuitively, it's not clicking.

Capacitors, for some reason, make much more intuitive sense to all of us (my teachers, authors, etc.).

Why aren't there any good explanations for inductors?

 

[pankti]

regarding inductor I have one more question, that why in purely inductive sircuits, current lags by voltage by exactly 90 degree? again mathemetical explanation is not enough. what exactly happens when, ac supply voltage is given to purely inductive circuit?

 

[berkeman]

You folks need to all be more explicit about what you are driving the test inductor with. You can't talk about the current through and voltage across the inductor as it responds to some nebulous drive source.

The current through an "ideal" inductor (zero resistance) will not level off if driven by a voltage source with a zero output impedance. It *will* level off if driven by a real voltage source with a set output resistance. The leveling off is due to the stabilization of the voltage across the inductor at zero Volts, and all the supply voltage dropping across the output resistance. The final DC current through the inductor is V/R.

Capacitance is defined by the voltage that you get in some geometry when you charge it up with some amount of charge Q = C V The relationship that you refer to ("no calculus") for current and voltage in a capacitor is I = C \frac{dV}{dt} The current is proportional to the change in voltage with respect to time, and that proportionality coefficient is the capacitance.

Inductance is defined by the flux linkage that you get in some geometry when you pass a current through it \wedge = L I The relationship that you refer to ("no calculus") for current and voltage in an inductor is V = L \frac{dI}{dt} The voltage across an inductor is proportional to the change in current with respect to time, and that proportionality coefficient is the inductance.

So to answer pankti's question (with a quiz question), if you put a voltage across an inductor where V = sin(\omega t), then what is the equation for the current, and what is the phase shift between the sin and cos functions?

And to the OP's question, if the calculus is not intuitive enough for you, think of it this way:

-- You put a voltage across an inductor
-- That voltage causes a current to start to flow
-- The change in current generates a back EMF voltage to be generated
-- The higher the inductance, the higher the back-EMF that is generated
-- The back-EMF voltage opposes the externally impressed voltage somewhat, which slows down the possible rate of changes in the current.
-- The change in current is proportional to the voltage across the inductor, and when there is a series resistance in the overall circuit, the current increases (more and more slowly) to the point where all the voltage drop of the source power supply is across that series resistor (and any real coil resistance of course).

 

[Urmi Roy]

Sorry to pull this thread out after such a long time...but I've been thinking and thinking about what the op said about the capacitor...

...It's because the more electrons that move from one plate to the other, the harder it is for the ones next in line to do so. Why harder? Because there are now more -ve charges repelling them at the destination plate, and more +ve charges attracting them at their originating plate.

It seems to me the 'levelling off' of the current curve i.e its sinusoidally varying nature is a direct consequence of how the voltage applied (which is sinusoidally varying) demands the current to change in order to match up the potential across the capacitor in order to equal the applied voltage.
If I'm right,there is no place for the description that the op gives. Please tell me if I'm wrong.

 

[Urmi Roy]

Perhaps this post supports my point??

I think of it more simply as the voltage difference between the source and the capacitor is decreasing so the current decreases so the number of charges per second flowing onto the cap decreases.

 

[berkeman]

Perhaps this post supports my point??

Yes, that is the better way to think of it. The voltage source by itself is happy to supply all the "force" you need to herd those pesky electrons. But when the delta-V gets lower and lower, you get less current, so less electron movement...

 

[Urmi Roy]

That's a relief!

 

[Urmi Roy]

Hi everyone..
I think I'll make use of this thread to get myself clear about inductors in ac...
I've been thinking and I've come up with an explanation...ot's just that I'm not sure about it...could someone tell me if my explanation is okay?

I'll put it in the next post.

 

[Urmi Roy]

Referreing to my diagram,the horizontal axis is time t and the verical axis is a meaure of the voltage applied and current I. (PLease take some time to go through my post)

Now, at t=o,the applied voltage is 0 but at that instant,it is changing to a higher value dv.

At the point dv,a potential has been established in the direction of dv and so instantly, a current is produced.

However,due to the inductor in the circuit,senses this rise in current from 0 to a certain value corresponding to dv,as instantly produces a back emf equal to dv,and so there is no net current in the circuit.

at V applied= dv+dv' (dv' is the increment on dv after a time dt..and its not equal to dv due to sinusoidal nature of curve),again the applied potential has increased,which is supposed to increase the current due to Vappl(Applied voltage)...howevever,again,due to the inductor,there is no net current in the circuit.

At t= 2T-dt,the voltage is instantaneously decreasing at the greatest rate,so induced emf is largest but value of Vappl is smallest(zero)..so the maximum induced emf has the dominant effect,and the current in the circuit is maximum.

This continues till T. At T-dt,there was again a certain value of Vappl,but it failed to produce any current due the equal back emf by the inductor.

At T,there is no change in cuurent(The V curve is flat),so there is no tendency of back emf in the inductor.

However,at T+dt, there is a lower value of Vappl= Vmax-dv,so there is a reduction in current in the circuit due to the applied voltage,so there is a reduction in flux linked to the coil due to the applied voltage...hence,the inductor now tries to push current in the direction of the applied voltage,to prevent this reduction in fluxed lined to it (Lenz's law).

Now,there is a small current in the direction of the applied voltage due to both the inductor's induced emf and the Vappl.

At T+2dt,the value of Vappl is further decreased to Vmax-dv-dv'. Thus,there is a larger decrease in Vappl from T + dt to T+2dt than there was from T to T+dt(due to sinusoidal nature of curve,for equal intervals dt,there are uneven changes in Vappl).

Due to this larger drop in Vappl,there is a larger induced current in coil at T+2dt than at T+dt.

At successive increments of dt,the drop in dv is greater (due to sinusoidal nature of curve) and so induced emf is greater at each successive instant.

However,there is decrease in Vappl at each point,but due to the increasing induced emf,the net current in direction of applied voltage increases.

 

[Urmi Roy]

Okay,if no one wants to read my huge post,then could someone atleast clarify this one point..then I could try to check my understanding on my own...

Suppose we have an ac supply,and only an inductor is in the circuit...initially,when the applied voltage is switched on,there should be no current (due to cancellation of applied emf by back emf)...this 'no current' condition continues till the appl. voltage becomes maximum(peak of voltage sinusoid).

Now,when the applied voltage starts decreaseing(from peak value of voltage applied),a certain value of cuurent is seen in the circuit...is this current solely due to the induced emf due of the inductor?...that is,does the appl. voltage not have any role in producing this current?

(Note: in the graph of current and applied voltage for ac circuit with only inductor,the current reaches its maximum value when the appl. voltage is zero! So I guess whatever current there is,must be due to the induced emf of the inductor).

 

[berkeman]

Okay,if no one wants to read my huge post,then could someone atleast clarify this one point..then I could try to check my understanding on my own...

Suppose we have an ac supply,and only an inductor is in the circuit...initially,when the applied voltage is switched on,there should be no current (due to cancellation of applied emf by back emf)...this 'no current' condition continues till the appl. voltage becomes maximum(peak of voltage sinusoid).

Now,when the applied voltage starts decreaseing(from peak value of voltage applied),a certain value of cuurent is seen in the circuit...is this current solely due to the induced emf due of the inductor?...that is,does the appl. voltage not have any role in producing this current?

(Note: in the graph of current and applied voltage for ac circuit with only inductor,the current reaches its maximum value when the appl. voltage is zero! So I guess whatever current there is,must be due to the induced emf of the inductor).

Yeah, sorry that the post was a bit too long for my limited attention span

But what you are saying here in the quote seems wrong. The equation relating the current and applied voltage is always true for an inductor:

v(t) = L \frac{di(t)}{dt}

When you say

this 'no current' condition continues till the appl. voltage becomes maximum(peak of voltage sinusoid).

that doesn't fit with the equation, does it?

 

[Urmi Roy]

I was referring to the situation in which the applied voltage is just switched on.

I found this in http://www.tpub.com/neets/book2/4.htm

Again,since I'm trying to explain the equation you gave to myself (might be quite useless,all the same it's absorbing!),I found that it the fact that the applied voltage is zero at the point the current is maximum must mean that the current in the circuit must be due to induced emf due to the coil.

However,at this point(V appl. is zero),the rate of change of applied voltage is maximum on the sinusoidal graph,so the current due to it should be changing at the max. rate,thus induced emf due to coil is max.(please confirm)

 

[Urmi Roy]

The only clue left to find the 'truth behind inductors in an ac circuit' seems to be as to whether the current flow at the instant of zero applied potential is only due to the inductor (which continues for the rest of the tme too)...could someone just say yes or no?

 

[vk6kro]

I tried a Spice simulation of the current in an inductor.

http://dl.dropbox.com/u/4222062/current%20in%20inductor.PNG

The current scale is at the right of the graph.

It seems the current in an inductor is zero when power is applied (assuming it is applied at the zero crossing point of the sinewave) and it continues for several cycles to be offset from the zero current line in the direction of the first input cycle.
Eventually, the current swings equally on either side of the zero current line.

This was done with a 1 KHz sinewave input voltage and a 1 mH inductor. I also put a very small resistor (0.1 ohm) in series with the inductor.

However, I am cautious (suspicious) of simulators and reproduce this result to see if others get similar results. I never liked the idea of a current flowing before there was a voltage, though, so this may be more elegant.

 

[Urmi Roy]

Thanks for the post vk6kro ...but I'm still at a loss...as I said,I was trying to explain the graph for myself...and the only conclusion I could reach (especialy to address the fact that at all points for which voltage is zero,the current is max)...is that the current is due only to the induced emf of the inductor.

It would be really nice if you explained on the lines I'm already on,as I've got used to it.

 

[Urmi Roy]

By the way,does the fact that the current in the first few cycles does not swing equally in both directions have any special significance in the idea of what actually happens in the circiut?

 

[berkeman]

I was referring to the situation in which the applied voltage is just switched on.

I found this in http://www.tpub.com/neets/book2/4.htm

Again,since I'm trying to explain the equation you gave to myself (might be quite useless,all the same it's absorbing!),I found that it the fact that the applied voltage is zero at the point the current is maximum must mean that the current in the circuit must be due to induced emf due to the coil.

However,at this point(V appl. is zero),the rate of change of applied voltage is maximum on the sinusoidal graph,so the current due to it should be changing at the max. rate,thus induced emf due to coil is max.(please confirm)

You are being confused by incomplete graphs (the ones you posted, not vk6kro's). They are not showing the full inductor current waveform. It does not magically start at the point where the voltage is max at 90 degrees. If they drew that graph completely, it would have been swinging up from a negative value and crossing zero when the voltage is max. It's unfortunately that a poorly drawn I-V diagram has caused you so much confusion. The equation that I wrote holds true for all time. Just do the differentiation to get the I(t) waveform.

V(t) = L \frac{dI(t)}{dt}

 

[Urmi Roy]

Even in vk6kro's graph,it shows the current to be max when applied voltage is zero...also I don't uderstand what the 'I' in the equation is...its equating the applied voltage to the change in current (not the absolute value of the current)...so is that current the current due to the net effect of the applied voltage and the inductor's emf?

 

[berkeman]

Even in vk6kro's graph,it shows the current to be max when applied voltage is zero...also I don't uderstand what the 'I' in the equation is...its equating the applied voltage to the change in current (not the absolute value of the current)...so is that current the current due to the net effect of the applied voltage and the inductor's emf?

Because of the differential equation that I'm showing you, the current sinusoidal waveform lags the voltage sinusoid by 90 degrees. That's what happens when you differentiate the current sinusoidal waveform. Are you familiar with differentiation? Does the equation that I wrote make sense to you?

In vk6kro's simulation, the voltage and current are starting at 0 at t=0. That's the initial condition for the simulation. Since the voltage swings one way first, that provides an offset in the current sinusoid. Notice how the current sinusoid waveform is slowly coming down near the right side of the plot? It will eventually settle down to where it has zero offset, just like the driving voltage sinusoid.

It may be helpful for you to think about how the current and voltage act in a parallel LC "tank" resonant circuit. In that circuit, the energy stored in an oscillating waveform goes back and forth between the inductor (its current), and the capacitor (its voltage). At the time when the inductor current is max, the capacitor voltage happens to be zero. When the capacitor voltage is max, the current in the inductor (and in the cap ovbiously) is zero.

 

[The Electrician]

By the way,does the fact that the current in the first few cycles does not swing equally in both directions have any special significance in the idea of what actually happens in the circiut?

Yes, it does.

If there is any resistance in the circuit at all, and there always is in the real world, then the differential equation for the circuit becomes:

V(t) = L* \frac{di(t)}{dt}+R*i(t)

The solution to this equation has two parts: a transient part and a steady-state part.

The transient part is a decaying exponential of current, and its magnitude depends on when you suddenly connect a sine wave to the inductor (which also has some resistance, remember). I've attached an image with three plots showing what goes on.

The first plot shows a sine wave of voltage suddenly applied to an inductor plus resistance. It shows two cases; the blue and red traces are the applied sine wave of voltage. The blue shows the sine wave being applied just as it goes through zero volts with a positive slope. The red trace shows the applied sine wave being applied at the positive peak of the sine wave.

I have inverted (multiplied by -1) the current traces to make them easier to distinguish from the voltage traces.

The green trace shows the current due to the application of the blue voltage. The magenta trace shows the current due to the application of the red voltage.

The second plot is the same as the first plot, but showing a longer time.

Notice that the green trace doesn't have equal swings about the zero axis at first. This is because it's not possible for the current in an inductor to change instantaneously, so when the blue voltage is suddenly applied, the current in the inductor must be zero to start with.

A transient current, shown in black, is present in the inductor to make this true. If you've ever studied differential equations, you will recognize this as the initial condition, which results in the transient. This transient current gradually dies out, as can be seen clearly in the second plot. After it does die out, the inductor has equal swings above and below the zero current axis.

The third plot shows what would happen if there were no resistance in the circuit. The transient current would never die out, and the green current wave would never become re-centered on the zero current axis. This can never happen in the real world, of course. It's only possible mathematically.

The transient current is zero if you apply the sine wave of voltage at the right time, as in the red trace (and it corresponding current, the magenta trace).

So, when you first apply the sine wave of voltage, the current may not be such that V(t) = L*di/dt at first, depending on just when you apply the voltage. The transient current (if there is one) must die out first.

This is all just plain old circuit theory.

Search on bing.com with the search terms: "transient analysis" inductor "natural response"
and you'll find lots of discussion of transient analysis.

 

[Urmi Roy]

I understand that vk6kro's post was based on a circuit with an inductor,plus some resistance,after comparing it to The Electrician's plots.

I didn't understand why the nature of the current in the circuit varies with the instant you apply voltage,though...

Also,before I go on to a circuit with both inductor and reistance,I'd like to know more about an ideal purely inductive circuit,that has reached its steady state...like the red and magenta voltage-current plot pair.

As I said earlier,In the differential equation describing the ideal inductive circuit i.e


V(t) = L \frac{dI(t)}{dt}

the applied voltage is equal to the change of the current w.r.t time...by Ohm's law,the current should be due to the applied emf..not the change of current.

Also,as I asked earlier...what is the current 'I' in this equation?

Is it the net current due to applied emf and inductor back emf?

 

[The Electrician]

I didn't understand why the nature of the current in the circuit varies with the instant you apply voltage,though...

Because the current through an inductor cannot change instantaneously. The rate of change of current in an inductor is proportional to the applied voltage. In order for the current to change instantaneously, you would need to apply infinitely great voltage for a short time--an impulse of voltage, in other words. That's not what you're doing when you suddenly apply a voltage of finite value.

If the applied sine wave of voltage is anything other than its peak at time tzero, then there will be a transient component of voltage, because in the steady state, at any time other than the peak of the applied voltage, the current in the inductor will be non-zero. Since the current in the inductor was zero just before we applied the voltage, it can't change instantaneously to a non-zero value just after we apply the sine of voltage.

Also,before I go on to a circuit with both inductor and reistance,I'd like to know more about an ideal purely inductive circuit,that has reached its steady state...like the red and magenta voltage-current plot pair.

As I said earlier,In the differential equation describing the ideal inductive circuit i.eV(t) = L \frac{dI(t)}{dt}

the applied voltage is equal to the change of the current w.r.t time...by Ohm's law,the current should be due to the applied emf..not the change of current.

How does it follow that if:

V(t) = L \frac{dI(t)}{dt}

then:

"the current should be due to the applied emf..". Do you mean that you think that inductor current should be proportional to the applied voltage? Algebraically, something like this:

i(t) = K*v(t)

where K is a constant of proportionality? If that's what you think, you are mistaken.

The three common circuit components, resistors, inductors and capacitors have three different terminal characteristics:

Resistor:

v(t) = R*i(t)

Inductor:

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

Capacitor:

v(t) = \frac{1}{C}\int i(t)\;dt

If you want current in terms of voltage, then the relationships become:

Resistor:

i(t) = \frac{v(t)}{R}

Inductor:

i(t) = \frac{1}{L}\int v(t)\;dt

Capacitor:

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

Also,as I asked earlier...what is the current 'I' in this equation?

Is it the net current due to applied emf and inductor back emf?

The current in the inductor is the integral of the applied voltage, in the steady state.

If a sine wave of voltage is suddenly applied to the inductor, then there may be a transient component of current which will eventually die out, but this transient component will be just what is needed to keep the inductor current from changing instantaneously, while allowing L*di/dt to be equal to the applied voltage at time t=0.

In the steady state, when the sine wave of applied voltage is going through zero in the positive direction, the inductor current will be a negative peak. But that can't happen when we first apply the sine wave of voltage at that point in the cycle, because the current in the inductor was zero just before we applied the voltage (tzero minus) and it must also be zero just after we apply the sine of voltage (tzero plus). The way this can happen is to have a transient current as shown in my plots. We must have both i(t) be zero (at time tzero),
and L*d i(t)/dt equal the suddenly applied voltage. The transient current bridges the gap between the initial conditions and the steady state.

But that transient isn't needed if the sine wave of voltage is applied at its peak, because then the current in the steady would normally be passing through zero at this time. No transient is needed to make both i(t) be zero and L*d i(t)/dt be equal to the suddenly applied voltage.

 

[Urmi Roy]

"the current should be due to the applied emf..". Do you mean that you think that inductor current should be proportional to the applied voltage? Algebraically, something like this:

i(t) = K*v(t)

where K is a constant of proportionality? If that's what you think, you are mistaken.

Inductor:

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

When we first did inductors...plain and simple,we said that there is a back emf at the instant that the current through the inductor changes...here the current means the applied voltage's current...like,for example in a dc circuit,when we switch the power on...at the first few instants,when the current is trying to reach a certain constant value,...thus,in the equation
Vapplied= L \frac{d\;i(t)}{dt}

for an inductor in a dc circuit(in the situation I referred to about),the i is the current that the applied voltage establishes...and that changes value...hence changing the value of magnetic flux linked to the inductor...and the back emf in the inductor is a result of this...is that okay?

The current in the inductor is the integral of the applied voltage, in the steady state.

If a sine wave of voltage is suddenly applied to the inductor, then there may be a transient component of current which will eventually die out, but this transient component will be just what is needed to keep the inductor current from changing instantaneously, while allowing L*di/dt to be equal to the applied voltage at time t=0.

In the steady state, when the sine wave of applied voltage is going through zero in the positive direction, the inductor current will be a negative peak.

This is exactly what I've been getting at...when the sine wave of applied voltage is going through zero,the inductor current is at the peak...so the current in the plot,at steady state,when Vappl. is zero is due to the inductor,and its back emf.

So it must be similar throughout the rest of the cycle also...i.e the current on the plot must be due to the inductor.

 

[The Electrician]

So it must be similar throughout the rest of the cycle also...i.e the current on the plot must be due to the inductor.

When you say "the current on the plot must be due to the inductor", you are being so vague as to convey almost no information.

Of course the steady state current is "due" to the inductor. The source of energy is the EMF of the voltage source, and since the only other thing in the circuit is the inductor, it goes without saying that the current is "due" to the inductor.

If the voltage source were driving only a resistor, then the current in the circuit would be "due" to the resistor, and also if only a capacitor were connected to the voltage source, the current would be "due" to the capacitor.

It is universally true that if a voltage source drives a single component, that component will determine the current by its current vs. voltage terminal characteristic.

It seemed to me that no explanation was needed for this rather general fact.

But for a more particular description of the current that will result when a sine wave of voltage drive a two terminal circuit component, we have to resort to a description of the component's terminal characteristic.

But you said you didn't understand "...why the nature of the current in the circuit varies with the instant you apply voltage."

Explaining that was the main thrust of my post. Do you understand that now?

 

[Urmi Roy]

When you say "the current on the plot must be due to the inductor", you are being so vague as to convey almost no information.

Of course the steady state current is "due" to the inductor. The source of energy is the EMF of the voltage source, and since the only other thing in the circuit is the inductor, it goes without saying that the current is "due" to the inductor.

Please don't get frustrated with me.

There seems to be a slight misunderstanding here.

When I said current due to the inductor,I meant the current on the plot is due to the back emf of the inductor (any value of current must be achieved due to a source of potential..in an inductive circuit,there are two sources of potential...applied voltage and the back emf)

 

[The Electrician]

Please don't get frustrated with me.

There seems to be a slight misunderstanding here.

When I said current due to the inductor,I meant the current on the plot is due to the back emf of the inductor (any value of current must be achieved due to a source of potential..in an inductive circuit,there are two sources of potential...applied voltage and the back emf)

This is not a useful point of view. What determines the flux in the core is Faraday's law; the flux is proportional to the integral of the applied voltage. The current is then proportional to flux if the core is linear, such as air.

If "...there are two sources of potential...applied voltage and the back emf)", then why do you say only that "...the current on the plot is due to the back emf..."? What about the applied voltage? Doesn't it have any effect on the current?

Have you considered the possibility that it's not the "back EMF" that determines the current, but rather the current (more precisely, the rate of change of the current) that determines the "back EMF".

Furthermore, the "back EMF" is equal to the applied voltage. If they're the same, then how can they cause any current at all?

If you apply 1 volt from an perfect voltage source (zero internal resistance, unlimited current capability) to a perfect 1 henry inductor (no internal resistance, no saturation of the core), the current will increase forever, in a linear ramp, at the rate of 1 amp per second. The "back EMF" will be exactly 1 volt, never increasing, never decreasing. How can that constant "back EMF" be responsible for a current that ramps up forever?

 

[b.shahvir]

The physics of inductors ain't easy to understand without audio visual source material. However below is a link to the thread wherein I've attempted to put forth my understanding of the concept;

https://www.physicsforums.com/showthread.php?t=355159

Regards,
Shahvir

 

[Phrak]

I disagree. There is a very simply fluid model of an inductor that mimics all the stored energy characteristics, and phase variation of voltage and current

A conducting wire is equivalent to a pipe containing fluid. An inductor is as a turbine or blade mounted within in the pipe. This turbine in equipped with a fly wheel. The stored energy of an inductor resides in the magnetic field. In the turbine-flywheel model, the energy resides in the flywheel.

It might be evident after a little scibbling with pencil and paper, that a sinusoidal variation in fluid pressure (voltage) should result in a sinusoidal variation in turbine speed and fluid flow (electric current), 90 degrees out of phase.

 

[Urmi Roy]

If "...there are two sources of potential...applied voltage and the back emf)", then why do you say only that "...the current on the plot is due to the back emf..."? What about the applied voltage? Doesn't it have any effect on the current?

I did consider the effect of applied voltage,which should have its own effect of causing current and all that...I put all that in my enormous post on the last page...but after analysing all this,with my limited intelligence and knowlege,I couldn't come up with much.

In regard to The Electrician's last post...but then how do we explain the zero applied voltage and maximum current points?

I haven't yet read through b.shahvir and phrak's points of view...please give me some time to read through...then I'll get back to you if I have any doubts.

 

[The Electrician]

In regard to The Electrician's last post...but then how do we explain the zero applied voltage and maximum current points?

The current in an inductor is proportional to the integral of the applied voltage. So when the voltage reaches zero, the maximum current is the result of the accumulated integral of the just past quarter cycle of voltage. The energy that flowed into the inductor during that quarter cycle is stored up (integrated), and manifests itself as a peak current.

This is similar to what happens if you apply a sine wave of current to a capacitor. The capacitor integrates the current and when the previous quarter cycle of current comes to an end as the current reaches zero, the voltage across the capacitor reaches a maximum.

This explains the phase shift between current and voltage with the inductor (the integral of sine is -cosine), and also explains this:

If you apply 1 volt from an perfect voltage source (zero internal resistance, unlimited current capability) to a perfect 1 henry inductor (no internal resistance, no saturation of the core), the current will increase forever, in a linear ramp, at the rate of 1 amp per second. The "back EMF" will be exactly 1 volt, never increasing, never decreasing. How can that constant "back EMF" be responsible for a current that ramps up forever?

This can't be explained by "back EMF".

 

[Urmi Roy]

b Shahvir,I read through your post on the other thread...please clarify some points about it...

Dear Sudar,

The voltage and current waveforms do not immediately phase shift by 90 deg as depicted by textbook diagrams. Consider positive half cycle of ‘voltage versus current’ waveform for an ideal inductor. At ‘t (time)’ = 0, as the voltage wave just rises above zero, the current wave rises in-phase with it. However as the voltage wave further progresses at ‘t (time)’ > 0 towards it’s peak value, the rate of change of magnetic flux becomes more pronounced ad the current wave now starts going out of phase w.r.t the voltage wave. Eventually as the cycle progresses, the current wave now completely goes 90 deg out of phase w.r.t the voltage wave (assuming an ideal inductor).

If you concentrate on the negative half cycle of the voltage vs. current waveform, you will find that the phase shifting is due to the ‘discharge’ of electrical energy by the inductor back to the power source as the magnetic field collapses (or rises, whatever may be the case), which precisely occurs when the input voltage wave has already reversed itself. This charge/discharge action of an inductor occurs due to the finite time taken my the magnetic field to rise or collapse……as force fields cannot change it’s states abruptly at ‘t = 0’ as per the laws of Physics or Mother Nature (if it would, the consequences would be disastrous). This in turn causes a delay in the flow of current thru the inductor and results in the phase shift between the voltage and current waveform. This delaying property of an inductance is also sometimes known as ‘Magnetic Inertia’.

‘t (time)’ > 0 towards it’s peak value, the rate of change of magnetic flux becomes more pronounced ad the current wave now starts going out of phase w.r.t the voltage wave.
...why does the rate of change of magnetic field become more pronounced as soon as it becomes t>0...like at t+dt?

If you concentrate on the negative half cycle of the voltage vs. current waveform, you will find that the phase shifting is due to the ‘discharge’ of electrical energy by the inductor back to the power source as the magnetic field collapses (or rises, whatever may be the case), which precisely occurs when the input voltage wave has already reversed itself. ...does that mean the discharge of the inductor occurs only when the applied voltage reverses itself?

This charge/discharge action of an inductor occurs due to the finite time taken my the magnetic field to rise or collapse……as force fields cannot change it’s states abruptly at ‘t = 0’ as per the laws of Physics ...PLease explain this in a little more detail...I'm finding it difficult to visualise,even after reading part B of your post.

 

[Urmi Roy]

I found this on allaboutcircuits (http://www.allaboutcircuits.com/vol_1/chpt_15/1.html)

"If a source of electric power is suddenly applied to an unmagnetized inductor, the inductor will initially resist the flow of electrons by dropping the full voltage of the source. As current begins to increase, a stronger and stronger magnetic field will be created, absorbing energy from the source. Eventually the current reaches a maximum level, and stops increasing. At this point, the inductor stops absorbing energy from the source, and is dropping minimum voltage across its leads, while the current remains at a maximum level. As an inductor stores more energy, its current level increases, while its voltage drop decreases. "

In the first few instants when the applied voltage (lets say dc source,as in the site),it says that the inductor completely opposes the flow of current by dropping a potential equal and opposite to the applied one...but no current has started flowing yet...so where does the inductor get the energy to oppose current from?

The current,even due to a dc source takes a finite time to increase to its constant value...in all the time instants between t=0 and this point (at which the current has reached a constant value)...the inductor keeps trying to oppose the current due to applied voltage...then how is the current established at all?

(In other words,I can't picture what happens in the first few instants after the voltage ha been applied till the establishment of current.)

 

[Phrak]

This describes an R/L circuit. See the graphs of voltage and current:

http://en.wikipedia.org/wiki/RL_circuit#Time_domain_considerations"

 

[The Electrician]

(In other words,I can't picture what happens in the first few instants after the voltage ha been applied till the establishment of current.)

Do you understand how it is possible for the current to be zero, but at the exact same instant of time the rate of change of the current can be non-zero?

If the current begins a linear ramp up of current, starting out at zero amps at time tzero, the "back EMF" can be non-zero even if the current is zero.

The value of the current and the value of its rate of change don't have to be the same. This is a critical thing for you to understand. It is at the root of your puzzlement.

 

[Urmi Roy]

...just figuring things out...by the way,the value of Ldi/dt doesn't have to be equal to the applied voltage at that instant,does it?(if it was,there would never be any current during the charging of inductor in dc circuit...but there is..the only effect if inductor here is to inhibit the current,not to cancel it

This analysis,after what The Electrician said in the last post and readind the page I referred to,seems similar to the charging of inductor in a dc circuit...except here,the voltage is 'switched on and off' repeatedly.

 

[The Electrician]

...just figuring things out...by the way,the value of Ldi/dt doesn't have to be equal to the applied voltage at that instant,does it?(if it was,there would never be any current during the charging of inductor in dc circuit...but there is..the only effect if inductor here is to inhibit the current,not to cancel it

In fact, L*di/dt is always equal to the voltage impressed across the inductor. If there is no resistance in the circuit, then L*di/dt is a constant if a constant DC voltage is applied to the inductor. In that case, the current in the inductor increases in a linear ramp forever.

One of the quotes you gave from the AAC book, says:

""If a source of electric power is suddenly applied to an unmagnetized inductor, the inductor will initially resist the flow of electrons by dropping the full voltage of the source. As current begins to increase, a stronger and stronger magnetic field will be created, absorbing energy from the source. Eventually the current reaches a maximum level, and stops increasing."

The lead-up to this doesn't explicitly say that the example involves a circuit with no resistance at all, but just prior they did give an example of an inductor shorted on itself with superconducting wire. They are aware that current would circulate forever.

If an ideal DC voltage source (one with zero internal resistance and that can supply infinite current) is applied to an inductor with no internal resistance, and if there is no other resistance in the circuit, it's not true that "Eventually the current reaches a maximum level, and stops increasing." This only happens if there is resistance in the circuit somewhere.

If there is a resistor in series with the inductor, then it's still true that the voltage applied to the inductor is equal to L*di/dt at all times. But in this case, as the current in the inductor increases, the resistor drops more and more of the voltage, so that the voltage left to be applied to the inductor is less, and the current approaches a limit, a "maximum level".

At tzero, the resistor drops no voltage, because the current is still zero, and the total source voltage is applied to the inductor, and L*di/dt is exactly equal to the applied voltage; the slope of the current is initially the same as it is when there is no resistor. But the slope immediately begins decreasing, and the current waveform in this case is the usual exponential approaching a limit.

Rather than trying to make sense of the point of view that the "back EMF" is opposing the applied voltage and therefore determining the inductor current, it's better to take the current as the thing which determines the "back EMF".

The current assumes that value that just makes the "back EMF" equal to the applied voltage. The current is proportional to the integral of the applied voltage, and assumes the exact value needed to produce a "back EMF" equal to the applied voltage. This is because saying that the current is proportional to the integral of the applied voltage is the same thing as saying that the "back EMF" of the inductor is proportional to the di/dt of the inductor current.

 

[Urmi Roy]

In fact, L*di/dt is always equal to the voltage impressed across the inductor. If there is no resistance in the circuit, then L*di/dt is a constant if a constant DC voltage is applied to the inductor. In that case, the current in the inductor increases in a linear ramp forever.

The fact that the "current can increase for ever in the absence of a resistance" must be due to the fact that the zero resistance of the circuit can ideally allow infinite current...thus,even if we have a fixed potental applied,to the circuit,the current may be infinite..like in superconductors...right?

One of the quotes you gave from the AAC book, says:

If there is a resistor in series with the inductor, then it's still true that the voltage applied to the inductor is equal to L*di/dt at all times. But in this case, as the current in the inductor increases, the resistor drops more and more of the voltage, so that the voltage left to be applied to the inductor is less, and the current approaches a limit, a "maximum level".

That means since the voltage across the ideal inductor decreases,this inductor,with zero resistance cannot allow for infinite current anymore...and the final current is determined but the resistance as per Ohm's law.

At tzero, the resistor drops no voltage, because the current is still zero, and the total source voltage is applied to the inductor, and L*di/dt is exactly equal to the applied voltage; the slope of the current is initially the same as it is when there is no resistor. But the slope immediately begins decreasing, and the current waveform in this case is the usual exponential approaching a limit.

This decrease in slope is due to the limiting effect of the reseistance..right?

Rather than trying to make sense of the point of view that the "back EMF" is opposing the applied voltage and therefore determining the inductor current, it's better to take the current as the thing which determines the "back EMF".

We could put it like this,perhaps...the applied voltage causes the current to increase from zero independantly (meaning the role of applied voltage is not affected by anything else)...and when this current reaches the inductor,it produces a magnetic field...at the same time,the current has a certain instantaneous rate of change (di/dt),which determines the back emf of the inductor...cause and effect sort of thing.

 

[Urmi Roy]

Tell me one thing,In the equation for an ideal inductor woth no resistance,

V is proportional to the rate of change of current.

In a circuit with only a resistance,

the applied voltage is proportional to the current produced...

the fact that the voltage is proportional to the rate of change of current...is it due to the fact that no resistance can allow for the current to reach infinite value,so appliying even a constant voltage means that the current is changing all the time.

This could explain the fact that the current at any time is proportional to the integral of V...it would mean that the current at the moment is the result of the total work done by the voltage source over a period of time.

In other words,in an inductor,work is done to increase the current and not to overcome the resistance (as in a resistance)...and the current established at any time is the result of the total work done over some time.

 

[The Electrician]

...the applied voltage causes the current to increase from zero independantly (meaning the role of applied voltage is not affected by anything else)...and when this current reaches the inductor,it produces a magnetic field...at the same time,the current has a certain instantaneous rate of change (di/dt),which determines the back emf of the inductor...cause and effect sort of thing.

Delays in the current flow out of the source are of the order of nanoseconds, and such delays play no part in ordinary network analysis. They are not considered at all in this context.

The applied voltage will not cause any current unless the inductor is connected to it. There's no such thing as a delay until the "...current reaches the inductor...". Current doesn't begin to flow out of the wire connected to the voltage source independently of the inductor, and after a delay, "...reaches the inductor...". Current won't flow out of the source unless and until the inductor is connected to it.

 

[The Electrician]

Tell me one thing,In the equation for an ideal inductor woth no resistance,

V is proportional to the rate of change of current.

In a circuit with only a resistance,

the applied voltage is proportional to the current produced...

the fact that the voltage is proportional to the rate of change of current...is it due to the fact that no resistance can allow for the current to reach infinite value,so appliying even a constant voltage means that the current is changing all the time.

The fact that the current is unbounded with time doesn't necessarily imply that the voltage is proportional to di/dt; it could be proportional to SQRT(di/dt), or some other functional relationship.

The terminal characteristic of an inductor is v(t) = L*di/dt and that implies that if a constant voltage is applied, the current will increase without limit. The fact that the current will increase without limit doesn't imply any particular functional relationship.

This could explain the fact that the current at any time is proportional to the integral of V

As I said, the fact that the current is unbounded with the application of a constant voltage doesn't necessarily imply that v(t) = L*di/dt; it could happen with other functional relationships.

...it would mean that the current at the moment is the result of the total work done by the voltage source over a period of time.

In other words,in an inductor,work is done to increase the current and not to overcome the resistance (as in a resistance)...and the current established at any time is the result of the total work done over some time.

But, as it happens, the current is proportional to the integral of the applied voltage, and this does mean that knowing the current at any time, we can calculate the total work done; it's work = L*i(t)^2/2, energy stored in the inductor.

Everything you need to know about inductors is contained in the terminal characteristic:

v(t) = L*di/dt

Trying to find reasons why this is so is to enter the realm of electronic philosophy, which I generally find not to be helpful.

Better to just accept that v(t) = L*di/dt and learn how circuit behavior follows inevitably from that.

 

[Urmi Roy]

The fact that V proportional to di/dt,in any way, does seem to say that 'the appiled voltage causes a change in current' (putting it simply,without going in too much into the reasons,as you advised)..even if the functional relation had been different,there would still have been a direct relation between the voltage and current,implying that their existence depend directly on each other ...similarly,in a pure resistor,the fact that voltage proportional to current does seem to say that the voltage causes current...if I put it this way,somehow it becomes easy for me to visualise.

If this this is not entirely wrong,I think I might have proceeded to understand inductors,a little bit.

Also,after reading the article in AAC,I formed a better understanding of how ain inductor behaves in a plain dc circuit ...and using what you said,about the current in a purely inductive circuit increasing forever for a fixed voltage to fine-tune the concepts provided by AAC,...I think I could imagine a purely inductive ac circuit as a dc circuit being turned on and off successively (in a fancy sinusoidal way)...do you think this okay?

 

[b.shahvir]

b Shahvir,I read through your post on the other thread...please clarify some points about it...



‘t (time)’ > 0 towards it’s peak value, the rate of change of magnetic flux becomes more pronounced ad the current wave now starts going out of phase w.r.t the voltage wave.
...why does the rate of change of magnetic field become more pronounced as soon as it becomes t>0...like at t+dt?

If you concentrate on the negative half cycle of the voltage vs. current waveform, you will find that the phase shifting is due to the ‘discharge’ of electrical energy by the inductor back to the power source as the magnetic field collapses (or rises, whatever may be the case), which precisely occurs when the input voltage wave has already reversed itself. ...does that mean the discharge of the inductor occurs only when the applied voltage reverses itself?

This charge/discharge action of an inductor occurs due to the finite time taken my the magnetic field to rise or collapse……as force fields cannot change it’s states abruptly at ‘t = 0’ as per the laws of Physics ...PLease explain this in a little more detail...I'm finding it difficult to visualise,even after reading part B of your post.

Dear Urmi,

As I had mentioned earlier, it is quite difficult to visualize working of an inductor as the parameters involved are magnetic field & current flow (unlike capacitors which involve accumulation of charge and voltage!)

It would be easy if you mention (in brief) exactly what factors about inductor behavior makes you uneasy the most...or else the thread would go on till infinity.

In passing i wud again mention, that if a pure sinusoidal voltage is applied to an 'ideal' inductor, there will be absolutely no phase difference or delay between applied volts and current as at t = 0, the current is just starting to flow and would be negligibly small to cause any significant creation of magnetic field. Also at t = 0, both applied volts and current would be immediately in phase as enough magnetic field is not developed to cause 'magnetic inertia.' As time (and hence voltage wave progresses further, the magnitude of voltage wave increases causing more current to flow, which in turn now causes significant increase in magnetic field) This increase in magnetic field creates back emf opposing further flow of source current causing cyclic current charge discharge process(both in positive and negative halves just like in an ideal capacitor)

Now, this cyclic rhythm slowly progresses with time, eventually forcing a complete 90 deg phase shift (delay, in layman terms) between the applied voltage and current waves.

As for part B, I suggest you go thru it once more slowly and you might understand...although I must state that my post might not be scientifically accurate or relevant as this is based on personal way of understanding a theoretical concept. Actually I've attempted to suggest that change of state cannot happen at t = 0, but requires a finite elapse of time i.e at t > 0, etc.

Pls. feel to discuss further if reqd.

Regards,
Shahvir

 

[Urmi Roy]

As I said, I seemed to have reached near a conclusion,especially by comparing what happens in a dc circuit with only inductor while switching on and off (I used the word seems,as I am not sure of anything yet!)...please refer to post 43.

 

[Urmi Roy]

I think I've found an analogy to explain why the inductor's back emf does not cancel the effect of the applied voltage at any instant...suppose we have an elastic band,and we attach one end to a wall...we pull at it and at any instant,the applied force is equal to the resisting force in the band.
However,the work done by the applied force to increase the length of the band is slowly getting stored in the band...which gets released as soon as we withdraw the applied force...in the same way,then the applied voltage causes an increase in current,the back emf is always opposing this,but the applied voltage,in the course of time is doing work in buiding a current...and as soon as we withdraw the applied voltage (reduce it to zero),the accumulated work,in the form of current flows back to us.

Is that okay?

 

[The Electrician]

I think I've found an analogy to explain why the inductor's back emf does not cancel the effect of the applied voltage at any instant...suppose we have an elastic band,and we attach one end to a wall...we pull at it and at any instant,the applied force is equal to the resisting force in the band.
However,the work done by the applied force to increase the length of the band is slowly getting stored in the band...which gets released as soon as we withdraw the applied force...in the same way,then the applied voltage causes an increase in current,the back emf is always opposing this,but the applied voltage,in the course of time is doing work in buiding a current...and as soon as we withdraw the applied voltage (reduce it to zero),the accumulated work,in the form of current flows back to us.

Is that okay?

Mechanical analogs can be helpful in understanding electrical components. There are two standard analog systems. See:

http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Analogs/ElectricalMechanicalAnalogs.html

 

[b.shahvir]

I think I've found an analogy to explain why the inductor's back emf does not cancel the effect of the applied voltage at any instant...suppose we have an elastic band,and we attach one end to a wall...we pull at it and at any instant,the applied force is equal to the resisting force in the band.
However,the work done by the applied force to increase the length of the band is slowly getting stored in the band...which gets released as soon as we withdraw the applied force...in the same way,then the applied voltage causes an increase in current,the back emf is always opposing this,but the applied voltage,in the course of time is doing work in buiding a current...and as soon as we withdraw the applied voltage (reduce it to zero),the accumulated work,in the form of current flows back to us.

Is that okay?

I think this is a good analogy ...but then again, understanding goes as far as the power of an individual's imagination. This being especially true for electrical technology!

 

[Urmi Roy]

I think this is a good analogy ...but then again, understanding goes as far as the power of an individual's imagination. This being especially true for electrical technology!

That's good!

Noone said anything about post 43...

 

[b.shahvir]

The fact that V proportional to di/dt,in any way, does seem to say that 'the appiled voltage causes a change in current' (putting it simply,without going in too much into the reasons,as you advised)..even if the functional relation had been different,there would still have been a direct relation between the voltage and current,implying that their existence depend directly on each other ...similarly,in a pure resistor,the fact that voltage proportional to current does seem to say that the voltage causes current...if I put it this way,somehow it becomes easy for me to visualise.

If this this is not entirely wrong,I think I might have proceeded to understand inductors,a little bit.

Also,after reading the article in AAC,I formed a better understanding of how ain inductor behaves in a plain dc circuit ...and using what you said,about the current in a purely inductive circuit increasing forever for a fixed voltage to fine-tune the concepts provided by AAC,...I think I could imagine a purely inductive ac circuit as a dc circuit being turned on and off successively (in a fancy sinusoidal way)...do you think this okay?

Yes this is OK. However, sinusoids are cyclic waves causing continuous voltage to be impressed across the inductor whenever flow of electrons (current) is persistantly opposed by back emf developed in the inductor during every cyclic variation of voltage, current and magnetic field.

 

[The Electrician]

Also,after reading the article in AAC,I formed a better understanding of how ain inductor behaves in a plain dc circuit ...and using what you said,about the current in a purely inductive circuit increasing forever for a fixed voltage to fine-tune the concepts provided by AAC,...I think I could imagine a purely inductive ac circuit as a dc circuit being turned on and off successively (in a fancy sinusoidal way)...do you think this okay?

I'm not sure what you're asking. You say "I think I could imagine a purely inductive ac circuit as a dc circuit being turned on and off successively (in a fancy sinusoidal way)", and then you ask "do you think this okay?". Do I think it's okay in what sense?

You can imagine any voltage waveform you wish, applied to an inductor, and you can solve the differential equation for the current waveform. I suppose that would be okay, wouldn't it?