# how a phase delay occurs in capacitors/inductors with visual images

Capacitors provide a phase delay between the current and voltage. Current leads the voltage by 90 degree. I was taught these only with the equations. But I want visual intuition, what happens in the capacitor that causes phase delay.

Alas, most educators seem to focus on rote memorization of equations, rather than intuition and understanding. The best "intuitive" explanation of capacitors I've seen so far comes from William Beaty. Here is an image from that explanation:

Beaty talks a lot about how things really work, and distinguishes between the Euclidean or "Greek viewpoint" involving memorizing equations; vs. the "Babylonian viewpoint" where concepts are far more important than equations. He tries to get pictures and analogies to give a visual and gut-level understanding of something.( a )

Most such "intuitive" descriptions of electric devices use a hydraulic analogy.

With both capacitors and inductors, energy can be stored "in" the device. Often we rapidly go back and forth between pumping energy into the device and pulling the energy back out of the device.

Whenever we have something in a box that stores something -- energy, rice, water, marbles, etc. -- and whenever we go back and forth between gradually putting something into the box and gradually taking something out of the box -- the peaks (maximum) of the "amount of stuff in the box" always lag the start of putting something into the box. Also, the lowest point in the valleys (minimum) "amount of stuff in the box" always lag the start of taking stuff out of the box. The time when we have the maximum amount of stuff in that box happens at the instant we stop putting stuff into the box, and start taking stuff out of the box.

We add more energy to a capacitor by pushing electrons in one side (and pulling the same number of electrons out the other side). The voltage ("pressure") across the capacitor can be used to calculate how much energy is stored in the capacitor.

We add more energy to an inductor by forcing an electromotive force across its terminals. The flow of electrons ("current") through the inductor can be used to calculate how much energy is stored "in" the inductor.

• Upvote + special thanks for sharing William Beaty's website! The "Babylonian" approach to electronics is exactly what I was looking for. For future readers if the original website goes down, here is an archived copy: archive.fo/9Fgux – akhmed Dec 12 '16 at 1:53

Think in this simple manner:

1. a capacitor is made of two plates and there's no voltage between them at the initial state.

2. if external voltage is applied, there still isn't voltage between the plates.

3. capacitor starts charging to reach the potential of the terminals; doing so it will consume current, initially maximum value, then less and less as it charges to full state.

And here's the picture to help:

The blue waveform is the input step voltage, the black and red ones are the voltage and the current through the capacitor. The formulas are the ones known to you.

From here, on, you can extrapolate with what Curd said, and then further on.

Well, this is the kind of thing everybody should understand in his/her own intuitive way. One of the ways of picturing this delay is this:

The equation describing the capacitor: $${I}_{C}= C\cdot \frac{d}{dt}{U}_{c}$$ So we have a derivative in there. Suppose our capacitor's voltage is a sine wave. What's the derivative of a sine wave? A cosine. And vice versa.

Now, I understand that you already know as much as this. To get more of a feeling of what's actually happening with the current and voltage, try this quite visually pleasing electronic simulator. There are tons of more examples. You can build your own circuit and observe current and voltage in an appealing way, instead of looking at graphs.

The person who made this simulator also has a lot of interesting physics java applications. They help understanding magnetism, accoustics, and many others.

If I've made any mistakes below I hope someone more knowledgeable will correct me, down-vote me or whatever. :)

First check out these two sources for some nice graphs of voltage and current for caps and inductors.

Inductors: http://www.tpub.com/neets/book2/4.htm

Now for a capacitor you charge it up by applying a voltage source in this case your sine wave. Well the rate that a capacitor charges is directly related to the rate of change of the voltage source you apply (see sources below). When you begin to apply your sine wave at T(0) you are at the maximum rate of change and and thus the capacitor is storing charge at the maximum rate (for this applied sine wave anyway). So here you are charging up as fast as you can which means current is screaming into the capacitor. In truth electrons are flowing into one plate and out of the other all while creating an electric field between the two plates. So current flows but it does not flow between physically through the space or dialectric between both plates.

Now as you go along your sine wave the rate of voltage change is decreasing T(1) in diagram so less current is flowing.

When you get to T(2), the zero point there's no rate of change so no current is flowing.

So that's why there's that phase difference.

Now for the inductor. As current flows into an inductor a magnetic field is created around it. The creation of that field is opposed by the space around it, and further the material that is in that space changes the amount of push back it will feel (so think iron cores etc). Now at the beginning of your sine wave (T(0) in the diagram) you are trying to change the current and the inductor is pushing back on you saying no current shall pass. The voltage is trying to push current through which is creating our magnetic field which is getting pushed back on by the space around it, so ultimately the flow of current is held up. Basically the inductor pushes back with a voltage drop that stops the flow of electrons.

Right as the voltage begins to come down in the second quarter of a cycle the field you just created begins to collapse and current flows out of the inductor (T(1) in the diagram). The ammount of current flow increases until the voltage hits the zero mark, here maximum current flows. As you continue along the rate of voltage change increases, and the inductor starts to choke off the flow of current until again you hit your maximum rate of change and there is no flow at all.

Hope that helps, also I used this other site as a reference too: http://www.allaboutcircuits.com/vol_1/chpt_15/1.html It has a nice explanation of inductors.

I don't provide you visual images but a somehow visual description:

As you can see a resistor as a device that "turns" current into voltage (as well as voltage into current) you can see a capacitor as a device that turns charge into voltage.

Charge is current integrated over time.

So if current is flowing through a capacitor you don't see voltage instantaneously but only after an amount of charge has accumulated.

So current through a capacitor is proportional to the rate of change of the voltage, i.e. it is proportional to the steepness of the voltage-vs-time curve.

Given that current and voltage signals are sinus signals yields the phase shift because

cos(wt) = sin(wt + 90°)

and

d/dt sin(wt) = w * cos(wt)

In a very intuitive way, for a capacitor to rise its voltage, it needs to accumulate charge, so there must be some current going on for a while first. Therefore, voltage lags current in time.

Additionally, voltage will rise if current is positive (because charge accumulates), and will decrease (and eventually become negative) when current is negative. Consequently, the voltage peaks happen when the current crosses zero (changes sign), because it flips from incrementing to decrementing, and vice versa. This explains the exactly 90 degree phase.

The inductor behavior can be explained in a very similar way, but reversing the roles. The reasoning would start by saying that for current to build up through an inductor, a voltage would need to be present for a while, therefore current lags voltage in time. Etc.

For real understanding, we should discard the mathematics and focus the physical things.
Think that any capacitor is just a structure made of a plate-pair (with a dielectric material sandwiched between the plates, like l:l). It is not usable unless it is charged i.e. one plate is laden with +ve charge (and other plate is with -ve charge).

During charging, each plate starts getting the respective charges and gradually the plate builds up with more and more charges. The rising of charges is actually the rise of charge concentration, also called rise of potential (Vc).

To gradually rise the charge, the charging supplier has to be more potential than that of capacitor If the supplier is of constant potential (Vs), then gradually the potential difference (Vs-Vc) reduces that results gradual fall in current. Finally Vc attains Vs and the current stops. If we analyse the situation, the charging current is falling with the capacitor potential (Vc) rising.

If the charging supplier is AC source, the supplier potential is gradually rises in the first quarter and falls in the second quarter and so on. During 1st quarter, the capacitor gets charge and gradually attains source voltage. During 2nd quarter, the capacitor discharge back to supplier as the latter's potential is lower than the former. This phenomena repeats in each half-cycle. The charging is considered as the reference phenomena. The charge inflow to capacitor is the first action and the potential build-up is the subsequent result. We call the potential (voltage) lags the current.