LR Time Constants

11/20/2001

Rising Current through the Inductor

• To begin with, taking advantage of "a picture is worth a thousand words", let's see just how easy it is to make a basic chart that will show us how this all works. Sure, I could give you a chart all made up, but you will be able to learn and understand all this a lot easier if you do it as we go - believe me! Pull up and print TC_Grid (54Kb PDF file), and then we'll see just how easy it is to plot and use important information for those Time-Constants.
• As we start to put information points on this chart, here is what they represent. We have an inductive element that is placed in series with a resistance that is in series with a voltage source.
• What we will discover is that the current increases through the inductor at a non-uniform rate, and that more specifically, there are points in time (called Time Constants) where the rate of current through the inductor will increase as time goes on. Theoretically, the current through the inductor really never reaches the full amount allowed by the source and the series resistance combination!
• Let's assume a source of 100V DC and a series resistance of 1K ohms, which would give us a maximum current of 100ma .
• What we will find is that at the 1st Time Constant, the current through the inductor will increase to 63.2% of the maximum current provided by the source, and limited by the series resistance. Note that the remaining current, which is the difference between that flowing through the inductor at this instant and the maximum current, is 36.8ma (100ma-63.2ma).

Note: The exact percentage change is a little above 63.2%, as actually 63.21206%, which accounts for the apparent number discrepancies you will find below. To plot the correct values use the RED values shown.

• Next we will find that at the 2nd Time Constant, the current through the inductor will increase by 63.2% of the remaining current (that 36.8ma), and above the existing current value(63.2ma). We find that this is therefore 63.2% of that 36.8ma, which will give us 23.26ma to add on to the currently existing 63.2ma to give us a new current through the inductor of 86.7ma . We would calculate our remaining current as 13.5ma
• Next we will find that at the 3d Time Constant, the current through the inductor will increase by 63.2% of the remaining current (that 13.5ma), and above the existing current of (86.7ma). We find that this is therefore 63.2% of that 13.5ma, which will give us 8.5ma to add on to the currently existing 86.7ma to give us a new current through the inductor of 95.0ma . We would calculate our remaining current as 4.98ma
• Next we will find that at the 4th Time Constant, the current through the inductor will increase by 63.2% of the remaining current (that 4.98ma), and above the existing current value (95.0ma). We find that this is therefore 63.2% of that 4.98ma, which will give us 3.14ma to add on to the currently existing 95.0ma to give us a new inductive current value of 98.14ma . We would calculate our remaining current as 1.83ma
• Finally we will find that at the 5th Time Constant, the current through the inductor will increase by 63.2% of the remaining current (that 1.83ma), and above the existing inductive current value (98.14ma). We find that this is therefore 63.2% of that 1.83ma, which will give us 1.16ma to add on to the currently existing 98.14ma to give us a new current though the inductor of 99.33ma . We would calculate our remaining current as 0.67ma
• Now, if you happen to have a "French Curve" or such variable curve drawing tool (actually, you could just about free hand it), you can draw in an approximation of this curve, at least close enough to show the ideas that we want to present.

These facts should be apparent:

1. The current through the inductor will get closer and closer to the 100% mark, but as you can see: 63.2% of what's left, and 63.2% of what's left .....never really gets there. Folks simply tell you that after 10 of these Time-Constant time frames, who will actually know just how close we might be. It's like being told to go half way to the wall, and then half way to the wall, and half-way to the wall ... when will your nose touch the wall??
2. The growing current through the inductor is definitely not linear with time.
3. Note especially that the current increase at the very beginning is very rapid, and then gradually slows down as the current begins to reach its maximum value.
4. Notice that nothing has been said about how much time is involved! That is because this principle is consistent, regardless of the actual time involved.
5. Now is when it's proper to say that the element of time is related to the value of the series resistance and the value of inductance as a simple quotient of the two. I.e. an inductance of 1,000 millihenrys divided by a resistance of 1Kohms will give us a quotient of 1.0, which represents a time factor of simply 1 second per Time-Constant.
6. With these values in mind and looking at the chart we just drew out, that simply means that in 1 second the current through the inductance will reach 63.2% of the maximum current allowed by the source and series resistance, and that in about 3 seconds it will reach 95% of the maximum current. Simple after all, isn't it?

Falling Opposing Voltage (CEMF) of the Inductor:

• An equally important consideration is that as the current through the inductor rises, the opposing voltage will diminish in the same logarithmic fashion, following the same curve, except that it is inverted and falls. This opposing voltage, created across the inductor, is called "Counter-Electromotive-Force" (CEMF), which starts out high and then diminishes (rapidly at first).
• To help understand a couple of key issues here, we need to grasp an often overlooked feature. Where it's easy to see that the current through the inductor is initially zero when we begin, most of us fail to realize that this same inductor is initially an open circuit. To have an open circuit of course, means that the inductor must create an opposing voltage equal and opposite to that of the source.
• As time progresses, and the current though the inductor rises toward the 100% mark, the opposing voltage diminishes toward 0% from 100%.

Inductive Reactance:

• Inductors do not like Current Changes, and they "REACT" by "creating opposing voltages to the change across the inductor" when the current through the inductor is increased, and then the inductor "gives back" the energy when the current through the inductor is decreased, to attempt maintaining the current at a constant value.
• When you consider this basic feature of the inductor to absorb or give back this energy, remember that the "time element" is a key factor. We saw where the initial changes were those that had the most impact on current changes.
• The rapidity of changes will involve this "time element" by appearing as lots of initial changes in a short time frame, in that the faster these occur, the greater the energy changes within that time frame (remember how the greatest changes occured at the beginning).
• Considering that the more often these events re-occur, the greater will be the average current flow of the inductor. This translates to an apparent "Higher Ohms", as a resistor of higher ohms would inhibit more current from a voltage source.
• The formula for Inductive Reactance X_L=2*pi*fr*L) shows that when the frequency (fr) is increased, the X_L (expressed in ohms) will increase. In the same respect, increasing the amount of inductance for any given frequency will cause X_L to increase in ohms ("pi" is ~ 3.14159).
• Note that although we are using a set of curves (one for the rising current and the other for the falling voltage), a facet of truth is that we can look at a small segment of a curve as a slope.
• This slope can be related to frequency, in that low frequencies have a low rising slope, and high frequencies have a high rising slope.
• Compare with Capacitive Reactance.

Phase Angles

• When we realize that with these two curves, where the increasing current starts at 0ma, and the opposing voltage (CEMF) starts at the maximum value, this is equivalent to a current sine wave and a voltage sine wave that are 90° out of phase with each other. At the instant that the voltage sine wave is at it's maximum voltage, the current sine wave is at it's minimum value.
• With that same concept, with these same two curves, where the increasing current has peaked, the opposing voltage (CEMF) has diminished, this is equivalent to a current sine wave and a voltage sine wave that are again 90° out of phase with each other. At the instant that the current sine wave is at it's maximum current, the voltage sine wave is at it's minimum value.
• If we are dealing only with pure inductance, then the resulting phase angles between voltage and current are at the 90° indicated, but if any resistance is involved, the resulting phase angle (for the circuit as a whole) will be altered from the 90° of pure Inductive Reactance to something less than the 90°.
• Compare with Phase Angles for Capacitance

How this all fits in with Impedance

• Impedance here is the result in the Ohms of the Inductive Reactance that may exist, and the combination with any Resistance. Remember that the Inductive Reactance in Ohms is a result of the frequency applied to the amount of Inductance.
• Since there is a phase angle of the current through the inductor vs. the opposing voltage across the inductor of 90°, and the phase angle of the voltage across the resistor and the current through the resistor is always 0°, the combination of these two will result in ohms that are not simple series or parallel computations. The phase angle will always be some value between 0° and 90°.
• This is why, on a Vector Chart, that the values associated with the capacitance are at 90° to those values associated with the resistance.
• This simply means that with any given LR circuit, that the resulting Impedance and Phase Angle will vary with the applied frequency.

Imaginary Numbers:

This description makes you feel like someone is pulling your leg, but here's where the concept comes from:

• When you either apply voltage across a resistance, or shove current though a resistance, it causes a "power loss" as lost energy.
• Suppose though that you press down on a spring, or compress air into a device, you are "storing energy" (neglect for the moment that there is a miniscule amount of loss in this process). This stored energy can be recovered later.
• Now, we need to remember that an inductor "stores energy" in its "Electromagnetic Field". Remember also that voltage and current go hand-in-hand here to accomplish this.
• When you try to increase the current through an inductor, it draws energy from the current (as to increase the magnetic field) to accomplish this. This "stored energy" is in the form of energy stored in the electromagnetic field, and if there is no change, there is no opposing voltage to the change. CEMF Voltage across the inductor would only exist during changes of current. If the inductor is applied to a resistive load, then current would now flow into that resistive load as the electromagnetic field collapses.
• At some time in the past, someone (or a collection of "some ones") decided that where the energy is stored and not lost, the numerical values represented for that action should be represented by "Imaginary Numbers", rather than "Real Numbers" (Real Loss of Energy).

Now, for a real trick:

Special - How we can utilize Constant Current Sources for a special effect.

• Remember our discussions about Constant voltage Sources (Thevinin's Equivalent Circuits), and Constant Current Sources (Norton's Equivalent Circuits)?
• A very interesting phenomenum occurs when we use a constantly changing current source to an inductor, and watch what happens to the CEMF across that inductor as we change the current.
• We will find that if we use a "Linearly" changing current through the inductor we will observe that the voltage will step to a determined value.
• The " Rate of Change " of this current through the inductor will create a proportional CEMF across the inductor depending on the value of the inductance.
• In fact, "One Henry" is defined where a "CEMF" of 1V will be produced by a change of 1A per second through the inductor.
• The reason for bringing this up is because it allows us to think of what's happening across the inductor (as CEMF) in terms of the resulting current change.
• Look at "Special Effects" caused by variable Constant Current Sources

  Return to Introduction to Time Constants