First we'll examine a simple circuit consisting of a battery (voltage source), and inductor (a simple coil of wire) and a switch. The assembly is wired together as in the schematic to the left.
The wire will have some resistance R, quite possibly small, but we include it here anyway. When the switch S is closed, current will flow through the inductor L and resistance R. Were it not for the magnetic field generated by the inductor, and the to-be-described interaction of the inductor with that magnetic field, the current I would instantly rise to a value determined just by the battery voltage Vo and the wire resistance R.
I = Vo/R
For a small wire resistance, that current would be extremely high.
But Faraday's law tells us that when a changing magnetic field B passes through a loop of wire with area A and N turns (we'll assume the direction of the magnetic field is pointing perpendicular to the plane of the loop), that an electromotive force EMF will be generated in the wire, and appear across the terminals of the loop. In the present circuit, the loop that "sees" the magnetic field happens to also be the source of the field. This EMF will appear in opposition to the applied battery voltage, so as to nearly exactly cancel the applied voltage Vo.
This EMF will have a value of
EMF = N A dB/dt
and since the value of the magnetic field created by the inductance L is related to the current I through the inductance, the EMF will be
EMF = (1/L) dI/dt
Just after the switch is closed, the EMF developed within the inductor will be the same as the applied voltage Vo and the current I through the inductor will increase linearly at a rate equal to dI/dt = Vo/L.
Because the EMF produced within the inductance is applied in opposition to the applied battery voltage Vo, the EMF produced within the inductance is referred to as Back EMF.
|If we wait for some time T1, the current will have reached a value (Vo/L) x (T1-T0). Energy has been provided by the battery (area under a graph of current x voltage = 1/2 Vo x Vo x (T1-T0)/L), and this energy is stored in the magnetic field (because we have assumed that the resistance of our circuit is zero, and hence resistive heating losses are zero).|
We may add some additional parts to our circuit. We add a second switch S2 and a capacitor C, ignore the voltage drop across R, and we restart the whole process from the beginning.
As before, when the switch S1 is closed, the current through the inductance will increase. No current will flow to the capacitor. Again, at a time T1, we open the switch S1 and immediately close switch S2. This time, rather than the current falling immediately to zero, the EMF generated by the change in current that occurs when the switch S1 opens will be applied across the capacitor C. We'll start off with the capacitor sufficiently charged (to 12 volts) so that point "C" is at zero volts, just as point "B" is when S1 is closed. Now, when S1 is opened and S2 closed, the current that had been flowing through the inductor will continue to flow, instead, into the capacitor C, transferring energy stored in the magnetic field in the inductor into the electric field in the capacitor. The voltage across the capacitor will be
V = Q/C where Q is the charge stored in the capacitor,
but dQ/dt = I, the current flowing in the inductor.
So dV/dT = I/C
and from Faraday's Law the voltage across the inductor will be V = (1/L) dI/dt or dV/dT = (1/L) d2I/dt2
Solving together yields
d2I/dt2 = I /(LC)
whose solution is (starting at a time t = 0 at the switch opening)
V = Vo*(T1-T0)(1/LC )^1/2* sin (wt)
I = Vo*(T1-T0)/L * cos (wt)
where w = 1/sqt(LC),
where we've assumed that the initial current in the inductor is limited by L, not R
and we've made the approximation that most of the initial energy is stored in the inductor .
We plot these below.
|The upper graph
the voltage at point "B" (not actually the voltage across
capacitor) and the lower graph is the inductor
oscillate sinusoidally with a period 2pi*sqt(LC)
At the risk of getting overly consumed by equations, let's see if we can get an intuitive feel for what this all means.
Let's examine the peak voltage
across the capacitor that results from the magnetic field resonantly "discharging" into the capacitor.
First we see that it depends linearly on Vo, the battery voltage. This makes sense because we expect that all of the voltages in the circuit should increase as the supply voltage increases.
Secondly, we see that it depends on how long we've been "charging" the inductor, in other words, how long the switch is closed (T1-T0) at the beginning.
And finally, for a given inductance L, a smaller capacitance C results in a higher peak capacitor voltage.
with a component that does the same thing as "S2
closing when S1 opens"; a simple diode D.
As S1 opens and the voltage at "B" begins to rise rapidly, the diode D will be forward biased and the current that was flowing through L will be diverted through the diode and into the capacitor. When S1 is again closed, and the voltage at "B" is pulled to near zero, the diode will be reverse biased (because C will have charged) and therefore stops conducting, just the same as the behavior of S2 in Figure 6.
with more modifications to the circuit.
We may add an additional coil of wire exactly like the first coil of wire, and wind the two coils in such a way so that the magnetic field created by current through the first coil is completely "enclosed" by the second coil. Then any changes in the magnetic field will be seen by both coils and the same EMF will be induced in both coils. And even more importantly, components added to the second coil will act as if they were attached to the first coil.
However, for the student familiar with some of the ideas associated with a "transformer", in advance, we emphasize that we are NOT making a transformer. It is more accurate to call what we are making a "coupled inductor"; it's main purpose is to store ENERGY during an initial "charging" period, after which the energy will be transfered to a second coil, unlike a transformer whose principle purpose is to transfer POWER in real-time.
So let's modify our circuit diagram with a second coil. Further, let's move the diode and capacitor from the first coil to the second.
Again, when the switch is closed, current builds up in the first coil. The magnetic field builds up, and this is seen by both coils. Within the first coil, a constant back EMF appears leading to a constant, linear increase in the current in the first coil. But the same back EMF is induced in the second coil, too. We have installed the diode onto the second coil in the same way as onto the first coil. So while the switch is closed, just as before, the diode is reverse biased, and no current flows through the diode and into the capacitor. Note the dots on the two coils. These dots indicate the "phase" of the winding. In other words, this indicates that both coils were wound clockwise (for example) and that the terminals with the dots both represent the start of a clockwise winding.
replace the mechanical switch S1 with an electronic switch,
a MOSFET transistor, that can be rapidly turned on and off
circuitry. Additionally, we see that the capacitor
so that the lower terminal is positive with respect to the
terminal: it's often desired to have the output
with respect to a common system "ground" terminal, so we
second coil symbol upside down and create a common ground
connection. Finally, we remove R from the circuit,
would like to minimize this as much as possible. We
circuitry used to turn the MOSFET switch off and on, and
measure the output voltage and to stop the MOSFET switching
output voltage on the capacitor has reached the desired
a conventional power supply used to supply a continuous
duty cycle (the "on" time) of the MOSFET would be adjusted
a constant output voltage under varying load
used as a capacitor charger, the MOSFET is operated at a
cycle until the capacitor is charged, then the MOSFET is
|Details, Details - The Flyback
While the description up to this point provides all of the basic fundamentals of operation, some extra details are helpful. The first detail is that the coils are usually wound onto a ferrite or powdered iron core of high permeability material, configured into the shape of a doughnut or toroid (or a closed rectangular approximation of a toroid), and provided with a small air gap, or slice, removed from the core. The first function of the core material is to localize the magnetic flux created by the current flow in the loop. In essence, this is to insure that the energy stored within the magnetic field will actually be enclosed completely by both coils and can thereby be extracted by both coils. The presence of the magnetically permeable core means that more energy can be stored at a lower inductor current (although the measure of stored energy is proportional to the voltage applied to the inductor times the length of time applied).
The gap in the core is included to prevent the magnetic material from saturating (magnetic field reaching a plateau as the current increases) and this actually means that most of the magnetic energy is stored within this air gap. It is as if the purpose of the magnetic core is to localize the magnetic energy to within a confined space, rather than allowing it to be distributed out beyond the center of the coils.
important consideration is that the use of a core will
reduce, but not
eliminate, the leakage
inductance. This is as if there were an
additional coil of
added in series with the first coil whose magnetic field
energy is not
available for transfer to the second coil when the current
in the first
coil is turned off. So we create an equivalent circuit
elements replacing our coupled inductor: a
Leakage inductance Li, a Magnetizing inductance Lm, and an
transformer of some turns ratio (for example, 1:10) that
"reflects", or "transforms" the voltage across the
inductance to the secondary
(and vice versa) multiplied by the turns ratio.
If the transformer is properly designed, the leakage inductance is minimized, and the majority of the energy is then stored in the magnetic field within the air gap which can be considered to be part of the Magnetizing inductance. However, when the MOSFET switch is turned off at the end of the "charging period", the period in which the current through the primary coil increases, the energy stored in the leakage inductance will not be transferred to the secondary coil and will instead induce a back EMF across the MOSFET switch. Current in the leakage inductance will continue to flow, charging the small capacitance that exists across the MOSFET to a high voltage. Some scheme is needed to prevent this.
"snubber" capacitor and additional diode
"switch" are added so that the leakage inductor current will
flow and charge the snubber capacitor to a lower voltage
to damage the
MOSFET. Afterwards, capacitor is relatively slowly
through the snubber resistor. Therefore, the energy
the leakage inductance is first transferred to the snubber
and then safely dissipated in heating the snubber
resistor, rather than damaging the MOSFET switch.
There are a variety of ways to select the snubber
is a typical application note that describes the procedure.