Overview
This is the third blog post in the series of fundamental concepts of electric circuits and signals with the Tek TBS1202B oscilloscopeTek TBS1202B oscilloscope. In my previous two posts, I discussed circuit analysis laws and capacitors. In this post, I will show an intuitive and analytical view of inductance and inductors. I will show how energy is stored in inductors and how that energy is released. Then I will analyze how inductors function in steady state conditions. I will be using the new TBS1202B-EDU Oscilloscope from Tektronix to illustrate these concepts.
What is Inductance
Most of us are familiar with the inductance of coils and how it relates to geometry, number of turns, and magnetic permeability of the core material. We also learned that Maxwell equations can be used to solve the magnetic fields of any given structure. However, for all other than very basic geometrical objects, the direct application of Maxwell equations becomes very complicated. So, in practice, software tools using numerical methods are used to solve the magnetic fields.
While these tools can provide very precise results, the overall accuracy depends mainly on setting up the boundary conditions. This becomes significantly complicated and requires many approximations, especially in the power distribution circuits on chip, package, and PCB. Thus, the accuracy of the field solver tools simulations depends primarily on the human factor. Therefore, when using field solver tools, it is important to understand the physical principles governing the magnetic fields and induction in power distribution circuits. This helps us analyze the circuits, make better trade-off decisions, and improve the performance of our designs. So in the following sections, I will present an intuitive approach for the analysis of inductance.
Electric Current Flowing Through a Wire
An electric current flowing through a wire generates circular magnetic field lines around the wire as illustrated in the figure (a) below.
To determine the direction of the magnetic field, we point the right hand thumb towards the current flow and the fingers curl in the positive direction of the magnetic field lines (b). Alternately, we can imagine rotating a screw to advance in the direction of the electric current, and the rotation shows the direction of the magnetic field lines (c).
The magnetic field increases with electric current and decreases rapidly with distance, as shown in the following figure.
Conductive Wire Encircled by Magnetic Field
Let’s see now what happens when magnetic field lines surround a wire, as illustrated in the figure below.
If the magnetic field is constant in time, the electric charge inside the wire is not affected. However, if the magnetic field varies, then the electric charge displaces inside the wire inducing a voltage. Figure (a) above illustrates the charge displacement and the induced voltage, VINDUCED, generated by an increasing transient magnetic field, B.
To determine the polarity of the induced voltage, we relate the charge displacement to an induced current, Ii as shown in part (b) of the figure. We then set the direction of this induced current so that its own generated magnetic field, Bi, opposes the direction of the initial magnetic field B.
Time Varying Current Flowing through a Wire
We have learned so far that an electric current generates a magnetic field, and a time-varying magnetic field surrounding a conductor induces a voltage in the conductor. We will now combine these two effects to understand what happens when a time-varying current, called here excitation current, flows through a wire.
First, the excitation current generates circular magnetic field lines around the wire, and the intensity of the magnetic field is proportional to the intensity of the current. Consequently, a time-varying excitation current generates a time-varying magnetic field, which then produces charge displacement and induced voltage back in the same wire.
If we determine the directions of the magnetic field and the induced voltage, we find out that the induced charge displacement opposes the variation of the excitation current. If the excitation current increases, the induced charge displacement current flows in the opposite direction, subtracting from the excitation current. Similarly, if the excitation current decreases, the induced displacement current flows in the same direction, adding to the excitation current.
The “always opposing” charge displacement current can be seen as a reluctance of the wire to changes in the electric current. This reluctance is measured by the quantity called inductance. The inductance uses the symbol L, and it is measured in Henries (H). Analytically, the inductance is defined as the ratio between the magnetic flux and the current generating that flux:
For a single wire the inductance is referred to as self-inductance.
When a variable current flows through a wire, the self-induced voltage into that wire is equal to:
Self-Inductance of a Straight Wire Segment
Let’s start with a very thin (filamentary) wire, which carries an electric current. The magnetic field B is calculated using the equation:
Let’s now calculate the self-inductance using as the ratio between magnetic flux and electric current. Knowing the magnetic field from the equation above, we can calculate the total flux by integrating over the area through which the magnetic field lines are passing. The self-inductance will be then equal to the total flux divided by the intensity of the electric current. While this seems relatively easy and straightforward, in reality things are more complicated.
First, the magnetic flux cross-section area expands all the way to infinity, and since the magnetic field depends on materials, we will need to know the magnetic permeability of all the materials surrounding the wire.
Second, for thicker non-filamentary wires the current is distributed within the volume of the wire and generates circular magnetic field lines, both inside and outside the wire. While constant and low frequency currents typically flow uniformly within the wire, high frequency currents concentrate closer to the surface due to the skin effect. This changes the distribution of the magnetic field inside the wire, and complicates the calculation of the magnetic flux. Notice that due to the skin effect, the magnetic flux inside the wire depends on frequency, and thus the self-inductance depends also on frequency.
As a result, the integration of magnetic flux often becomes impractical, and thus, in the power distribution analysis we often use approximation formulas, which provide reasonably accurate results. We look at self-inductance as a sum of an internal inductance due to the magnetic field lines inside the wire, and an external inductance due to the magnetic field lines outside the wire. The internal inductance of a cylindrical straight wire segment can be calculated as:
Notice that the internal inductance does not depend on the wire diameter. This formula is valid only at low frequencies, where the electric current is assumed uniformly distributed inside the wire. At high frequencies, due to the skin effect, the current flows closer to the surface. For very high frequencies, we can assume that all the current flows only on the surface and, thus, no magnetic field exists inside the wire. In this case, the internal inductance becomes zero.
The external inductance can be calculated as:
The self-inductance of the wire is the sum of internal and external inductance, LSELF = LINT + LEXT. The self-inductance depends on the wire geometry, magnetic permeability of the wire, and magnetic permeability of the surrounding materials. The table below compares the relative magnetic permeability of some materials.
Notice that the inductance can increase by a factor as large as 10,000, depending on the material of the wire and the materials that surround it.
For non-ferromagnetic wires, the self-inductance is dominated by the external component. Thus, in most applications, the self-inductance can be approximated with only the external component, which makes it independent of frequency. We also need to keep in mind that the inductance of a wire increases with length and decreases with thickness. When the wire or surroundings are made of ferromagnetic materials, the inductance increases significantly.
Wanted and Unwanted Inductors
I like to categorize inductors as “wanted” and “unwanted”. I will start first with unwanted inductors: these are the parasitic inductors of various circuits, for example the power distribution to an integrated circuit. Transient currents generated on the silicon chip flow through the power distribution network on the package and printed circuit board (PCB) and due to the parasitic inductance generate transient voltages following the formula V=LdI/dt. These transient voltages are seen as power supply noise by the circuits connected to the same power/ground rails. So when designing the power distribution network of an integrated circuit, we want to lower the inductance as much as possible.
In other applications inductors are “wanted”, for example in radio frequency (RF) circuits or switching power supplies, inductors are used as part of the active circuits. In these applications we need much higher inductance than we can obtain from a simple wire. One method is to shape the wire geometry in a coil. The figure below shows the magnetic field generated by electric current flowing through a coil.
Notice: all the way to the left, the circular magnetic field line surrounding the straight wire, as we discussed earlier. As current flows through the coils, these circular magnetic field lines overlap through the center of the coil, creating a high intensity magnetic field inside the coil and increasing the inductance. To further increase the inductance, ferromagnetic materials are placed inside the coil. The inductance of a coil can be calculated using the equation:
L=u0*ur*N^2*A/length
L=inductance (Henries (H))
u0=permeability of free space (Wb/A*m)
ur = relative permeability of the material inside the coil
A= cross-section area inside the coil (m^2)
length = length of the coil (m)
Inductors have also parasitic capacitance between coils and series resistance, which makes them RLC resonant circuits at high frequencies.
Energy Stored in an Inductor
Inductors store energy when electric current flows through them. The energy stored can be expressed by the formula:
W = ½*L*I^2
Based on this formula, higher inductance stores more energy and higher current flow stores more energy. So what happens with this energy when the current stops flowing? Let’s say we have a switch that can interrupt the current flow through the circuit. When we turn the switch OFF, the energy stored in the inductor will maintain the current flow at the output of the inductor until all energy dissipates. However, since the electric circuit is open, this current has no place to go. Intuitively, we can think that electric charge is “pushed” into that node of the circuit and accumulates there. This results in an increase of voltage to very high values, much higher than the nominal voltage supply of typical circuits. As an example, electromagnetic relays have this problem when switching. This is why there is a reverse-biased diode mounted in parallel with the coil. When the current through the coil turns off, the energy stored in the coil dissipates through that diode, avoiding high voltages that would typically damage semiconductor devices.
Let’s take a look now at an experiment that stores energy in an inductor and then releases it. The setup is shown in the figure below:
On the breadboard, there is a circuit made of a 100uH inductor in series with a 3.9kOHm resistor and a switch. The schematic is shown, annotated in yellow on the picture. The Vout node is probed by channel 1 (yellow trace) of the Tektronix TBS1202B-EDU oscilloscope. The current through the circuit is probed as voltage drop on the 3.9kOhms resistor using channel 2 (blue trace) of the oscilloscope.
As current flows through the circuit, energy stores in the inductor. Then, when I push the switch button, the flow of electric current is interrupted and the energy stored in the inductor dissipates in the form of electric current pushed into the Vout node. The resulted transient charge accumulation at Vout then dissipates into voltage source node (Vdc=4V node on the schematic) through the resonant RLC equivalent circuit of the inductor, as we can see by the resonant oscillation on the yellow waveform.
I have also captured this experiment in the video below:
Inductors in Steady-State Conditions
Steady-State condition means that the voltages and currents in the circuit have reached their “settled” values (all transient effects when turning ON the circuit or applying a perturbation have been settled). So let’s look at an experiment where a sinusoidal voltage of 1V amplitude is applied to a circuit made of a 47mH inductor in series with a 10kOhm resistor. This setup is shown in the figure below:
The circuit schematic is shown in yellow on the bottom right corner of the picture. The frequency of the sinusoidal voltage is about 2kHz. The waveforms at nodes Vin and Vout are probed with the Tektronix TBS1202B-EDU oscilloscope (Vin = channel 2 – blue trace, and Vout = channel 1 – yellow trace).
Notice that at 2kHz frequency, the amplitude and phase of the output signal, Vout, are almost equal to the amplitude and phase of the input signal. This is because that at low frequencies an inductor acts like a very small resistor in the circuit.
Let’s see now what happens at higher frequencies. In the picture below, I show the same experiment with the frequency of the sinusoidal signal increased to 15.2kHz.
Notice that the amplitude of the output signal, Vout, has decreased to 700mV. In this case, the inductor functions as a reactance which looks similar to a resistance, but it is not— since instead of dissipating energy through heat, the inductor stores it as magnetic field and then releases it.
The reactance, X, can be calculated using the formula: X = 2*pi*f*L, where f is the frequency and L is the inductance. So at 15.2kHz, the reactance of the inductor is equal to X=2*3.14*15,200*47e-3 = 4.48 kOhms.
So Vout can be calculated using the resistor divider formula as Vout = Vin*(10k/(10k+4.48k)=690mV (close to the measured 700mV considering the +/-10% variation on component values from nominal.)
Notice that this circuit functions as a low pass filter, similarly to the low pass RC filter I described in my previous blog post. The frequency of 15.2kHz is equal to the cutoff frequency, defined as fc=R/(2*pi*L).
If we increase the frequency even more, Vout amplitude continue to decrease, as I am showing in the following picture for the frequency of about 200kHz.
Similar functionality to RC high-pass, filters can be obtained if we switch the inductor with the resistor in the circuit above. A significant difference from RC filters is the output impedance, which is very low at low frequencies since the inductor acts like a short circuit.
To conclude, in this blog post I have covered fundamental concepts of inductance, inductors, energy storing, and functionality in steady state conditions. In the following blog post, I will talk about amplifiers and comparators.
Best Wishes,
Cosmin
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