Inductor's Equivalent Circuit Yields True Picture of Losses

Nov. 1, 2002
An inductor's apparent inductance varies with the measurement method and frequency.

Inductors range from RF coils used in communications through switchmode inductors used in power supplies and differential and common mode inductors in EMI filters. As switchmode power supply designs extend their operating frequencies, it is important to understand how an inductor appears to the circuit in which it operates. Also, at RF frequencies an inductor may not present the inductance one might expect. A thorough understanding of an inductor is necessary to achieve the results expected when it is used in a circuit.

The predominant effect on measuring inductance is the change in core permeability as a result of flux level and frequency. Although free space has virtually a linear and constant permeability, all other materials used to provide a low reluctance flux path and increase inductance have a nonlinear permeability. The permeability of the material is different at different flux densities. Since flux density is directly proportional to the voltage across the coil and inversely proportional to the frequency, it can be seen that frequency and voltage can affect the inductance in this manner. Also, as frequencies increase, the eddy currents increase. These currents create a flux that opposes the coil flux and inhibits its penetration into the core material. This increases the reluctance of the flux path resulting in less inductance.

The inductor's equivalent circuit also affects the measured inductance, which occurs in all types of coils. The change of inductance as a result of the equivalent circuit is called apparent inductance (La). The inductance that does not include these changes is called true inductance (Lo).

You can represent core loss in a circuit as a dc resistance. This loss is the result of eddy currents, hystersis loss, and residual losses within the core and is dependent on core material, frequency and core flux level.

Winding dc resistance depends on the wire material, wire gauge, length of wire and temperature. Copper has a resistance of 10.371 (ohms/circular mil)' feet at 20°C. The temperature change in resistance is 0.393% per °C and increases with temperature.

Winding capacitance, referred to as Cd, is the result of the voltage gradient in the coil. This difference in voltage causes an electrostatic field that creates a capacitive effect on the coil. It is sometimes convenient to think of capacity as an electrostatic field just as it is to think of inductance as a magnetic field. For example, in a transformer, the leakage flux opposes changes in the input current. This can be equated or measured as an inductor in series with the transformer because of the circuit effect of the flux that opposes changes in the input current.

Various winding techniques can reduce winding capacitance. Voltage gradient creates an electrostatic field, so separating turns with the greatest voltage will reduce the winding capacity. For example, since the greatest voltage is between the first turn and the last turn of a coil, providing a gap between the start and finish on a toroid, will have a very positive effect on reducing the winding capacity. Also, since the electrostatic field is stored in the dielectric between the windings. The dielectric constant of the material increases the capacity by acting as a simple multiplier. Free space having a dielectric constant of one and varnish, wire insulations, and tapes having a dielectric constant greater than one, increase the winding capacity. This capacitance also causes a loss factor. Due to the loss in the dielectric associated with this capacitance, a loss occurs that becomes part of the loss component of the coil and is included in the equivalent resistance.

The true inductance is a result of the number of turns squared and the permeability of the medium in which the magnetic flux occurs. As with the resistance of a conductor, the reluctance of a magnetic medium is directly proportional to its length and inversely proportional to its cross sectional area. Therefore, the reluctance is:

Reluctance

(1)

The reluctance of this flux path can be dependent on frequency and voltage, which affects the flux level in a nonlinear flux path, resulting in a permeability change in the path.

To investigate how its equivalent circuit affects an inductor we have to see how it behaves in a circuit or as measured on an inductance meter. The equivalent circuit that includes these components is shown in Fig. 1.

The turn-to-turn winding capacitance is summed as Cd and the true inductance is Lo. The apparent inductance (La), is related to these two components by the following relation:

(2)

La = Apparent inductance

Lo = True inductance

F = Frequency in Hz

Cd = Distributed capacitance (sum of all turn-to-turn and winding capacitance)

The apparent inductance will increase with frequency until:

(3)

This defines the self-resonant frequency, Fo, of the inductor. The self-resonant frequency is also related by:

(4)

This is the resonant frequency formula for a tuned circuit. At the self-resonant frequency, the phase angle of the impedance is zero and the impedance it presents to the circuit is a maximum. Above the self-resonant frequency, the tuned coil appears capacitive. In other words, it has a leading phase angle and is no longer an inductor. A current through the inductor would no longer lag the voltage by some degree as it would in an inductor but would lead the voltage. As the frequency is further increased, the phase angle by which the current leads the voltage increases.

At resonance, the impedance of the capacitance is equal to the impedance of the inductance. That is XL = XC. This causes the phase angle to be zero. Below the self-resonant frequency the current predominately is conducted through Lo making the circuit appear inductive. Above the self-resonant frequency the current is predominately conducted through Cd making the circuit appear capacitive.

As seen from Equation (2) for La, inductance increases in proportion to the frequency squared, that is, exponentially. At low frequencies the inductance is fairly flat. For instance, at a frequency of 1/20 the self-resonant frequency the inductance is increased by 0.25%, 1/10 of self-resonant frequency 1%, 1/5 the self-resonant frequency 4% and finally at ½ the self-resonant frequency the inductance is increased 25%. At one-half the self-resonant frequency, the inductance vs. frequency curve has become fairly steep and rises rapidly until it reaches the self-resonant frequency where inductance is undefined and the inductor appears purely resistive.

The equivalent circuit for an inductor can contribute undesirably to the performance of a circuit. If the inductor is used in a filter circuit or as a common mode inductor, the frequencies above the self-resonant frequencies can pass through the inductor because the inductor appears capacitive. Frequencies are shunted through readily and the higher the frequency the less attenuation is achieved. In transformers often used in switch mode power supplies, capacitance of this nature can cause high peak currents during the input transition of the voltage pulse. This current is required to charge the capacity before any work can be done. In resonant circuits, the winding capacitance of the inductor results in less external capacitance needed to resonate the inductor. This is apparent and can be resolved in one of two ways. First, since the apparent inductance is higher, it takes less capacitance to achieve resonance as demonstrated by the resonant frequency formula. Secondly, the distributed capacitance (Cd) is in parallel with the inductor and would be in parallel with the external capacitor. Since capacitors in parallel add, the value needed for the external capacitor is reduced by the value of Cd. Both approaches are equivalent.

DC resistance, core loss, and winding capacitance dissipation of the coil is the loss portion of the circuit. It is entirely resistive and is lumped together as R. These components reduce the Q of the circuit and reduce the impedance and phase angle of the inductor. The impedance angle is arctan XL/R. It can be seen that since Q = XL/R, then Q is the tangent of the phase angle. Also, another term, although more often used with capacitor losses, is dissipation (D). Dissipation is related to Q by D = 1/Q; therefore, dissipation is the cotangent of the angle. These loss factors have no bearing on the inductance as read on an inductance meter. These are the only components of the equivalent circuit that consumes power and can cause component heating.

As mentioned, the loss factors of core loss, dc resistance, and winding Cd dissipation serve to reduce the circuit's Q. In a tuned circuit this broadens the bandwidth. The loss factors also cause coil heating and raise the component's temperature during operation. The dc resistance of the coil causes a voltage drop due to the current through the coil by the simple relation E = IR.

In measuring this inductance we have to examine the device that is doing the measurement and the manner in which it does it. The older conventional method of measuring inductance was with a bridge whose arms are adjusted for a meter null. The inductance was determined by the settings of the bridge arms used to obtain balance. Some styles of bridges were the Owens, the Maxwell and the Hay Bridge. These bridges adjust phase angle and impedance to obtain a null. From this, you can determine the Q and inductance. This is a frequency-limited measurement, but it does provide the inductor's apparent inductance (La). Fig. 2 is a plot of apparent inductance to true inductance.

Most modern inductance measuring devices are meters, not bridges. These inductance meters measure the impedance and phase angle of the inductor. From these measurements, a microprocessor determines the value of the parameters selected. These are multifunction devices and measure inductance, capacitance, dissipation, phase angle, and impedance.

If you attach any component to the meter, different and predictable results can occur. It will read the apparent inductance (La) of an inductor. That is, as the meter frequency increases, the inductance increases as you approach the self-resonant frequency. When the frequency surpasses the self-resonant frequency, the device is no longer an inductor and appears capacitive. This causes a negative reading on the meter because the phase angle is now a negative angle associated with capacitance. The magnitude of this negative reading is related to the product of measured impedance and the sine of its phase angle. This is the normal relation used to determine the inductive reactance when impedance and angle are known.

Inductor measurements with an impedance analyzer, also yields predictable results. As the frequency increases, the impedance increases because the XL increases and also because the apparent inductance increases. The angle increases because XL/R increases. At self-resonance, the impedance is maximum and the phase angle is zero. Beyond self-resonance, the angle becomes negative and the impedance declines. The angle appears negative because the circuit is now capacitive. The impedance declines because XC declines with frequency. Fig. 3 plots inductor impedance vs. frequency.

The inductor is more complex than expected when considering all parameters that constitute the inductor. The inductance and loss parameters of the inductor are dependent on voltage level and frequency. In a real way these parameters alter the results obtained when measuring the inductor and how the inductor behaves in a circuit.

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