Tag Archives: Electromagnetism

Linear and Non-Linear Inductors

An ideal inductor would have zero capacitance and zero resistance.

The Figure below shows a graph of inductive reactance X_{L} versus frequency f. Inductive reactance increases linearly with frequency.

Figure: Inductive reactance X_{L} vs frequency f (ideal inductor)

A real inductor can be modeled by the following elements:

  • a series inductor L
  • a series resistor R_{DC} or R_{S}
  • a parallel capacitor C_{P} or C_{d}: It is the distributed capacitance between the turns of the wire and is derived from the Self Resonant Frequency (f_{o}).
  • a parallel resistor R_{P}: It represents the magnetic core loss of the inductor core.

The figure below shows a real-life impedance vs frequency graph.

Figure: Inductive reactance X_{L} vs frequency f (real inductor)

Self-Resonant Frequency (SRF) or f_{o} in Hz: This is the frequency at which the inductance of the inductor L resonates with the inductor’s distributed capacitance C_{P}. Increasing L or C lowers f_{o}. Decreasing L or C raises f_{o}.

f_{o} = \frac{1}{2 \pi \sqrt{LC}}

At f_{o}

  • the inductor will act as a pure resistor,
  • the input impedance is at its peak,
  • the Quality factor of the inductor is zero,
  • the reactance of the inductor X_{L} is zero,
  • the capacitance is given by C_{P} = \frac {1}{(2 \pi f_{o})^2L_{o}}

At frequencies below f_{o} the reactance is inductive and increases as the frequency increases.

At frequencies above f_{o} the reactance is capacitive and decreases as the frequency increases.

Закон за пълния ток (Ampere’s Law)

Законът обвързва циркулацията на вектора на интензитета на магнитното поле H по произволен затворен контур G с пълния ток I_{\Sigma}, който преминава през ограничената от контура повърхност. За избраната посока на обхождане на G, I_{\Sigma} = i_{1} - i_{2} - i_{k}.

Когато контурът G обхвъща навивките на намотка с N навивки, през които протича ток i, пълният ток I_{\Sigma} = Ni=F_{m}, където величината F_{m}=Ni се нарича магнитодвижещо напрежение (magnetomotive force).

Ако пространството през което минава контурът G се раздели на M участъка, всеки с дължина l_{k}, сечение S_{k}, и магнитна проницаемост \mu_{k}, такива, че във всеки участък интензитета на полето  H_{k} има постоянна стойност, законът за пълния ток приема вида:

С използване на известните връзки между интензитета на магнитното поле H, неговата индукция B и създадения магнитен поток \Phi, магнитодвижещото напрежение на източника F_{m} се представя с израза:

Основна крива на намагнитване

Фиг. B=f(H)

Основна крива на намагнитване – това в зависимосста B_{m}(H_{m}) определена като съвкупност от положенията на върховете на симетричните хистерезисни цикли при различни стойности на B_{m} и H_{m}.

Αυτεπαγωγή L (self-inductance of the coil)

Όταν σ’ ένα κύκλωμα ένα ρεύμα μεταβάλλεται, η μαγνητική ροή που περνάει μέσα από το ίδιο κύκλωμα (coil) μεταβάλλεται και δημιουργείται, επάγεται όπως λέμε στο κύκλωμα μια ΗΕΔ. Υποθέτοντας σταθερή τη μαγνητική διαπερατότητα (permeability), η επαγόμενη ΗΕΔ είναι ανάλογη προς τη μεταβολή του ρεύματος ανά μονάδα χρόνου δηλ. u_{L}=L \frac{di}{dt}

Η σταθερή αναλογίας L καλείται αυτεπαγωγή του κυκλώματος και έχει μονάδα το Henry (H).

Η ΗΕΔ ενός πηνίου με Ν σπείρες είναι u_{L}=N \frac{d \phi}{dt} (Faraday’s law)

όπου N d\phi είναι η μεταβολή της πλεγμένης μαγνητικής ροής N \phi του κυκλώματος. Από τις παραπάνω δύο εξισώσεις έχουμε

L \frac{di}{dt}=N \frac{d \phi}{dt} απ’ όπου L=N\frac{d\phi}{di}  (Ldi=N d\phi)

Αν στη θέση του πυρήνα υπάρχει μόνο αέρας, η ροή και το ρεύμα συνδέονται γραμμικά L=N \frac{\phi}{i}=const.

Magnetic reluctance

Reluctance in magnetism (R_{m}) is analogous to resistance R in electricity. Just as a current travels along the path of least resistance, so the flux of a magnetic field takes the path of least relactance. Iron has very low reluctance.

(Ohm’s law: R= \frac{V}{I})

Δινορεύματα

Με το νόμο του Faraday εξηγούνται οι απώλειες εξαιτίας των δινορευμάτων (eddy currents). Μια χρονικά μεταβαλλόμενη μαγνητική ροή επάγει τάση μέσα στον ίδιο τον πυρήνα, όπως ακριβώς και στο τύλιγμα του. Αυτή η τάση προκαλεί στο εσωτερικό του πυρήνα ρεύματα με μορφή στροβίλων. Τα ρεύματα ονομάζονται δινορεύματα. Καθώς τα δινορεύματα διαρρέουν τον αγώγιμο πυρήνα καταναλώνεται ενέργεια. Η ενέργεια που χάνεται μετατρέπεται σε θερμότητα και θερμαίνει τον πυρήνα.

Αποδεικνύεται ότι το ποσό της ενέργειας που χάνεται εξαιτίας των δινορευμάτων είναι ανάλογο του μήκους της διαδρομής που αυτά διαγράφουν μέσα στον πυρήνα. Γι’ αυτό το λόγο οι σιδηρομαγνητικοί πυρήνες, στους οποίους είναι δυνατό να αναπτυχθούν τέτοια ανεπιθύμητα ρεύματα, χωρίζονται σε λεπτά φύλλα που ονομάζονται δυναμοελάσματα (laminations). Κατόπιν  τα δυναμοελάσματα ενώνονται για να σχηματίσουν το νέο πυρήνα. Ανάμεσα στα δυναμοελάσματα παρεμβάλλονται μονωτικές ρητίνες που περιορίζουν τις διαδρομές των δινορευμάτων. Η παρεμβολή ρητίνων δεν επιδρά σχεδόν καθόλου στις μαγνητικές ικανότητες ενός πυρήνα μια και έχει πολύ μικρό πάχος. Αντίθετα παίζει πολύ σπουδαίο ρόλο στη μείωση των απωλειών εξαιτίας των δινορευμάτων.

Inductor specification

  • Inductance L (\mu H) (tested at a certain test conditions for example 100 KHz, 0.1 V_{rms}, 0 A DC)
  • Inductive tolerance: It is the allowed amount of variation from the nominal value specified by the manufacturer (e.g. ±20%).
  • Rated operating voltage (across inductor)
  • DC Resistance (DCR): The resistance of the inductor winding measured using DC current. The resistance in a component due to the length and diameter of the winding wire used.

R_{DC}=\rho \frac{l}{s}

\rho \ - \ resistivity \ (material \ dependent \ factor), [\Omega m]

l \ - \ length,\ [m]

s \ - \ cross \ section, [m^2]

The DC resistance has a constant value. The bigger the cross section, the lower the DC Resistance (DCR), the lower the copper losses. The smaller the cross section, the higher the DC resistance (DCR), the higher the copper losses.

  • AC Resistance (ACR)
  • Maximum DC current I_{DC}: Maximum DC current is the DC current at which the inductance falls to 90% of its nominal value or until its temperature rise reaches 30 °C.

Figure: Inductance vs DC Bias Load (or DC Bias Characteristic)

DC Bias current relates to a constant current element that is added to the AC signal.

  • Incremental Current Rating: The DC bias current that causes an inductance drop of 5% from the initial zero DC bias inductance value.
  • I_{rms} or RMS current:
    • I_{rms} for a 20°C rise above 25°C ambient temperature
    • I_{rms} for a 40°C rise above 25°C ambient temperature
  • Saturation current I_{SAT}: The DC bias current that causes the inductor to drop by a specified percentage (e.g. 10% or 20%) from its value without current. See Figure Inductance vs DC Bias Load (or DC Bias Characteristic)
  • Q factor or Quality factor: The measure of the relative losses in the inductor.

Q=2 \pi \frac{maximum \ energy \ stored}{energy \ dissipated \ per \ cycle}=2 \pi \frac{\frac{1}{2}LI_{max}^2}{(\frac{I_{max}}{\sqrt{2}})^2R_{e}T}=2 \pi \frac{L}{R_{e} \frac{1}{f}}=\frac{2 \pi f L}{R_{e}}=\frac{X_{L}}{R_{e}}

Quality factor is defined as the ratio of the inductive reactance X_{L} to the effective resistance R_{e}. Both X_{L} and R_{e} are functions of frequency. The test frequency must be given when specifying Q .

  • Self-Resonant Frequency (SRF) or f_{o} in Hz
  • Curie temperature T_{C} (in degrees Celsius): It is the temperature at which the core material start to lose its magnetic properties.
  • Inductance temperature coefficient: The change in inductance per unit temperature change. Measured under zero bias conditions and expressed in parts per million (ppm).
  • Resistance temperature coefficient: The change in DC wire resistance per unit temperature change. Measured at low DC Bias (<1 VDC) and expressed in parts per million (ppm).
  • Magnetic saturation flux density B_{SAT}: At this value of flux density, all magnetic domains within the core are magnetized and aligned.
  • Shielding
    • with shield
    • without shield
  • Electromagnetic interference (EMI): It refers to the magnetic field radiated away from the inductor into space. The magnetic field may cause interference with other magnetically sensitive components.
  • Core material
    • Ferrite cores
    • Iron powder cores
  • Storage temperature range
  • Operating temperature range
    • Ambient temperature range not including self-temperature rise
    • Product temperature range including self-temperature rise. The operating temperature T_{Op} is equal to the ambient temperature T_{Amb} plus component’s self-heating \Delta T. The maximum allowable temperature for an inductor is the maximum ambient temperature plus the maximum temperature rise.
  • Moisture Sensitivity Level (MSL)