Category Archives: Electrical Engineering

Power inductor loss

  • Copper loss P_{Copper}=P_{DCR}+P_{ACR}
    • DC copper loss caused by the DC Resistance (DCR) of the windings. It increases proportionally to the square of the current.
    • AC copper loss caused by the AC Resistance (ACR). The AC resistance depends on the frequency and is based on
      • skin effect: The higher the frequency the lower the skin depth. As the frequency becomes higher, there is a tendency for the current flow to become concentrated in the area near the conductor surface and for the effective resistance value to increase.
      • proximity effect: The proximity effect increases the AC resistance as well. A current driven wire creates a magnetic field around the wire, which also creates a magnetic field around the wire, which also creates eddy currents.
  • Core loss
    • Hysteresis loss
    • Eddy Current loss. Eddy current loss is proportional to the square of the frequency.

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}.

Ανάλυση συζευγμένων κυκλωμάτων (analysis of coupled coils)

Θεωρούμε τα πηνία τυλιγμένα στον ίδιο πυρήνα.

Επειδή κάθε κύκλωμα περιέχει μια πηγή τάσεως, εκλέγουμε τα ρεύματα βρόχων i1 και i2 ώστε να έχουν ίδια φορά με τις πηγές και γράφουμε τις δύο εξισώσεις βρόχων σύμφωνα με το νόμο τάσεων του Kirchoff

Η πολικότητα των τάσεων αλληλεπαγωγής εξαρτάται απο τη φορά των περιελίξεων. Για να καθορίσουμε τα σωστά πρόσημα στις παραπάνω εξισώσεις εφαρμόζουμε τον κανόνα του δεξιού χεριού σε κάθε πηνίο. Καθορίζονται έτσι οι θετικές φορές των \phi_{1} και \phi_{2} όπως το σχήμα. Αν οι ροές \phi_{1} και \phi_{2} που οφείλονται στα υποτιθέμενα θετικά ρεύματα, έχουν ίδια φορά και άρα προστίθενται, τότε τα πρόσημα των τάσεων αλληλεπαγωγής είναι τα ίδια με τα πρόσημα των τάσεων αυτεπαγωγής. Ξαναγράφοντας τις εξισώσεις με σωστά πρόσημα, όπου οι \phi_{1} και \phi_{2} είναι αντίθετης φοράς έχουμε

Αν υποθέσουμε ότι οι τάσεις των πηγών είναι ημιτονοειδείς συναρτήσεις του χρόνου και ότι βρισκόμαστε στη μόνιμη ημιτονοειδή κατάσταση