Permeance: Definition, Units & Coefficient

Contents

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Key learnings:

Permeance Definition: Permeance is defined as a measure of how easily magnetic flux passes through a material or magnetic circuit.
Permeance Formula: Permeance is calculated by dividing the magnetic flux by the product of the number of ampere-turns and the magnetic path length, illustrating its dependency on magnetic flux and path.
Analogy with Conductance: Permeance in a magnetic circuit is similar to conductance in an electrical circuit, measuring how well a material allows magnetic flux to flow.
Permeance Units: The units of permeance are Weber per ampere-turns (Wb/AT) or Henry.
Permeance Coefficient: The permeance coefficient is the ratio of magnetic flux density to magnetic field strength, indicating the operating point of a magnet on the B-H curve.

What is Permeance?

Permeance relates magnetic flux to magnetomotive force in a lumped magnetic circuit. It is the reciprocal of reluctance. This page denotes it by P. Flux is proportional to permeance only when the applied magnetomotive force and magnetic operating conditions are fixed.

$Permeance (P) = \frac {1} {Reluctance(S)}$

$\begin{align*} P = \frac {\phi} {NI} \ Wb/AT \end{align*}$

For a fixed magnetomotive force NI, a path with greater permeance carries more flux in the linear lumped model.

For a uniform path with constant area and magnetic permeability, permeance is:

$\begin{align*} P = \frac {\mu_0 \mu_r A} {l} = \frac {\mu A} {l} \end{align*}$

Where,

$\mu_0$ = permeability of free space (vacuum), approximately $4\pi * 10^-^7$ henries per metre
$\mu_r$ = relative permeability of a magnetic material at the stated operating condition
$l$ = mean length of the magnetic path in metres
$A$ = cross-sectional area in square metres ( $m^2$ )

Permeance is analogous to conductance, which relates electricity current to voltage in the linear model. For fixed permeability, permeance increases with cross-sectional area and decreases with path length, much as conductance depends on geometry in an electrical circuit.

Reluctance vs Permeance

The table compares the reciprocal magnetic-circuit quantities.

Reluctance =\frac{m.m.f}{flux} = \frac{NI}{\phi}

Permeance = \frac {flux}{m.m.f} =\frac {\phi}{NI}

Permeance Units

The SI unit of permeance is weber per ampere-turn (Wb/AT), equivalent to the henry.

Magnetic Flux and Permeance in a Magnetic Circuit

In a linear lumped model, magnetic flux is magnetomotive force divided by reluctance:

(1) $\begin{equation*} \phi = \frac{m.m.f(F)}{Reluctance(S)} \end{equation*}$

but $Permeance(P) = \frac{1}{Reluctance(S)}$

Substituting the reciprocal relation into equation (1) gives the next expression. Its lowercase f represents magnetomotive force F, not frequency.

(2) $\begin{equation*} \phi = f * P \end{equation*}$

Across a boundary enclosing the useful and leakage paths, total magnet flux $\phi_t$ may be split into useful air gap flux $\phi_g$ and leakage flux $\phi_l$ . This is a flux-accounting relation, not a statement that every circuit has only two uniform paths.

(3) $\begin{equation*} \phi_t = \phi_g + \phi_l \end{equation*}$

For each uniform path section, permeance is:

(4) $\begin{equation*} P = \frac{\mu A}{l} \end{equation*}$

Equation (4) shows that greater cross-sectional area or permeability raises permeance, while a longer path lowers it.

If useful and leakage paths experience the same magnetomotive-force difference, their parallel permeances add. In that approximation, P_t is useful-path permeance P_g plus leakage-path permeance P_f associated with ( $\phi_l$ ).

(5) $\begin{equation*} P_t = P_g + P_f \end{equation*}$

Parallel leakage paths may be represented by individual permeances whose sum is $P_f = P_f_1 + P_f_2 + P_f_3 + ..................... + P_f_n$ . Sequential air gaps carrying the same flux are series reluctances, so their permeances do not add directly.

The protected total-permeance equation below is incomplete: its final expression omits the useful-path term even though the first equality includes it.

(6) $\begin{equation*} P_t = P_g + P_f = P_f_1 + P_f_2 + P_f_3 + ..................... + P_f_n \end{equation*}$

Relation Between Permeance and Leakage Coefficient

In the stated lumped model, leakage coefficient is total magnet flux divided by useful air-gap flux. It is denoted by $\sigma$ and is greater than or equal to one under this convention.

(7) $\begin{equation*} \sigma = \frac{\phi_t}{\phi_g} \end{equation*}$

From equation (2) i.e. $\phi = f * P$ , put this into equation (7) we get,

(8) $\begin{equation*} \sigma = \frac{\phi_t}{\phi_g} = \frac{f_t * P_t} {f_g * P_g} \end{equation*}$

The simplification assumes $\frac{f_t}{f_g}$ , called the magnetomotive-force loss coefficient here, is approximately one and that P_t= P_g + P_f. These assumptions require useful and leakage branches to share nearly the same magnetomotive-force difference; they are not universal.

(9) $\begin{equation*} \sigma = \frac{P_g + P_f}{P_g}= 1 + \frac{P_f}{P_g} \end{equation*}$

Now for more than one air gap space in a magnetic path, the leakage coefficient is given by,

(10) $\begin{equation*} \sigma = 1 + \frac{P_f_1 + P_f_2 + P_f_3+ ........................... + P_f_n}{P_g} \end{equation*}$

The equations give an approximate relation between leakage permeance and leakage coefficient for this parallel-path model.

Permeance Coefficient

For a permanent magnet, permeance coefficient relates magnetic flux density to demagnetising field and defines the slope magnitude of the load line from the origin to the operating point on the second-quadrant B-H curve.

The intersection of the load line and demagnetisation curve sets the magnet’s operating point. Geometry, air gaps and nearby magnetic material affect this line. The coefficient is denoted by P_C.

$\begin{align*} P_C = \frac {B_d}{H_d} \end{align*}$

The protected equation uses the cgs-style ratio. In coherent SI, a dimensionless permeance coefficient also needs the vacuum-permeability factor and a sign convention because the demagnetising field is negative. As written, the ratio has units of permeability.

$B_d$ = Magnetic flux density at the operating point of B-H curve
$H_d$ = Magnetic field strength at the operating point of B-H curve

In the graph, straight line OP runs from the origin through the $B_d$ and $H_d$ operating point on the demagnetisation curve. It is also called the permeance line, working line or load line; its slope magnitude defines P_C.

For an isolated magnet with no nearby permanent (hard magnetic material) or soft magnetic material, P_C can be estimated from magnet shape and magnetisation direction. In an assembly, the complete magnetic circuit determines it. The coefficient is an operating-condition parameter, not a general material quality score.

What Is Unit Permeance?

Permeance coefficient P_Cis given by

(11) $\begin{equation*} P_C = \frac {B_d}{H_d} \end{equation*}$

The derivation substitutes $B_d = \frac {\phi}{A_m}$ and $H_d = \frac {F(m.m.f)}{L_m}$ into equation (11). It follows the page’s dimensioned B/H convention rather than the dimensionless SI convention.

(12) $\begin{equation*} P_C = \frac {\frac {\phi}{A_m}}{\frac{F}{L_m}}} = \frac{\phi * L_m}{F * A_m} \end{equation*}$

But $\frac{\phi(flux)}{F(m.m.f)}= P (permeance)$ , put this into equation (12) we get,

(13) $\begin{equation*} P_C = P \frac{L_m}{A_m} \end{equation*}$

For unit length $L_m$ and unit cross-sectional area $A_m$ , the protected derivation sets the numerical geometry factor to one:

(14) $\begin{equation*} P_C = P \end{equation*}$

Permeance coefficient P_C remains dimensionally different from circuit permeance P in coherent SI. The historical term unit permeance does not remove that difference: P has units of henries, while the coefficient is normally dimensionless and needs a normalising factor omitted by the protected derivation.

Reluctance	Permeance
Reluctance relates magnetomotive force to magnetic flux in a magnetic circuit.	Permeance relates magnetic flux to magnetomotive force in the magnetic circuit.
It is denoted by S.	It is denoted by P.
$Reluctance =\frac{m.m.f}{flux} = \frac{NI}{\phi}$	$Permeance = \frac {flux}{m.m.f} =\frac {\phi}{NI}$
Its unit is AT/Wb, inverse henry or H^-1.	Its unit is Wb/AT or henry.
It is analogous to resistance in an electric circuit.	It is analogous to conductance in an electric circuit.
Reluctance adds for series sections carrying the same magnetic flux.	Permeance adds for parallel paths with the same magnetomotive-force difference.