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 is defined as a measure of how easily magnetic flux can pass through a material or magnetic circuit. It is the reciprocal of reluctance and is directly proportional to the magnetic flux. Permeance is denoted by the letter P.

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

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

From the above equation, we can see that the amount of magnetic flux for a given number of ampere-turns depends on permeance.

In terms of magnetic permeability, permeance is given by

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

Where,

$\mu_0$ = Permeability of free space (vacuum) = $4\pi * 10^-^7$ Henry/meter
$\mu_r$ = Relative permeability of a magnetic material
$l$ = Length of the magnetic path in meter
$A$ = Cross sectional area in square meters ( $m^2$ )

In an electric circuit, conductance measures how well an object conducts electricity. Similarly, permeance measures how well magnetic flux flows in a magnetic circuit. Permeance increases with larger cross-sections and decreases with smaller ones. This concept is similar to how conductance works in electrical circuit.

Reluctance vs Permeance

The differences between reluctance and permeance have been discussed in the table below.

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

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

Permeance Units

The units of permeance are Weber per ampere-turns (Wb/AT) or Henry.

Total Magnetic Flux (ø) and Permeance (P) in a Magnetic Circuit

The magnetic flux is given by

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

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

By using this relation into equation (1) we get,

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

Now, the total magnetic flux i.e. $\phi_t$ for an entire magnetic circuit is the sum of air gap flux i.e. $\phi_g$ and leakage flux i.e. $\phi_l$ .

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

As we know that the permeance for a magnetic circuit is given by

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

From the equation (4), we can say that for greater the cross-section area and permeability, and the shorter the magnetic path length, the greater the permeance (i.e. the smaller the reluctance or magnetic resistance).

Now permeance i.e. P_t for the entire magnetic circuit is the sum of air gap permeance i.e. P_g and leakage permeance i.e. P_f which is caused by leakage magnetic flux ( $\phi_l$ ).

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

When there is more than one air gap space in the magnetic path, the total permeance is expressed as a sum of air gap permeance and the leakage permeance of each magnetic path space i.e. $P_f = P_f_1 + P_f_2 + P_f_3 + ..................... + P_f_n$ .

Therefore, the total permeance is

(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

The leakage coefficient is the ratio of total magnetic flux generated by the magnet in the magnetic circuit to the air gap flux. It is denoted by $\sigma$ .

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

Now in equation (8) the ratio $\frac{f_t}{f_g}$ is the magneto motive force loss coefficient which is close to 1 and P_t= P_g + P_f , Put these into equation (8) we get,

(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 above equation indicates the relationship between permeance and leakage coefficient.

Permeance Coefficient

Permeance coefficient is defined as the ratio of magnetic flux density to magnetic field strength at the operating slope of the B-H curve.

It is used to express the “operating point” or “operating slope” of the magnet on the load line or B-H curve. Thus the permeance coefficient is very useful in designing magnetic circuits. It is denoted by P_C.

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

Where,

$B_d$ = 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 above graph, the straight line OP passing between the origin and the $B_d$ and $H_d$ points on the B-H curve (also called as demagnetization curve) is called the permeance line and the slope of the permeance line is the permeance coefficient P_C.

For the only single magnet, that is when there is no other permanent magnet (hard magnetic material) or soft magnetic material placed nearby, we can calculate the permeance coefficient P_C from the shape and the dimensions of the magnet. Therefore, we can say that the permeance coefficient is a figure of merit for a magnet.

What Is Unit Permeance?

Permeance coefficient P_Cis given by

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

But $B_d = \frac {\phi}{A_m}$ and $H_d = \frac {F(m.m.f)}{L_m}$ put these into equation (11) we get,

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

Now, when the length of the magnet i.e. $L_m$ and cross-section area i.e. $A_m$ is equal to the size of the unit, then in this condition

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

Thus, the permeance coefficient P_C is equivalent to Permeance P. It can be called as unit permeance.

Reluctance	Permeance
Reluctance opposes the production of magnetic flux in a magnetic circuit.	Permeance is a measure of the ease with which magnetic flux can be set up 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 or 1/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 in a series of the magnetic circuit.	Permeance adds in a parallel magnetic circuit.