In a bi-metal thermal circuit breaker, the elements expand in longitudinal directions, but the major or useful mechanical movement is in a lateral direction to the elements.

 Consider the two blade elements, 1 and 2 shown in Figure 3.15a.

A blade is a body with an elongated, rectangular cross-section, and with overall length L much greater than either the width b or thickness h.  Both blades have equal length L_o at a reference temperature T_o.  But, at temperature rise ΔT = T-T_o, blade 2 is longer than blade 1, L₂>L₁, because the difference in the two coefficients of thermal expansion, α_t2>α_t1.

If we now bond the two blades together while at the reference temperature, to form a single blade of thickness h = h₁ + h₂, we produce a bi-metal blade element see Figure 3.15b).  Now, at temperature rise ΔT, blade 2 will try to expand to length L₂, but will be “held back” by its bond to blade 1.  Blade 1 wants to expand only to length L₁, but it is “pulled” to a length greater than L₁ by its bond to blade 2.  Thus, blade 2 is subjected to an overall longitudinal compressive force P₂, while blade 1 is subjected to an overall tensile force P₁.  Since there is no movement of the total free body when the elevated temperature is held constant, these two forces must be equal and opposite along the length of the total composite blade.  The curvature seen in Figure 3.15b, will be concave (an inside curve) on the blade 1 side and convex (an outside curve) on the blade 2 side.

The formal definition of concave curvature is

We might (correctly) suspect that for small values the concave curvature of the composite blade would be proportional both to the temperature rise ΔT of the blade and to the difference in coefficients of thermal expansion of the two blade materials.  That is,

and K is a proportionality constant which would be a function of the elasticity of each blade material and the thickness of each blade.  Neglecting small end effects, the radius of curvature should be a constant along the length of the composite blade, so that K should not be a function of the original length L_o.  K should also not be a function of the width b, if L_o is several times larger than b, since the major bending will only be along the longitudinal axis.

The original derivation of the proportionality constant K is credited to Villarceau, who published his work in Paris in 1863.  The standard English language derivation of K is the classic 1925 paper of Steven Timoshenko, “Analysis of Bi-Metal Thermostats.”  In this paper Timoshenko not only derived the bending constant K, he also presented for the first time, the theory of “snap action” thermostat behavior. 

Timoshenko has shown that for an amazingly large range in elastic material constants, and for an amazingly large range in blade thicknesses, the bending constant K can be approximated as

Figure 3.16 illustrates that for a free blade, we can them solve for the maximum mid-blade deflection ό as follows:

The curved blade arc angle φ is given by

where ΔL is the composite blade terminal expansion.  But for small deflections, L_o/r is already small so that the term ΔL/r is of second order and can be neglected, thus, φ =~L_o/r.

The cosine of φ/2, by the construction shown in Figure 3.16, is

which for small φ can be approximated by

For a blade built into a fixed wall – a cantilever beam – we see from Figure 3.17 that the blade end deflection is exactly the same as the mid-blade deflection of a free blade of initial length 2L_o.  Thus, for a cantilever blade of initial length L_o, we have for the blade end deflection

Note that for equal lengths, the maximum deflection of a cantilever blade is four times that of the free blade.  We can now combine (3.14) and (3.15) with deflection Equations 3.16 and 3.17 to obtain:

Normally a creep bi-metal blade must not simply deflect - it must also exert a force on a latch release mechanism during a portion of, or during all of, its deflection trajectory. 

From the elementary theory of uniform beam deflections, we have for the configurations of Figure 3.18 that the deflections for a concentrated mechanical force P_m are

where E is the effective Young’s modulus of elasticity for the composite blade.  It follows that a bi-metal blade will exert a concentrated force P_m if it is constrained such that if it was free it would deflect by an amount ό_m.  Thus, to calculate the temperature rise ΔT needed to deflect a blade by an actual distance ό, and to exert force P_m, we use the equivalent deflection ό_eq,

This equivalent deflection would be used in place of the actual deflection ό in Equations 3.18 and 3.19.

Let us subtract from any free deflection ό_f that a blade would travel before exerting any mechanical force on a latch mechanism.  This free travel deflection would be subtracted from the total actual deflection ό, and the total equivalent deflection ό_eq.  We then have

where we can define

an equivalent free deflection while exerting a mechanical force on the latch mechanism and

an actual deflection while exerting a mechanical force on the latch mechanism.  So that

If we assume that the average amount of work, W_m, expended in tripping the latch mechanism is equal to a constant average mechanical force P_m, times the action distance ό’, using (3.18) through (3.21) for both free and cantilever blades, we have

where we differentiate between free and cantilever blades by the factor β, where

and ΔT’ is the temperature rise needed to deflect the blade a total equivalent distance of ό’_eq.  We can simplify (3.24) and place it in the form

where the constant W_k = 9/64 β⁴E (Δα_t)², m = ό_m/ό’_eq and L_obh is the total volume of the blade material.  Thus, for a given latch mechanism which requires energy W_m to trip, and for a given desired operating temperature rise ΔT’, we can determine the most economical blade design - (i.e. the minimum volume design) if we minimize the expression for the total blade volume

with respect to the normalized suppressed mechanical travel m = ό_m/ό’_eq.  By taking the derivative of (3.25) with respect to m, and setting it to equal zero, we find that the optimum design is one with

That is, the suppressed deflection should be equal to the actual deflection when the blade is in the process of exerting a mechanical force.  Any other choice of deflection ratios will result in an increase in the required volume of bi-metallic material, and thus an increase in material cost.  Note that the amount of free travel that the blade undergoes before it begins any mechanical work has no effect on the optimum choice of suppressed blade deflection.  This free travel only influences the total amount of temperature rise that is needed to complete the total equivalent deflection.

As a practical example of a minimum volume design, consider the use of Chace No.2400 bi-metal material in a cantilever blade latch release mechanism.

In our notation, the manufacturer supplied data for this representative material is given in Table 3.1.  Assume that the blade must, after some length of free deflection, push against a latch mechanism with an average force of one once.  And to trip the mechanism the blade must travel 0.05 in. while applying this force.  The work required to trip the latch is then

Further assume that we wish this work to be accomplished by a deflection due to temperature rise ΔT’ = 50^oF.  From Equation 3.25 with W_k = 5.74 x 10^-3 1bf/(in^2oF), the minimum volume of material that is capable of performing this work (i.e. the m = ½ design), is given by

A commonly available thickness value for bi-metal material is 0.008 inches.  If we choose this value for h, and assume an active blade length of ¾ inch, we see that we need a blade width of,

Note that this minimum volume design satisfies our original assumptions of a “blade” device, one with a length several times greater than width or thickness.

The time it takes for a creep bi-metal element to trip a latch release mechanism is the time it takes to heat the element to the required total temperature rise ΔT = ΔT_f,+ΔT’, where ΔT_f, is the temperature rise during any free deflection of the element before it begins any latch release work.  We can safely neglect any inertial effects due to the bi-metal element itself.  The stiffness of bi-metal material insures that the transient or natural mechanical response of the element is completely negligible in comparison to the driven mechanical response for any reasonable level of heat application.  This can be easily seen if one calculates the period of the fundamental frequency of natural vibration of a bi-metal element.  This period is much less than the time needed to raise the temperature of the element to the value dictated by the required total deflection.  Thus the time-current detection period characteristics of a directly heated bi-metal element can be determined exactly like those of a hot-wire element.  That is, we can use Equation 3.9, with suitable values for the constants: threshold current I_th, element resistance temperature sensitivity factor k, and element thermal time constant Ƭ. 

For an indirectly heated bi-metal element, the thermal equivalent circuit is more complicated than the simple two element model.  An indirectly heated element analysis must, at a minimum, involve two thermal time constants – one for the actual heating mechanism and one for the bi-metal element itself.

For a directly heated element we can solve for the time-current response using Equation 3.9 as follows:

1)      1) Choose a desired threshold temperature rise ΔT = ΔT_f + ΔT’.

2)      2) For the particular blade element material, find the resistance temperature factor k from

3)      3) Integrate (3.9) numerically.

As an example, let us solve (3.9) for Chace No. 2400 material.  If we choose a threshold total temperature rise of 200^oF, we have

The actual value of the long time threshold current I_th is best determined experimentally since it depends on the thermal resistance R_t, which is unknown and not easily theoretically determined.  In normalized form, the solution for stepped DC currents for the detection time t_d, the time required to reach T_th, is given in Figure 3.19.

In general, due to the differences in thermal capacity, the thermal time constants for hot wire breakers will be smaller than those of thermal creep-blade breakers.  Thus, the actual detection times, for equal ratios of operating current to threshold current, will be related by

T_d (hot wire)< t_d (creep-blade)

Finally we note that thermal “creep” type breakers must be derated at high ambient temperatures.  However, by special design, such as the addition of a second bi-metal (complimentary) mechanism, thermal breakers can be ambient temperature compensated.  These compensated mechanisms show little variation over their specified operating temperature range.