The theory of snap-action bi-metal elements is a special application of the more general theory of elastic stability of flexible structures.   Many of the principles outlined here are based on Mechanical Products’ published work, The Theory and Practice of Overcurrent Protection (1987). In 1744, in one of the first mathematical discussions of the theory of elastic stability, Euler showed that column structures under axial compressive loading were subject to failure by lateral buckling at certain critical values of the axial load.  In essence, this same theory applies to snap bi-metal elements.

Consider the pre-curved bi-metal blade structure shown on edge in Figure 3.20.

This pre-curved blade has thickness h, width b, and a slight reference temperature upward curvature, such that the mid-span y-axis deflection at reference temperature is ό₀.  The blade is held firm at ends by rigid frame members A and B, restricting any x-axis elongation.  If the higher expansion side of the bi-metal blade is on the inside of the pre-curve, the blade mid-span deflection ό will tend to diminish as the blade temperature is raised due to the bimetal bending movement.  As the blade’s curvature diminishes as the blade’s temperature is raised, there will develop with the blade a longitudinal compressive force P, since the rigid frame will not allow any extension in its x-axis separation distance L₀.   If the pre-curvature is large enough, and the temperature is raised high enough, the blade compressive force P will reach the Euler critical value and the blade will suddenly buckle downward in an attempt to relieve the force, and reach a new post-buckling locus, shown by the dotted line in Figure 3.20b.  This sudden buckling action is the snap we refer to when we describe a bi-metal element as a snap element. Snap-action behavior in bi-metal elements is fundamentally tied to elastic stability and compressive buckling theory (McCleer, 1987). 

As the temperature is raised beyond the buckling temperature, the blade’s downward curvature is simply increased.  There is no further buckling action.  If the temperature of the buckled blade is lowered, however, the downward deflection will diminish, and the compressive force P will start to rise again.  As the temperature is lowered still further, the compressive force P will again reach the Euler critical value, and the blade will suddenly reverse buckle.  This time, however, in the upward direction (i.e. the blade snaps back).

If we were to plot, for the buckling structure of Figure 3.20, the midspan deflection ό and the blade compressive force P versus the blade temperature rise ΔT, while being careful to observe the direction of ΔT change, we would find that the plots would be similar to those shown in Figure 3.21 (a) and (b).

The direction arrows on the ό and P trajectories indicate the direction of ΔT travel.  The loops in the ό and P curves are referred to as hysteresis loops and are commonly found in “thermostat” devices. Hysteresis characteristics are critical to stable switching performance in thermal control and overcurrent protection devices (McCleer, 1987). 

The hysteresis characteristics, like those shown in Figure 3.21, are ideal for an on/off or “bang-bang” control system.  For example, assume that the snap blade structure of Figure 3.20 is to be used as the temperature sensing and control unit for a household furnace.  A switching structure is coupled to the blade such that the furnace is on if the blade is curved upward, and off if the blade has buckled and is curved downward.  It should be clear that the temperature rise at the thermostat location will then oscillate between the buckling rise ΔT_B and the reverse buckling rise ΔT_RB, and will have an average value of ΔT_c – the temperature rise at the center of the hysteresis curve.  The thermostat designer has the job of making the average temperature rise ΔT_c adjustable, and the hysteresis temperature width ΔTB-ΔT_RB a reasonable value. Proper hysteresis control is essential for predictable device operation and prevention of unstable switching behavior (McCleer, 1987).  Shortly, we shall see how these goals are accomplished.

First, let us reconsider the structure of Figure 3.20.  This structure buckles because it can develop enough longitudinal compressive force P.  Are there other structures, or variations of this structure, which can also allow build up of compressive force to the Euler critical value in a bi-metal element?  The answer, of course, is yes.

Consider the flexible frame structure shown in Figure 3.22. 

This assembly differs from the fully rigid structure of Figure 3.20 only in the degree of rigidity in which frame members A and B are held at separation distance L_o.  In the flexible frame, any incremental increase in frame separation ΔL from the original length L_o incurs a spring retention force ϒΔL, where ϒ is the effective spring constant of the frame.  This retention force is then the bi-metal blade longitudinal compressive force.

The compression force P for the blade, including the frame extension ΔL, is now given by

where ℓ_o is the blade arc length at the reference temperature, ℓ is the blade arc length at temperature rise ΔT, and A = bh is the cross-sectional area of the blade.  Since P = ϒΔL, we can factor out the frame extension term and form

Note that the frame flexibility factor, which we will give the symbol q, is always less than or equal to unity.  It is equal to unity for a fully rigid frame.  Thus, if the frame is not too flexible, there is no basic difference between operation of a fully rigid frame and a flexible frame blade buckling structure.   Frame rigidity and compressive force development are key design variables in snap-action thermal device performance (McCleer, 1987). Though, for an equal change in blade arc length with temperature rise, the flexible frame compressive force P will not be as great as that for a rigid frame.

Let us assume that we wish to use a particular bi-metal material in a snap blade design.  Let the material thickness be 0.008 in., and the reference temperature frame separation L_o be 1.0 in.  In a snap blade design, the hysteresis center temperature rise ΔT_c and the hysteresis width temperature difference are to be pre-specified.  Let these quantities by 100^o F each. 

Now, using the material parameters given in Table 3.1, and the equations given in Figure3.23,

we have 

So, that the reference temperature mid-span deflection, due to pre-curvature, must be

Note that once the choice of material and the reference temperature frame separation has been made, the hysteresis center temperature depends solely on the amount of pre-curvature.  Now, since

the critical buckling slope S_B is equal to 0.5.  From Figure 3.23, we then have

so that the required frame flexibility factor is

If the frame in the above example was rigid rather than flexible, the hysteresis width temperature difference could not be pre-specified.  Once the required pre-curvature deflection ό_o has been determined for a rigid frame, we would have an unrealistic result, since the value of Δα_t is certainly not valid over such a temperature range.

and

And, thus

For a rigid frame, if one wishes to pre-specify both ΔT_c and ΔT_B – ΔT_RB, the reference temperature frame separation L_o must also become a variable.  If we stick to specified hysteresis temperature width and hysteresis center temperature, we must have S_B = 0.5.  And thus, a = 2.67.  We would then have to have

Or

The frame separation can then be solved for as

Or

There are several practical implementations of the bi-metal snap structure.  They all differ in the manner in which the blade longitudinal force P is formed.  They all operate, however, by principles similar to those we have developed for the simple compressive frame structure of Figure 3.22. Although implementations vary, the underlying principles of thermal expansion, compressive loading, and elastic instability remain consistent across snap-action device designs (McCleer, 1987). 

The circular disk structure in Figure 3.24 was, perhaps, the first snap bi-metal device.

A snap disk supplies its own longitudinal compressive force P due to its circular symmetric structure.   Snap-disc structures are widely used in thermal switching applications due to their compact geometry and repeatable snap characteristics (McCleer, 1987).  Any lateral deflection in that disk face increases or decreases circumferential tension.  This circumferential force results in a radially directed force, which can induce lateral buckling, just as in the simple blade structure.  A complete analysis of the stability of an isolated bi-metal disk structure has been given by Wittrick et. al.  This theory however, can only be approximately applied to practical disk structures, since real devices support the disk by an arm or frame.  This support destroys the circular symmetry and invalidates the solutions of Wittrick et.al.  Most practical disk devices are designed using the empirical method (i.e. “cut, try and remember”).

The Valverde thermostat structure is a flexible frame blade device which supplies its own longitudinal compressive force P by means of an ingenuous crimped outer frame.  As shown in Figure 3.25, the bi-metal blade is composed of three parallel sub-blades, joined at the ends. 

The structure can easily be formed from a single blade by punching two longitudinal slits, symmetric about the central sub-blade (see Figure 3.25a).  A length shortening crimp is made in the outer two sub-blades which pre-curves the center sub-blade and forms the compressive frame.  Since the structure supplies its own compressive frame, it need only be supported at one end, enabling the entire device to act as a snapping contact arm (see Figure 3.25c).   Any decrease in the center blade deflection, due to bi-metallic thermo-action, increases the longitudinal compression of the center blade.  At a sufficient level of center blade compression, the entire structure snaps to position of opposite curvature.  Since the crimped outer blades also experience lateral thermo-deflection, device analysis for this structure is somewhat more complicated than that for the simple, single blade, compressive frame structure of Figure 3.22.

The final snap structure we shall discuss is the Taylor blade, Figure 3.26.

This device, like the snap disc, develops the needed longitudinal compressive force P by means of blade self-confined quadrature forces.  The structure is held or restrained at one end only.  The free end is pre-curved, in a direction perpendicular to the blade length, such that the center blade – the contact blade – pushes a stationary fixed contact with a positive contact make force.  As the blade temperature is raised, the bi-metallic thermo-deflection increases this contact force.  Compressive buckling force in the longitudinal direction (longitudinal to the blade pre-curvature, but perpendicular to the blade length) is supplied by the blade itself.  Much like the disk structure, any decrease in lateral deflection at the free end is countered by restraint due to the anchored outer legs.  At the critical, or buckling temperature, the free end snaps to an upwardly curved position (See Figure 3.26c), and the center contact blade rapidly moves away from the fixed center contact.

The Valverde and the Taylor blade structure discussed above are near ideal overcurrent switching structures, since the contact mechanical “make” force increases as the operating or switching temperature is approached.  This characteristic is quite important since the making force is countered by an electromagnetic breaking force, which is proportional to the square of the current through the device contacts. Understanding the interaction between thermal, mechanical, and electromagnetic forces is fundamental to reliable overcurrent protection device design (McCleer, 1987). 

Reference:
McCleer, Patrick J., Ph.D., P.E. The Theory and Practice of Overcurrent Protection. Chelsea, MI: BookCrafters, Inc., 1987. 

Our concern, here, is overcurrent protection, so we will restrict our attention to one particular location within a circuit, the location, or potential location of an interruptive device.  We arbitrarily divide the total circuit into two sections:   a delivery, or source, electrical network; and a load electrical network (Figure 2.1).  These two networks are connected by two current paths, one of which contains the interruptive device. 

Many of the principles outlined here are based on Mechanical Products’ published work, The Theory and Practice of Overcurrent Protection (1987).  

If both the source network and the load network are linear – a common approximation which is entirely adequate for fault transient calculations – we can, by Thevenin’s theorem (2.1), replace each network by an equivalent source voltage, E, and equivalent network impedance, Z.  The equivalent source voltage for each network is the voltage which would be measured at the network terminals under open circuit (no load) conditions.  The equivalent network impedance is a mathematical description of the combination of resistances, capacitances and inductances that would be measured at the open circuit network terminals. 

This type of system modeling approach is foundational to circuit protection analysis and transient behavior evaluation (McCleer, 1987).  

The equivalent Thevenin networks for the source and load networks are shown in Figure 2.2.

 The voltages, E₁ and E₂, represent the equivalent voltage sources for the source network and load network, respectively.  The source network voltage, E₁, represents all the combined voltage generators within the source network.  For Example, if the source network is a feed from an electric utility network, then E₁ represents the thousands of generators connected to the utility network, as measured under open circuit conditions at the source network terminals.  But, if the source network is an equivalent for an aircraft 28 volt DC electrical network, E₁ represents the rectified alternator output, as measured under open circuit conditions at the source network terminals. 

The load network voltage source, E₂, represents all of the combined voltage generators within the load network.  Many times there are no voltage sources within the load network, so E₂ is zero.  But, if the load network contains a source, such as the electromotive force (emf) in the windings of a motor, E₂ represents the combined total load network voltage, as measured under open circuit conditions at the load network terminals.

The network impedance elements, Z₁ and Z₂, represent equivalent impedances for the source and load networks, respectively.  The impedance of the interrupting device itself can be arbitrarily lumped in with either the source or the load impedance, Z₁ or Z₂.

If the source network is an equivalent for a 120 volt wall outlet, then Z₁ represents the combined impedances of the thousands of transmission lines, transformers and generator windings which make up the feeding utility network; and the impedances of the local wiring network which connects the outlet.  If the source network is an equivalent for an aircraft network then Z₁ represents only the interior wiring within the airplane and the internal impedances of the alternator, as measured at the network terminals.

In a practical sense, the dominant contributors to Z₁ in a largesource network are only those network elements which are physically close to the source network terminals.  In a large factory electrical network, the dominant terms in Z₁ are the impedances of the wires which connect the source network terminals to the nearest substation transformer, and the impedance of the nearest substation transformer windings.  In smaller networks, such as an automotive network, all impedance terms must be accounted for in Z₁, since they all make significant contributions.

For example, in the simple automotive network shown in Figure 2.3, the internal resistance of the twelve volt automobile battery limits the initial short circuit current at the battery’s terminals to 2000 amps.  But, if the short is at the terminals of the starter motor, the initial short circuit current would be only 1500 amps.  Clearly the additional resistance of the wiring from the battery to the starter motor is responsible for this reduction in potential short circuit current.  The additional resistance is comparable in value to the internal resistance of the battery.

Contrast this case, where the impedances of all network elements are important, to the example of a large factory system.  In a large industrial system, it would make negligible difference to an individual user at a particular work station if one or even ten generators were added to the electric utility system which feeds the factory substation.  The value of a work station’s perceived system impedance, Z₁, would not change to any significant degree.

The equivalent load impedance, Z₂, represents the load network passive elements, as measured from the load network terminals.  Just as in the case of Z₁, only dominant elements need be accounted for when computing system currents. 

The equivalent network of Figure 2.2 can be simplified even further, if we make the following substitutions:

 We then obtain the simplified circuit of Figure 2.4.  Observe that a solution of the electrical network equations for this circuit will give us a solution for the breaker current, i_b.  Also observe that it does not matter whether this circuit represents the steady-state operation of our system, or the operation during a system transient due to some sudden change in the system network.

 We will use the equivalent network of Figure 2.4 to represent the transient operation of an electrical circuit, and from it derive the transient behavior of the breaker current, i_B.  Before the transient, assumed to start at time t=0, the breaker may be passing some value of pre-transient current, i_B0.  We can use this value as a boundary condition in our transient solution.  We can be assured, however, that i_B0 is within the rating of the breaker and/or circuit, since, by definition, we are asserting that only under transient conditions will any rated value of circuit current or voltage be exceeded.

We now give an example of circuit reduction to the form of Figure 2.4.  If we choose a thermal breaker within a plug strip as our study breaker, then the reduced networks – which are equivalent to the single-phase feed from the utility through the plant and laboratory wiring and the power supply load – can be represented as shown in Figure 2.5.

 The source side inductance L_W, and the resistance R_W, represent the inductance and resistance of the wiring from the utility substation transformer to the plug strip, and the inductance and resistance of the utility substation transformer itself.  We will also include with L_W and R_W any inductance and resistance within the breaker mechanism itself.  The source side capacitance, C_W, represents a lumped approximation for the distributed capacitance of the supply wiring to ground, and the transformer windings to ground.  The source voltage, E_S, is a 60 Hz sinusoidal voltage with RMS (root mean square) magnitude, E_m = 120 V, and a phase angle, o.  The power supply load is represented by a wiring resistance, R_c, a series connected on/off switch, S₁, a full wave diode rectifier, and a parallel R₀, C₀ load.

A transient overcurrent could occur any number of ways in this circuit.  Figure 2.6 illustrates two of the most common. A start-up transient is shown in Figure 2.6a.  Here, the on/off switch, S₁, is closed at t = 0, and the load capacitor, C₀, is assumed to be uncharged before t = 0.  Once C₀ is charged, the line current through the breaker settles down to its rated value or below, dependent on the equivalent load resister, R₀.  The circuit in Figure 2.6a neglects the rectifying action of the diode bridge and the load resistor, R₀, during the start-up transient.  Analysis of this particular circuit would be a conservative method of calculating the start-up transient behavior of i_B.  Note that the circuit in Figure 2.6a is of the same form as our general transient equivalent circuit, as given in Figure 2.4.

The second type of transient overcurrent which we will consider for the plug strip and computer power supply circuit is the worst case transient – a permanent short circuit at the input to the computer line cord.  The equivalent circuit for this condition is shown in Figure 2.6b.  Here, the computer load does not enter into the transient calculation, since it is completely shorted out by the equivalent shorting switch, S₂, which closes at t = 0.  Also shorted for all practical purposes, and thus irrelevant to the analysis, is the source shunt capacitance, C_W.  Note again that the resultant circuit is of the same topology as the general transient circuit of Figure 2.4.

Prior to the short, any prefault line current flowing through the breaker in Figure 2.6b is treated as a t = 0 boundary condition for i_B.  The solution for the breaker current is the complete solution, both transient and steady-state, to the differential equation which describes Kirchoff’s voltage law around the E_s-L_W-R_w loop. Namely:

Where w=2f is the system drive radian frequency, with f=60Hz.  The total solution to Equation 2.1, from the elementary theory of differential equations, is

 Where Ƭ is the source inductive time constant, L_W/R_W; Z_w is the inductive impedance at the driving frequency f,

 X_wis the source network inductive reactance, X_W=wL_W;  and the 0 is the angle by which the steady-state short circuit current lags the drive voltage E_s,

 The constant I_B must be determined from the boundary condition for i_B.  Since i_B=i_Bo at t = 0, we have

If we define the symmetrical RMS short circuit current magnitude as

 Then, for the complete time variation of the short circuit breaker current, we have

 In nearly all cases, the magnitude of the steady state short circuit current is much greater than the pre-fault current (which is within the rated value of the circuit).  So we can easily neglect the first term in Equation 2.2.  We then have

Equation 2.3 is plotted in Figures 2.7a through 2.7e, normalized to  , as a function of the product ft.  Note that

 In Figures 2.7a through 2.7e, the system impedance angle, 0, is specified, varying from 5^o to 85^o.  Since the fault would be a random event, the switching angle, φ, could be any value over a 360^o range.  We show individual i_B curves for discrete values of φ, varying from φ=180^o to +180^o in 30^o increments.  These sets of curves for the different values of φ then form waveform envelopes, in which any time variation of i_B will lie.

 Of interest, in the results shown in Figures 2.7a through 2.7e, is the transient bulge in the i_B envelopes for circuits with largely inductive (0>45^o) system impedances.  The more inductive the circuit, the greater the bulge.  This effect is due to the inertial property of the magnetic flux within the system inductance.  Mathematically, it is evidenced in the second term in Equation 2.3.  This term is referred to as the DC offset in the line current.  It decays exponentially, with a time constant equal to L_W/R_W, and has a maximum initial value when the angular difference between the switching angle, φ, and the system impedance angle, 0, is +/- 90^o.

The principal consequence of the DC offset is the potential for fault current levels through the breaker to be almost twice the level of the peak steady-state or sustained fault current level.  This transient amplification effect is a critical consideration in overcurrent protection device selection and rating (McCleer, 1987).  

The phenomenon is seen more clearly in Figure 2.8, where four cycles of an almost fully offset current waveform are shown.  As can be seen, the line current at one half cycle into the fault is approximately 1.75 times the peak fault current after the offset transient has died down.

 In general, low voltage (240/120V) circuits in industrial and household installations have enough wiring between the circuit and the utility system, so that the system X over R ratio is less than unity (i.e. 0<45^o).  Thus, these circuits do not experience offset fault currents. Fault currents, such as those shown in the waveforms of Figure 2.7 and 2.8, must pass non-destructively through the breaker mechanism.  The peak potential current (point A in Figure 2.8) in a fault waveform is termed the peak value of the prospective fault current.  This current value must be less than, or equal to, the interruptive capacity of the breaker (the maximum amount of current a breaker can interrupt, while still able to function if reset). Proper understanding of prospective fault current is fundamental to ensuring safe and reliable protection system performance (McCleer, 1987)  

If the breaker or the fuse is a “current limiting”  type device, then the peak prospective fault current is the peak  fault current that would flow if the limiting device were not present.  The actual peak fault current would be less than the peak prospective fault current, and would depend on the amount of additional circuit impedance that is inserted by an interruptive device mechanism.  Usually, this additional impedance is in the form of an elongated arc, such as that in a fuse or cooled arc chamber.  Thus, the waveforms of Figures 2.7 and 2.8 do not apply to fault currents which are limited by current limiting type breakers.  They apply only to the perspective fault currents for these types of breakers.

If the generalized circuit of Figure 2.4 represents a DC circuit wherein the source voltage is a simple DC voltage with magnitude E₀, the solution for the transient breaker current, i_B, is considerably simpler.  For the loop voltage equation, we have

 The solution for this equation, again from the theory of elementary differential equations, is

 where, as before, i_Bo is the prefault (t=0) breaker current and Ƭ is the source system time constant, L_W/R_W.  The steady state sustained fault current, I_mo, is simply the pure resistive current,

 As in the single-phase AC case, the steady-fault current is generally much larger than the prefault current, i_Bo, so that we can approximate the transient fault current by

This waveform is shown in Figure 2.9, normalized to I_mo.

Note that in this DC case there are no switching angles or timing considerations.  The fault current simply rises exponentially to its steady state value, governed by the system time constant Ƭ.  The peak prospective fault current is the steady-state fault current, I_mo, and there are no offset currents that add to this value.  The complication, if there is any in a DC circuit, is that the fault current is truly a unipolar current with no natural current-zero.  A DC breaker must then force a current-zero in order to interrupt the circuit.

Reference:
McCleer, Patrick J., Ph.D., P.E. The Theory and Practice of Overcurrent Protection. Chelsea, MI: BookCrafters, Inc., 1987.  

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