In a hot wire circuit breaker a wire of length x₀ at reference temperature T₀ will expand by length Δx=x₀a_tΔT for temperature rise ΔT.  This temperature rise, as in a fuse element, can be generated by the I²R loss in the wire itself. 

If a given wire extension Δx_trip trips a latch mechanism, which releases a contact separation mechanism (see Figure 3.13), the needed temperature rise within the wire at the detection threshold is then

We now have a situation exactly analogous to that of the detection mechanism in a fuse.  The exception is that we need only raise the element temperature to the value given by (3.8), not to the melting point.  The thermal circuit analysis Equations 3.1 through 3.4, with suitable notation changes, also apply to the hot wire thermal circuit

We define

A_r = the wire temperature coefficient of resistivity,

R_o = the wire resistance at the reference temperature,

R_t  = the thermal resistance of the wire to the surroundings,

I_th = the steady-state DC current which will raise the temperature by ΔT_th (i.e. the long trip time current),

k  = the wire resistance temperature sensitivity factor = a_rI_th²R_oR_t  and C_t = the wire thermal capacity.

Thus, just as in Equation 3.4 for a fuse, we have for the hot wire temperature rise ΔT, and for any current I (t)

where is the wire thermal time constant

As an example, consider the use of an 80% Nickel – 20% Chromium alloy wire, which has the trade name Nichrome V.  This alloy has the average properties:

Reference resistivity:  ϱ_o = 108 x 10^-6 Ω-cm (20^o)

Coefficient of resistivity:  α_r = 1.1 x 10^-4/⁰C (20-500^o C)

Coefficient of expansion:  α_t = 1.7 x 10^-5/^oC (10-1000^oC)

Specific heat:  C_p = 435 J/kg^oC.

Assume that the relative extension of the wire at the trip threshold Δx_trip/x_o is to be 0.5%.  Thus, by (3.8)

If we choose #30 wire (American wire gauge) we have the following data from the manufacturer:

Wire diameter:  0.010 in

Resistance at reference temp:  R_o = 6.500 Ω/ft(20^oC)

Mass:  2.86 x 10^-4 lbm/ft

Long time heating current for horizontal wire in air:

The wire thermal capacity per foot is then

If we use the free air heating-temperature data at I (RMS) = 1.21 A, we have for the wire resistance  R

Thus

The free air thermal time constant is then

The long time threshold current is

And the wire resistance temperature sensitivity factor k is

We can now, as we did for the equivalent fuse link, solve for the time it takes for the wire to reach its threshold temperature as a function of current.  These solutions are given in Figure 3.14.

Note that for Nichrome V the resistance temperature sensitivity factor is so low that we can, as a fair approximation, neglect the k multiplied term in (3.9) and solve the resulting equation analytically as

The time to reach the threshold temperature rise t_d is then given by

Which, for long (i/I_th)², reduces to