In our analysis of the 2D CEB array, we shall use the basic CEB model with a strong electrothermal feedback due to electron cooling[

9,

12,

13] and the concept of series arrays in a current-biased mode[

15,

19]. The operation of a CEB array can be analysed using the heat balance equation for a single CEB, taking into account the power distribution between the

*N* ×

*W* bolometers:

$\begin{array}{l}{P}_{\mathit{cool}}(V,{T}_{e},{T}_{\mathit{ph}})+\Sigma \Lambda ({T}_{e}^{5}-{T}_{\mathit{ph}}^{5})+\frac{{V}^{2}}{{R}_{j}}+{I}^{2}{R}_{a}\\ \phantom{\rule{1.5em}{0ex}}=\left[{P}_{0}+\delta P\left(t\right)\right]/W/N\end{array}$

(1)

Here,

$\Sigma \Lambda ({T}_{e}^{5}-{T}_{\mathit{ph}}^{5})$ is the heat flow from the electron to the phonon subsystems in the absorber,

*Σ* is a material constant,

*Λ* is the volume of the absorber,

*T*_{
e
} and

*T*_{
ph
} are, respectively, the electron and phonon temperatures of the absorber,

${P}_{\mathit{cool}}(V,{T}_{e},{T}_{\mathit{ph}})$ is the cooling power of the SIN tunnel junction,

*R*_{
j
} is the subgap resistance of the tunnel junction,

*R*_{
a
} is the resistance of the absorber, and

*P(t)* is the incoming RF power. We can separate Eq 1 into the time independent term,

$\begin{array}{l}{P}_{\mathit{SIN}0}(V,{T}_{e0},{T}_{\mathrm{ph}})+\Sigma \Lambda ({T}_{e0}^{5}-{T}_{\mathrm{ph}}^{5})\\ \phantom{\rule{1.5em}{0ex}}={P}_{0}/W/N\end{array}$

(2)

and the time dependent term,

$(\partial {P}_{\mathit{SIN}}/\partial T+5\Sigma \Lambda {T}_{e}^{4})\delta T=\delta {P}_{1}$

(3)

The first term in Eq. 3,

${G}_{\mathit{SIN}}=\partial {P}_{\mathit{SIN}}/\partial T$ , is the cooling thermal conductance of the SIN junction that gives the negative electrothermal feedback (ETF); when this term is large, it reduces the temperature response,

*δT* , because the cooling power,

*P*_{SIN}, compensates for the change of signal power in the bolometer. The second term in Eq. 3,

${\mathit{G}}_{\mathit{e}-\mathit{p}\mathit{h}}=5\Sigma \Lambda {\mathit{T}}_{e}^{4}$ , is the electron–phonon thermal conductance of the absorber. From Eq. 2, we can define an effective complex thermal conductance that controls the temperature response of the CEB to the incident signal power:

${G}_{\mathit{eff}}={G}_{\mathit{SIN}}+{G}_{e-ph}$

(4)

In analogy with TES[

5], the effective thermal conductance of the CEB is increased by electron cooling (negative ETF). Here, we assume that the SIN tunnel junctions are current-biased, and the voltage is measured by a JFET amplifier. The responsivity,

*S*_{
V
}, is described by the voltage response to the incoming power:

${S}_{V}=\frac{\delta {V}_{\omega}}{\delta {P}_{\omega}}=\frac{\partial V/\partial T}{{G}_{e-ph}+{G}_{\mathit{SIN}}}$

(5)

In the second term of Eq. 5,

${G}_{\mathit{SIN}}=\frac{\partial {P}_{\mathit{SIN}}}{\partial T}-\frac{\partial {P}_{\mathit{SIN}}}{\partial V}\left(\frac{\partial I}{\partial T}/\frac{\partial I}{\partial V}\right)$

(6)

is the cooling thermal conductance of the SIN junction, which provides some electron cooling and helps to avoid the overheating of the absorber.

Noise properties are characterised by the NEP, which is the sum of three contributions:

$\begin{array}{l}NE{P}_{\mathit{tot}}^{2}=N*W*NE{P}_{e-ph}^{2}+N*W*NE{P}_{\mathit{SIN}}^{2}\\ \phantom{\rule{4em}{0ex}}+NE{P}_{\mathit{AMP}}^{2}.\end{array}$

(7)

Here,

*NEP*_{
e-ph
} is the noise associated with electron–phonon interaction:

$NE{P}_{e-ph}^{2}=10{k}_{B}\Sigma \Lambda ({T}_{e}^{6}+{T}_{\mathit{ph}}^{6})$

(8)

In Eq. 7, NEP

_{SIN} is the noise of the SIN tunnel junctions. The SIN noise has three components: the shot noise,

*2eI/ S*^{
2
}_{
I
}, the fluctuation of the heat flow through the tunnel junctions and the correlation between these two processes:

$NE{P}_{\mathit{SIN}}^{2}=\frac{\delta {I}_{\omega}^{2}}{{\left(\frac{\partial I}{\partial V}Sv\right)}^{2}}+2\frac{<\delta {P}_{\omega}\delta {I}_{\omega}>}{\frac{\partial I}{\partial V}Sv}+\delta {P}_{\omega}^{2}$

(9)

Due to this correlation, the shot noise is increased by 30–50% in contrast to a CEB in voltage-biased mode, where strong anti-correlation decreases the shot noise.

The last term of Eq. 7 depends on the voltage

*δV* and the current

*δI* noise of a JFET

*,* which are expressed in nV Hz

^{-1/2} and pA Hz

^{-1/2}:

$NE{P}_{\mathit{AMP}}^{2}=(\delta {V}^{2}+{(\delta I*(Rd+Ra)/W*N)}^{2})/{({S}_{V}/W)}^{2}$

(10)

Estimations were made for the 7 THz channel of SPICA.