Let P, R, V, I be the time-dependent Joule power, resistance, voltage, and current of the TES. Let Pq, Rq, Vq, Iq be the quiescent values.

Define a signal power and signal current

Ps = Pq - P, Is = Iq - I

These are the magnitudes of the deviations from the quiescent values during a pulse. So Ps is the reduction in Joule power corresponding to incident power from phonons and Is is the measured current over baseline. We wish to express Ps in terms of Is and the assumed known constants Rq (TES biased resistance), Rsh (shunt resistance), and Ib (bias current).

P = VI = Rsh(Ib-I)I  (because the TES voltage can be determined by the current through the shunt resistor)
Pq = Rsh(Ib-Iq)Iq

Subtract these to get signal power

Ps = Rsh( IbIq-Iq^2 - IbI + I^2 )

Express I in terms of Is, I = Iq-Is, substitute above and simplify

Ps = Rsh( IbIs - 2IqIs + Is^2 )

The quiescent current Iq is known because the Rsh and Rq form a current divider

Iq = IbRsh/(Rq+Rsh)

Substitute this into the power expression

Ps = RshIbIs( 1 - 2Rsh/(Rq+Rsh) + Is/Ib)

The total energy can be found by integrating the above over the time of the pulse. Is is the time-varying quantity, the other quantities are constants.

Let P, R, V, I be the time-dependent Joule power, resistance, voltage, and current of the TES. Let Pq, Rq, Vq, Iq be the quiescent values.

Define a signal power and signal current

Ps = Pq - P, Is = Iq - I

These are the magnitudes of the deviations from the quiescent values during a pulse. So Ps is the reduction in Joule power corresponding to incident power from phonons and Is is the measured current over baseline. We wish to express Ps in terms of Is and the assumed known constants Rq (TES biased resistance), Rsh (shunt resistance), Rp (the parasitic resistance in series with the TES), and Ib (bias current).

P = VI = [Rsh(Ib-I)-RpI]I  (because the TES voltage can be determined by the current through the shunt resistor)
Pq = [Rsh(Ib-Iq)-RpIq]Iq

Subtract these to get signal power

Ps = Rsh( IbIq - Iq^2 - IbI + I^2 ) + Rp(-Iq^2 + I^2)

Express I in terms of Is, I = Iq-Is, substitute above and simplify

Ps = Rsh( IbIs - 2IqIs + Is^2 ) + Rp( -2IqIs + Is^2 )

The quiescent current Iq is known because the Rsh and Rq and Rp form a current divider

Iq = IbRsh/(Rq+Rsh+Rp)

Substitute this into the power expression

Ps = RshIbIs[ 1 - 2(Rsh+Rp)/(Rq+Rsh+Rp) + (1 + Rp/Rsh)Is/Ib ]

The total energy can be found by integrating the above over the time of the pulse. Is is the time-varying quantity, the other quantities are constants.