The miniature Stirling cryocooler driven by linear motor does not have fixed amplitude of motion as in the case of crank driven system. The amplitude depends upon the cooler and the motor characteristics. Therefore, it becomes necessary to analyze dynamics of the cryocooler and the motor characteristics coupled together.

Many researchers have described. free piston Stirling cryocoolers and their analysis. Bapat [1994] has done the analysis of free piston free displacer cryocooler analysis with linear motor in the ideal conditions without taking the losses into consideration. The different types of losses have been described by Chhatwal [1992], Choi [2004], Davey [1987] and Orlowska [1987]. No analysis, however, could be found out where the motor and the piston motions are coupled with losses. The present work is the analysis of the FPFD cryocooler with coupled linear motor taking the pressure losses across regenerator and in the bounce space of the cryocooler

Bapat [1994] have described and analysed the linear motor in detail and have calculated the power input and power output of the linear motor as given below in equations 1 and 2.

Input power to the linear motor

Power output from the linear motor

Where.

E₀ Impressed voltage Amplitude (V)

E_s Induced voltage Amplitude (V)

R_T Coil Resistance (Ω)

a_{n, Bn} Constants in the Fourier series hs Active coil height (m)

hg Total coil height (m)

ɸ Motor Phase Shift

The bounce space of the Free piston free displacer Stirling cryocooler has been considered to be at the same mean pressure throughout the cycle and the variation in the pressure due to the movement of the piston in the bounce space are neglected. The variations, however small, can affect the performance of the cryocooler. It acts as a pneumatic spring.

Assuming the piston to be at the mean position, the mean pressure in the bounce space is the equal to the charge pressure, P_av.If the piston is displaced by a small displacement X_p, The pressure in the bounce space change to P_av+ 5P. Where 5P can be positive or negative according to the movement direction from mean position. Assuming the constant temperature in the bounce space, the spring constant can be found out to be

Where, V_b is the volume of the bounce space

This spring acts in unison with the metallic spring. So, the effective spring constant is the sum of both the piston and the gas spring constants and can be taken as

While deriving the expressions of power input cooling power and coefficient of performance, the instantaneous pressure throughout the whole cryocooler had been assumed to be same. As the regenerator causes restriction to the flow, there is a pressure loss across the regenerator, which had not been considered earlier. The loss of pressure across the regenerator can be taken into account in the form of flow loss coefficients, which is defined by Bapat [1994], While calculating the cooling power and coefficient of performance these coefficients were neglected. The flow loss coefficients are F_pc, F_de' F_peand F_dc,

Where, F_pc is defined as the flow loss coefficient in the compression space due to the movement of the piston; F_de is the flow loss coefficient in the expansion space due to the movement of the displacer; F_pe is the flow loss coefficient in the expansion space due to the movement of the piston, and F_dc is the flow loss coefficient in the compression space due to the movement of the displacer. The pressure loss is the product of the coefficient and the velocity of the piston or the displacer. After the consideration of the flow losses the assumptions made in the analysis of the cryocooler are as follows:

The working diagram of FPFD Stirling cryocooler is shown in Fig. 2. The instantaneous pressures in compression and expansion spaces after considering the flow losses across the regenerator can be expressed as follows.

Where,

Z Pressure position coefficient (Nm⁻³)

X_p, X_d Amplitude of piston and displacer

The pressure drop across the regenerator is given by

Where, F_p=(F_pc-F_pc) and F_d=(F_de-F_dc)

F_p and F_d are called flow loss coefficients. It can be assumed that two pressure position coefficients Z_{p &} Z_d and the two flow loss coefficients F_p & F_d are constant i.e. they are independent of the piston and displacer positions X_p & X_d and the phase shift (0) between them.

Refrigeration effect Q_eo is given by

Putting the values of P_e and p_c

The expression for work done or power input to FPFd is given by:

Coefficient of performance is given as

The Fig. 2 shows a schematic diagram of a FPFD split Stirling cryocooler, along with the free body diagram of the piston and displacer. For FPFD split Stirling cryocooler various force acting on the piston are Inertia force pressure force (P_c-P_av), Spring Force (KpXp) and Electro Magnetic force (Bg Iv i) by linear motor.

The governing equation of motion is represented as

Where,

M_p

P _av

P_c

B_g

i

Pressure force

Electro magnetic force

Where,

Now substituting these values in the balance force equation 12 and comparing the coefficients of Sinβ and Cosβ and simplifying, we get

Motorized Displacer dynamics

Various forces acting on the displacer are:

Now,

Pressure force acting on the displacer

Electromagnetic force

Balancing the forces acting on the displacer

Substituting the respective values of the different forces in the balance eqn.21 and comparing the coefficient of sin α & cos α, we get

from equations 22 and 23

From the (25 and 16)

Cross multiplying and simplifying gives

Where,

Here also the motor phase shift (ɸ is independent of applied voltage but is strongly dependent on the resonance conditions. The displacer amplitude X_d is directly proportional to applied voltage Eo’.

Free Displacer dynamics

The equation of motion of displacer is

Various forces acting on the displacer are same as derived earlier for motorized displacer except electromagnetic force.

Substituting this value of the forces in balance force equation 30 and comparing the coefficients of sin α and cos α, we get

Since in this case the displacer is driven by gas force, its amplitude is directly proportional to the piston amplitude and the Cosine of the phase difference.

and

The expression found includes the pressure loss coefficients across the regenerator. If these coefficients are put equal to zero, the expressions are same as obtained without flow loss coefficients by Bapat [1994].

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