Replacing a time varying inductor with constant inductor: In the transmission line model, transmission line parameters vary as derived below [6, 18] --------- (1).

Where *r*
_{0} is the vessel radius, *E* is the Young's modulus of the vessel wall, *h* is the vessels wall thickness, ρ is the mass density of blood, and Η is the blood viscosity. We notice that the transmission line model has an inductive element that varies with pregnancy time. It is difficult to make an inductance using active elements [10]. So, we removed this variation in inductance by mapping the change in inductance to other circuit elements. This was achieved through some mathematical manipulations, as explained below.

*The transfer function for RLC-circuit implementation of transmission line model is:*

where *I*
_{
out
}is the output current; *I*
_{
in
}is the input current; *R, C* and *L* are transmission line resistance, capacitance and inductance respectively; and *R*
_{
l
}and *C*
_{
l
}are load resistance and capacitance respectively. In order to make the inductance of transmission line independent of pregnancy time, we make the following transformations in the circuit.

*L* → 1; *C* → *LC*; *C*
_{
l
}→ *LC*
_{
l
}; *R* → *R/L*; *R*
_{
l
}→ *R*
_{
l
}/*L*

Note that this does not alter the transfer function of the TL model. Thus, equations for the transformed circuit elements become:

*As, h = 0.1 r*
_{
0
}
*for normal pregnancy, the capacitance becomes independent of r*
_{
0
}
*. This enables us to use a passive capacitor to represent the transmission line capacitance. It may be observed that with this transformation, the transfer function is effectively represented in terms of time constants of the circuit.*

This transformation resulted in an equivalent model for the micro-vascular system. But the disadvantage of this exercise is that the model is no more directly analogous to the biological system. For example, in the modified model, the load capacitance is a function of radius of the vessel as well as the mass density of blood. This has no direct analogy in the biological system. Also, there is no longer a direct analogy between pressure in biological system and voltage in the model. But the resulting simplification in implementation justifies this transformation. Thus, in the modified model, inductance remains constant throughout the pregnancy period. The value of inductance is kept at 10 uH.

### Implementation of circuit elements

#### Transmission line resistance

The transmission line resistance is implemented using channel resistance of an NMOS. Channel resistance has the following expression [17], when the NMOS is in triode region.

Where *k* is gain factor of NMOS, *W* is width of channel, *L* is length of channel, *V*
_{
GS
}is gate voltage, *v*
_{
ds
}is drain voltage, *V*
_{
T
}is threshold voltage of MOSFET.

Assuming low *v*
_{
ds
},

Thus, if the gate voltage were made to vary proportional to pregnancy time,

the resistance of uterine artery would decrease as pregnancy matures. This is as expected, in normal pregnancy. Assuming a linear variation of radius in equation (3) with pregnancy time, we get

where, *t* refers to pregnancy time (expressed as periods of 8 weeks) and the vessel radius is assumed to vary as (*a* + *bt*).

In normal pregnancy, vessel radius varies from 0.14 cm to 0.20 cm in 25 weeks. Thus, *b* is small. Therefore,

(*a* + *bt*)^{2} ≈ *a*^{2} + *2abt* = *A* + *Bt*

*Substituting the above expression in equation. (6), we get*

Comparing equation (5) and (7), we get an expression for V_{GS} in terms of pregnancy time. Thus, the transmission line resistance is directly controlled with a voltage that is in turn proportional to pregnancy time.

#### Transmission line capacitance

A passive capacitance is used to model the capacitance in transmission line. Its value is proportional to density of blood and radius of vessel, and inversely proportional to Elasticity of arterial wall and the thickness of arterial wall.

#### Transmission line inductance

A passive inductance is used to model inductance in transmission line.

#### Load resistance

The load resistance is also implemented using an NMOS. Load resistance decreases as pregnancy matures.

#### Load capacitance

The variation in transmission line inductance is mapped onto the load capacitance. Load capacitance now varies as, it is implemented using a bank of capacitors, consisting of passive capacitance in parallel, with MOS switches connecting them (Fig. 10). The control voltage decides the gate voltage of MOSFETs. This gate voltage, in turn, decides the number of capacitors that add in parallel in the circuit. Thus, a voltage-controlled capacitance is obtained. Since load capacitance decreases with time, therefore the control voltage for this capacitive bank is decreased with pregnancy time. Also, since the decrease in capacitance is inverse square w.r.t. Radius of the vessel, and the control voltage is taken to be proportional to pregnancy time; therefore, the passive capacitance values decrease inverse-squarely in the bank.

#### Matching resistance

Variation in matching resistance is neglected.

#### Matching capacitance

Matching capacitance is also implemented using a bank of capacitors. This capacitance varies linearly with pregnancy time.

#### Matching Impedance (C_{m}, R_{m})

The matching impedance is needed at source to absorb the waves that come after reflection from the load.