The variation of the threshold voltage with the applied bulk-to-source voltage is typically observed by plotting the square root of the drain current as a function of the source-to-drain voltage. This is because the threshold voltage is a key parameter that affects the behavior of a MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor).

Explanation:

Threshold Voltage:
The threshold voltage, denoted as Vth, is the minimum voltage required to create a conducting channel between the source and drain terminals of a MOSFET. It determines whether the transistor is in an "on" state or an "off" state. When Vth is exceeded, the transistor turns on and allows current to flow between the source and drain.

Bulk-to-Source Voltage:
The bulk-to-source voltage, denoted as Vbs, is the voltage difference between the bulk (or substrate) terminal and the source terminal of a MOSFET. It affects the threshold voltage and influences the device's performance.

Drain Current:
The drain current, denoted as Id, is the current flowing from the drain terminal to the source terminal of the MOSFET. It is a crucial parameter that characterizes the transistor's behavior and performance.

Square Root of Drain Current:
By plotting the square root of the drain current (sqrt(Id)) as a function of the source-to-drain voltage (Vds), we can observe the variation of the threshold voltage with the applied bulk-to-source voltage. This is because the square root of the drain current is proportional to the square root of the ratio of the drain current to the threshold voltage (sqrt(Id/Vth)).

In MOSFETs, the threshold voltage is affected by the bulk-to-source voltage, which can modify the carrier concentration in the channel region. As the bulk-to-source voltage changes, the threshold voltage also changes, resulting in different levels of drain current. By plotting the square root of the drain current, the variations in the threshold voltage can be observed more clearly and accurately.

The plot of sqrt(Id) vs. Vds allows us to analyze the impact of the bulk-to-source voltage on the threshold voltage. It helps in understanding the behavior and characteristics of the MOSFET under different operating conditions, which is essential for designing and optimizing circuit performance.