Question

A square-law device is one whose output is proportional to the square of the input. A junction field-effect transistor (JFET) (Fig. \(12-13\) ) is such a device. The current \(I\) that will flow through an \(n\) -channel JFET when a voltage \(V\) is applied is $$I=A\left(1-\frac{V}{B}\right)^{2}$$ where \(A\) is the drain saturation current and \(B\) is the gate source pinch-off voltage. (a) Solve this equation for \(V\) (b) A certain JFET has a drain saturation current of \(4.8 \mathrm{mA}\) and a gate source pinch-off voltage of \(-2.5 \mathrm{V} .\) What input voltage is needed to produce a current of \(1.5 \mathrm{mA}\) ?

Short answer

Solve for V: First, isolate the term with V, then take the square root, isolate V, and substitute known values to find V = 2.21 V or V = -2.5 V.

Step-by-step solution

- Isolate the term containing V

To solve the equation for V, start by isolating the term containing V on one side. This will involve moving the other terms to the opposite side. To do this, divide both sides of the equation by A:

- Remove the square by taking the square root

Since the term with V is squared, take the square root of both sides to remove the square. This will give you two potential solutions because the square root has both positive and negative outcomes.

- Solve for V

After taking the square root, isolate V by multiplying both sides by B and then subtracting B from both sides to find the value of V.

- Substitute given values and solve for V

Now substitute the given current I and the constants A and B into the equation derived in Step 3. Then calculate the value of V for the given current I.

Key concepts

Square-Law Device

Understanding the behavior of a square-law device is fundamental in fields such as electronics and signal processing. It relates to an element whose output—to be specific, the output current—is proportional to the square of its input, typically a voltage. In the context of a junction field-effect transistor (JFET), the output current flowing through the device is dependent on the input voltage applied to it in a quadratic manner.

Imagine the JFET as a water pipe; as you adjust the tap (input voltage), the flow of water (output current) doesn't increase linearly, but rather as the square of the voltage change. This relationship is encapsulated in the formula: \[I = A\left(1 - \frac{V}{B}\right)^{2}\] where 'A' represents the maximum current (when voltage V is zero) and 'B' stands for the voltage at which the current becomes zero, known as the pinch-off voltage. This quadratic relationship is akin to throwing a ball upwards, where the height (current) is related to the square of the time (voltage) until it starts falling back down due to gravity.

Drain Saturation Current

The drain saturation current, denoted as 'A' in our JFET equation, is a crucial parameter that defines the maximum achievable current for a particular JFET when the gate-source voltage is zero. Think of it as the widest possible opening of our earlier water pipe analogy—this is the point where the pipe can't allow more water to flow regardless of increases in pressure (voltage).

When the gate-source voltage (V) equals zero, the formula simplifies to \[I = A\] indicating that 'A' is indeed the drain saturation current. This parameter is determined by factors including the physical dimensions of the JFET and the properties of the semiconductor material used. In practice, having knowledge of 'A' helps predict the upper limit of current that the device can handle and is crucial when designing circuits to ensure they operate within safe limits.

Gate Source Pinch-Off Voltage

The gate source pinch-off voltage, symbolized as 'B' in our formula, is another significant characteristic of the JFET. It provides the negative voltage at which the JFET channel is 'pinched off', and the current drops to zero. This voltage is akin to closing the tap on our hypothetical water pipe completely—the point where no more water can flow, regardless of how much force you apply.

This pinch-off phenomenon occurs due to the electric field from the gate voltage, which constricts the conductive channel in the semiconductor, eventually preventing any current from flowing through. The gate source pinch-off voltage is an intrinsic property of the JFET and greatly influences its operation, particularly in its ability to act as an electronic switch or amplifier. Depending on whether a JFET is normally-on or normally-off type, the value of the pinch-off voltage can imply the necessity for a negative or a positive gate voltage to turn off the current flow.

Solving Quadratic Equations

When faced with a quadratic equation like the one derived from the JFET's operation, understanding the process of solving quadratic equations becomes invaluable. A quadratic equation is typically in the form \[ax^{2} + bx + c = 0\] and it usually has two solutions, which arises from the fact that a square term can have both a positive and negative root. These solutions can be real or complex depending on the discriminant \[b^{2} - 4ac\].

To parse any given quadratic equation, it's possible to isolate the variable squared (often 'x' or in our case 'V') by moving other terms to the different side of the equation, followed by extracting the root. However, with quadratic equations, there is always a pair of answers to contemplate—the root operation essentially asks 'what two numbers, when multiplied by themselves, give me this value?'. This pair of solutions, found using methods like completing the square, factoring, or applying the quadratic formula, encapsulates both the positive and negative potential inputs that could lead to the given output.