is a lead compensator ifSelect one:a. a = 3, b = 2b. a = -3, b = -1    c. a = 3, b = 1d. a = -1, b = -2

To determine if the given options represent a lead compensator, we need to analyze the form of a lead compensator and the values of $a$ and $b$.

Understanding Lead Compensators

A lead compensator can generally be expressed in the form: $C(s) = K \frac{s + a}{s + b}$ where $K$ is a constant gain, $a$ represents the position of the zero, and $b$ represents the position of the pole. For a compensator to be classified as a lead compensator, the following conditions must hold true:

1. Real Parts: The pole $-b$ must be more negative than the zero $-a$.
2. Magnitude: $b > a$.

Step 1: Break Down the Problem

We will evaluate each set of values $(a, b)$ to see if they abide by the lead compensator conditions.

Step 2: Evaluate Each Option

1. Option A: $a = 3, b = 2$

   • Here, $-a = -3$ and $-b = -2$
   • Condition: $-b > -a \Rightarrow -2 > -3$ (True)

2. Option B: $a = -3, b = -1$

   • Here, $-a = 3$ and $-b = 1$
   • Condition: $-b > -a \Rightarrow 1 > 3$ (False)

3. Option C: $a = 3, b = 1$

   • Here, $-a = -3$ and $-b = -1$
   • Condition: $-b > -a \Rightarrow -1 > -3$ (True)

4. Option D: $a = -1, b = -2$

   • Here, $-a = 1$ and $-b = 2$
   • Condition: $-b > -a \Rightarrow 2 > 1$ (False)

Step 3: Verification

From our evaluations:

• Options A and C satisfy the conditions of lead compensators.
• Options B and D do not.

Final Answer

The choices that represent a lead compensator are:

• Option A: $a = 3, b = 2$
• Option C: $a = 3, b = 1$