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 In bilinear transformation, the left-half s-plane is mapped to which of the following in the z-domain?
  • a)
    Entirely outside the unit circle |z|=1
  • b)
    Partially outside the unit circle |z|=1
  • c)
    Partially inside the unit circle |z|=1
  • d)
    Entirely inside the unit circle |z|=1
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
In bilinear transformation, the left-half s-plane is mapped to which o...
Explanation: In bilinear transformation, the z to s transformation is given by the expression
z=[1+(T/2)s]/[1-(T/2)s].
Thus unlike the backward difference method, the left-half s-plane is now mapped entirely inside the unit circle, |z|=1, rather than to a part of it.
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In bilinear transformation, the left-half s-plane is mapped to which o...
The bilinear transformation is a method used to convert continuous-time systems into discrete-time systems. It maps points from the left-half s-plane in the Laplace domain to points in the z-plane in the z-domain. The z-plane is the complex plane in the discrete-time domain.

To understand how the left-half s-plane is mapped to the z-domain, let's consider the transformation equation:

s = (2/T) * ((z-1)/(z+1))

where s is the Laplace variable, z is the z-transform variable, and T is the sampling period.

Mapping of the left-half s-plane to the z-domain:

1. Left-half s-plane:
- In the s-plane, the left-half s-plane consists of points with negative real parts.
- These points are located to the left of the imaginary axis.

2. Bilinear transformation:
- The bilinear transformation maps the s-plane to the z-plane using the equation mentioned above.
- It maps the imaginary axis (s = jω) to the unit circle in the z-plane (|z| = 1).

3. Mapping of left-half s-plane:
- Since the left-half s-plane lies to the left of the imaginary axis, its points will be mapped to the left side of the z-plane.
- The bilinear transformation preserves the order of points, meaning that points that are closer to the imaginary axis in the s-plane will be closer to the unit circle in the z-plane.

4. Mapping to the z-plane:
- The bilinear transformation maps the entire left-half s-plane to the interior of the unit circle (|z| < 1)="" in="" the="" />
- This means that all points in the left-half s-plane will be mapped inside the unit circle in the z-plane.

Therefore, the correct answer is option 'D': Entirely inside the unit circle |z| = 1. All points in the left-half s-plane will be mapped to the interior of the unit circle in the z-plane.
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In bilinear transformation, the left-half s-plane is mapped to which of the following in the z-domain?a)Entirely outside the unit circle |z|=1b)Partially outside the unit circle |z|=1c)Partially inside the unit circle |z|=1d)Entirely inside the unit circle |z|=1Correct answer is option 'D'. Can you explain this answer?
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