Lecture 6 - Modeling Discrete Time Systems by Pulse Transfer Function - Electrical Engineering (EE) PDF Download

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Lecture 6 - Modeling discrete-time systems by pulse transfer function, Control Systems

1 Relationship between s-plane and z-plane 

In the analysis and design of continuous time control systems, the pole-zero configuration of the transfer function in s-plane is often referred. We know that:

• Left half of s-plane ⇒ Stable region.
• Right half of s-plane ⇒ Unstable region.

For relative stability again the left half is divided into regions where the control loop transfer function poles should preferably be located.
Similarly the poles and zeros of a transfer function in z-domain govern the performance characteristics of a digital system.
One of the properties of F ^∗(s) is that it has an infinite number of poles, located periodically with intervals of ±mw_s with m = 0, 1, 2, ....., in the s-plane where w_s is the sampling frequency in rad/sec.
If the primary strip is considered, the path, as shown in Figure 1, will be mapped into a unit circle in the z-plane, centered at the origin. The mapping is shown in Figure 2.
Since
e^(s+jmws)T = e^Tse^j2πm
= e^Ts
= z
where m is an integer, all the complementary strips will also map into the unit circle.

1.1 Mapping guidelines 

1. All the points in the left half s-plane correspond to points inside the unit circle in z-plane.

2. All the points in the right half of the s-plane correspond to points outside the unit circle.

Figure 1: Primary and complementary strips in s-plane

Figure 2: Mapping of primary strip in z-plane

3. Points on the j w axis in the s-plane correspond to points on the unit circle |z | = 1 in the

z-plane.

s = jw
z = e^Ts

= e^jwT ⇒ magnitude = 1

1.2 Constant damping loci, constant frequency loci and constant damping ratio loci 

Constant damping loci: The real part σ of a pole, s = σ + j w, of a transfer function in s-domain, determines the damping factor which represents the rate of rise or decay of time response of the system.

•  Large σ represents small time constant and thus a faster decay or rise and vice versa.
• The loci in the left half s-plane (vertical line parallel to j w axis as in Figure 3(a)) denote positive damping since the system is stable
• The loci in the right half s-plane denote negative damping.
• Constant damping loci in the z-plane are concentric circles with the center at z = 0, as shown in Figure 3(b).
• Negative damping loci map to circles with radii > 1 and positive damping loci map to circles with radii < 1.

Figure 3: Constant damping loci in (a) s-plane and (b) z-plane

Constant frequency loci: These are horizontal lines in s-plane, parallel to the real axis as shown in Figure 4(a).

Figure 4: Constant frequency loci in (a) s-plane and (b) z-plane

Corresponding Z-transform:

z = e^Ts
= e^jwT
When w = constant, it represents a straight line from the origin at an angle of θ = wT rad, measured from positive real axis as shown in Figure 4(b).

Constant damping ratio loci: If ξ denotes the damping ratio:

where wn is the natural undamped frequency and β = sin⁻¹ ξ . If we take Z-transform

For a given ξ or β , the locus in s-plane is shown in Figure 5(a). In z-plane, the corresponding locus will be a logarithmic spiral as shown in Figure 5(b), except for ξ = 0 or β = 0^o and ξ = 1 or β = 90^o.

Figure 5: Constant damping ratio locus in (a) s-plane and (b) z-plane