Lec.6 Digital Control (Z- transform and Properties And Theorems Of Z-Transform)( S domain-cosine)

Digital Control Course: Lecture 6 - Z-Transform of Cosine Function

Introduction to the Z-Transform

• Peace, Mercy, and Blessings of God be upon you. The lecture focuses on finding the Z-transform for a cosine function defined as x(t) = cos(omega t) for t geq 0 , and zero otherwise.

Working with the Cosine Function

• The function x(t) = cos(omega t) is analyzed only in its positive domain. The cosine graph is discussed, emphasizing that sinusoidal functions can be transformed using exponential forms.

Exponential Representation of Cosine

• Two key equations are introduced:

• e^jomega t = cos(omega t) + jsin(omega t)

• e^-jomega t = cos(omega t) - jsin(omega t)

These equations will be used to derive the Z-transform through addition.

Deriving the Z-Transform

• By adding the two exponential equations, we isolate cosine:

• Resulting in e^jomega t + e^-jomega t = 2cos(omega t) .

This leads to calculating its Z-transform by substituting into known formulas for exponential terms.

Applying the Z-Transform

• The transformation yields:

• For e^jomega t : Z(e^jomega) = 1/1-e^jomegaz^-1

• For e^-jomega t : Z(e^-jomega) = 1/1-e^-jomegaz^-1

Substituting these results allows us to express the overall transform in a unified form.

Finalizing the Expression

• After unifying denominators and simplifying, we arrive at an expression involving cosine:

• The final result simplifies down to a fraction where both numerator and denominator contain terms related to cosine.

This process illustrates how algebraic manipulation leads us back to our original function's properties within the context of digital control systems.

Conclusion and Next Steps

• The derived formula is compared with previous results from sine transformations, confirming consistency.

Students are encouraged to attempt deriving tangent transformations independently or seek assistance if needed. This concludes our exploration of cosine's Z-transform; further discussions will include polynomial calculations next time.

Z-Transform and Laplace Domain Conversion

Summation and Initial Formula

• The discussion begins with substituting values into a summation from k = 0 to infinity, leading to the expression 1 + Az^-1 + A^2z^-2 + A^3z^-3 .

• It is noted that dividing this series by itself yields z^-1 , resulting in the formula z/1 - Az^-1 .

Transitioning Between Domains

• The speaker emphasizes the need to transition from the s-domain to the z-domain, requiring a review of inverse Laplace transforms.

• The method of partial fractions is introduced as a technique for simplifying expressions involving constants over denominators.

Finding Constants through Denominators

• To find constant 'A', substitute s = 0 , yielding a result of 1. For constant 'b', substituting s = -1 results in -1.

• This leads to an expression of the form 1/s - 1/s+1 , which will be referenced later.

Utilizing Transformation Tables

• The speaker refers to transformation tables where specific terms like 1/s correspond to unit step functions and exponential forms.

• It is reiterated that these transformations are essential for manual calculations, despite existing shortcuts available in engineering contexts.

Final Transformations and Adjustments

• Converting back to z-domain involves using previously established relationships, such as unifying denominators.

• Further simplifications lead to expressions involving common factors, ultimately refining the equation into manageable components.

Conclusion on Z-transforms

• The final adjustments yield an expression that retains key elements while simplifying others, demonstrating effective manipulation of mathematical forms.

• The session concludes with anticipation for further exploration into different Z-transforms and their properties.