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[SOLVED] VCO Laplace Transform not making sense

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dpkdpk

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Hi everyone, I've been stuck on this one. Here's an excerpt from my reference source:

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The input voltage Vo to the VCO controls its output frequency, which can be expressed as:

1.png

where omega_0 is the center angular frequency of the VCO, k1 is the gain of the VCO, and u(t) is a unit step function, which is defined as:

2.png

The phase angle of the VCO can be obtained by integrating Equation (1) as:

3.png
---------------------------

My problem is that I don't understand getting from equation 3 to equation 4. Ignoring the constant k1, the integral of a unit step function u(t) is a ramp function t*u(t). The Laplace transform of t*u(t) is 1/s^2. Just to confirm, I typed this equation into the Wolfram Laplace Transform Calculator:

4.png

Another way to approach this problem is using the relationship:

5.png

Setting k1 aside, g(t) is the unit step function u(t). The Laplace Transform [G(s)] of a unit step function is 1/s. So then G(s)/s = (1/s)/s = 1/s^2.

Everywhere else confirms that the transfer function in the s-domain of a VCO is k1*1/s, and the rationale is that 1/s is the Laplace transformation of an integration. However, I wish I could have a derivation that makes sense.

I hope you guys can help me out. Thanks!
 

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The Laplace transform is used for the VCO transfer function. Such a function is defined for linear conditions only.
Therefore, it applies to the PLL under locked condition only - and this is the key for an answer to your question:
The PLL linear transfer function - hence, also the VCO function - is defined in the PHASE domain only.
And for the voltage-phase relation we find the expression PHI(s)/V(s)=Ko/s.
 
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    dpkdpk

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The Laplace transform is used for the VCO transfer function. Such a function is defined for linear conditions only.
Therefore, it applies to the PLL under locked condition only - and this is the key for an answer to your question:
The PLL linear transfer function - hence, also the VCO function - is defined in the PHASE domain only.
And for the voltage-phase relation we find the expression PHI(s)/V(s)=Ko/s.

What is the relationship between the s-domain and the phase domain, mathematically? Is it just F(phase domain) = F(s)/Vo? How do you get from k/s^2 (Laplace domain) to k/s (phase domain)?
 

Apart from miswriting (mathematically speaking) in post #1, your pictures of derivation of Laplace transform are wrong i.e. equation 4 is wrong.

You are correct. The result is PHI(s)=k1·1/s^2

- - - Updated - - -

How do you get from k/s^2 (Laplace domain) to k/s (phase domain)?

LvW's expression is PHI(s)/V(s)=Ko/s where V(s) = 1/s. Do the math yourself.
 
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    dpkdpk

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Apart from miswriting (mathematically speaking) in post #1, your pictures of derivation of Laplace transform are wrong i.e. equation 4 is wrong.

You are correct. The result is PHI(s)=k1·1/s^2

- - - Updated - - -

LvW's expression is PHI(s)/V(s)=Ko/s where V(s) = 1/s. Do the math yourself.

OK, makes sense now.

What did I miswrite in my original post?
 

The limits of an integral and the variable you are integrating should be different.

Your integral, for mathematical correctness, its limits should be "t" (time) and the variable you are integrating should be a different one e.g. "tau".

Integral from 0 to "t" of [ k1·u(tau) d(tau) ]
 
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