Derivation of closed-loop transfer function of the type-II PLL

[promach]

I am reading on the section on derivation of closed-loop transfer function of the type-II PLL

I found that expression (9.14) is not the differentiated version. It should have the apostrophe ( ' or ') character to imply that it is the result of differentiating Eq. (9.13) with respect to time

In this case, I really suspect if expression (9.15) is correct or not since H(S) = (9.14) * (Kvco / s) / (1 + (9.14) * (Kvco / s) )

Could anyone suggest if I overlook something ?

Note: The book screenshots below are from Razavi "RF microelectronics" 2nd edition

 

[rmanalo]

u(t) is a unit step function. tu(t) is a ramp function. The derivative of a ramp function is a unit step function. the laplace transform of a unit step function is 1/s.

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edit:
in your derivation, you can ignore the impulse function because it only exists at the origin an basically what's left is L{u(t)} -> 1/s

 

[promach]

u(t) is a unit step function. tu(t) is a ramp function. The derivative of a ramp function is a unit step function. the laplace transform of a unit step function is 1/s.

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edit:
in your derivation, you can ignore the impulse function because it only exists at the origin an basically what's left is L{u(t)} -> 1/s

I have already derived all you have stated in your post.

Please check carefully my question.

I found that expression (9.14) is not the differentiated version. It should have the apostrophe ( ' or ') character to imply that it is the result of differentiating Eq. (9.13) with respect to time

In this case, I really suspect if expression (9.15) is correct or not since H(S) = (9.14) * (Kvco / s) / (1 + (9.14) * (Kvco / s) )

 

[rmanalo]

It should have the apostrophe ( ' or ') character to imply that it is the result of differentiating Eq. (9.13) with respect to time

Differentiating Eq. (9.13) with respect to time, normalizing to ΔФ, and taking the Laplace Transform, we have

Eq. (9.14) is in fact the result of differentiating Eq. (9.13) and then transformed into its laplace equivalent. i.e. Vcont'(t) -> Vcont(S).

 

[promach]

Vcont'(t) -> Vcont(S)

Did you use any laplace properties ? What happen to the apostrophe ( ' or ') character AFTER laplace transform ?

 

[rmanalo]

Did you use any laplace properties ?

unit step functions is 1/s

What happen to the apostrophe ( ' or ') character AFTER laplace transform ?

I think you're mixing up time domain and s domain. Besides, the apostrophe you're referring to is just a symbol describing dVcont(t)/dt.

 

[promach]

laplace transform of x′(t) gives s*X(s)

Vcont'(t) -> Vcont(S)

Please check very carefully your own post statement

 

[rmanalo]

laplace transform of x′(t) gives s*X(s)

the exact statement is x'(t)=s*X(S)-x(0)

Vcont'(t) -> Vcont(S)

I apologize. I did not mean its exact equivalent but its transformation from time domain to s domain.

Please check very carefully your own post statement

 

[promach]

the exact statement is x'(t)=s*X(S)-x(0)



I apologize. I did not mean its exact equivalent but its transformation from time domain to s domain.


View attachment 148537

However, how are you going to arrive at expression (9.15) using your handwritten computation result ?

 

[rmanalo]

how are you going to arrive at expression (9.15)

RF circuits is not my field. I only assumed your problem is with (9.14). But Fig. 9.27 is in feedback which explains eq 9.15.

 

[promach]

Your handwritten computation result just proves otherwise, that expression (9.14) is not correct. Please double check very carefully again.

Vcont(s) has the denominator of quadratic power while expression (9.14) only have denominator of linear power

But Fig. 9.27 is in feedback which explains eq 9.15.

I really suspect if expression (9.15) is correct or not since H(S) = (9.14) * (Kvco / s) / (1 + (9.14) * (Kvco / s) )

 

[promach]

Have a look at the explanation given below:

What do you think ?

 

[rmanalo]

Like I said before, RF circuits is not my field. I'm only familiar with the math.

Vcont(s) has the denominator of quadratic power while expression (9.14) only have denominator of linear power

PLLs are a topic where RF touches control engineering.

I agree with him. And in control engineering they view the system in blocks where each block contributes poles or zeroes depending on how they are modeled
e.g. X(S)->[block contributing pole or zero]->Y(S).
Here I assume the VCO block contributes a pole, i.e. KVCO/s.

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I took a quick scan on razavi's book. Try looking at Eq.(8.177) if this checks out.