10/22/2010 Voltage Controlled Oscillator.doc 1/4

Jim Stiles The Univ. of Kansas Dept. of EECS

Voltage Controlled Oscillators

A voltage controlled oscillator is a rather simple device in theory—it’s simply an oscillator whose frequency is related to a control voltage. In other “words”:

( ) ( )vcovco C

d θ t ω f vdt

= =

Thus, if control voltage Cv is a constant with respect to time, the oscillator frequency will likewise be a constant—the oscillator will produce a “pure” tone of the form:

( ) ( )0vco vcov t cos ω t θ= + Conversely, if the control voltage is time-varying, the oscillator frequency will also change with respect to time.

The result is a frequency modulated output signal!

( )Cv t ( ) ( )vco vcov t cos θ t=

10/22/2010 Voltage Controlled Oscillator.doc 2/4

Jim Stiles The Univ. of Kansas Dept. of EECS

Ideally, the relationship between control voltage Cv and oscillator frequency vcoω is very simple; easily expressed as a first-order polynomial:

( )0

vco C

v C

ω f vK v ω

=

= +

where constant vK obviously has units of radians/ sec volt⋅ , sometime expressed as 2π Hz/volt .

The result is thus an equation of a line, with slope vK and y-intercept 0ω : Now, consider the case where we frequency modulate this oscillator signal. The control voltage will change as a function of time (i.e., ( )Cv t ), and so the VCO output frequency will likewise:

( ) ( ) 0vco v Cω t K v t ω= +

Cv

vcoω

vK

0ω

10/22/2010 Voltage Controlled Oscillator.doc 3/4

Jim Stiles The Univ. of Kansas Dept. of EECS

Q: What then is the VCO output signal ( ) ( )vco vcov t cos θ t= ??? A: Remember, the frequency function ( )vcoω t is the time derivative of the phase function ( )vcoθ t . So to determine

( )vcoθ t , we must integrate ( )vcoω t !

( ) ( )

( )

0

0 00

t

vco vco

t

v C

θ t ω t dt

K v t dt ω t θ

′ ′=

′= + +

∫

∫

Thus, the relationship between two of the four important PLL parameters has been established. The phase function ( )vcoθ t is determined by integrating control voltage ( )Cv t . Typically, this relationship is mathematically described using the Laplace Transform!

( ) ( ) ( )0

stvco vco vcoθ s θ t θ t e dt

∞−= =⎡ ⎤⎣ ⎦ ∫L

Of course, it is now obvious to you that:

( ) ( ) 0 02

vvco C

K ω θθ s v ss s s

= + +

Now, to simplify the math a bit, most PLL mathematics ignore the last two terms of the expression above. Essentially, it is assumed that 0 0ω = and 0 0θ = .

10/22/2010 Voltage Controlled Oscillator.doc 4/4

Jim Stiles The Univ. of Kansas Dept. of EECS

Although this is not explicitly true, this simplifying assumption will not ultimately affect our final conclusions, but will make the math a whole lot easier. So, we can state that the VCO is mathematically described as:

( ) ( )vvco C

Kθ s v ss

=

vKs

( )vcoθ s ( )Cv s