74HC4046A Datasheet, PDF (31/34 Pages) NXP Semiconductors – Phase-locked-loop with VCO

Philips Semiconductors
Phase-locked-loop with VCO
Product specification
74HC/HCT4046A
PLL design example
The frequency synthesizer, used in
the design example shown in Fig.32,
has the following parameters:
Output frequency: 2 MHz to 3 MHz
frequency steps : 100 kHz
settling time : 1 ms
overshoot
: < 20%
The open-loop gain is
H (s) x G (s) = K_p × K_f × K_o × K_n.
Where:
K_p = phase comparator gain
K_f = low-pass filter transfer gain
K_o = K_v/s VCO gain
K_n = 1/n divider ratio
The programmable counter ratio
K_n can be found as follows:
N_min. = _f--fs--^o-_t--^ue--^tp- = _{1---2-0---0-M-----kH---H--z--z-} = 20
N_max. = _f--fs--^o-_t--^ue--^tp- = _{1---3-0---0-M-----kH---H--z--z-} = 30
The VCO is set by the values of R1,
R2 and C1, R2 = 10 kΩ (adjustable).
The values can be determined using
the information in the section
“DESIGN CONSIDERATIONS”.
With f_o = 2.5 MHz and f_L = 500 kHz
this gives the following values
(V_CC = 5.0 V):
R1 = 10 kΩ
R2 = 10 kΩ
C1 = 500 pF
The VCO gain is:
K_v = -0---.--9----2_–----f--₍^L--V--^×---_C^˙--2-_C----^×-_–----^π-0---.--9----₎- =
= ^{1-----3-M---.-2-H----z--} × 2π ≈ 2 × 10⁶ r/s/V
The gain of the phase
comparator is:
K_p = _4-V----×-^C---^C-_π-- = 0.4 V/r.
The transfer gain of the filter is
given by:
K_f = _1-----+-----1-₍---_τ-⁺--₁---^τ-₊-²---s-_τ---₂--₎----s- .
Where:
τ₁ = R3C2 and τ₂ = R4C2.
The characteristics equation is:
1 + H (s) × G (s) = 0.
This results in:
s² + -1----⁺-----K-----^p--₍--^×τ---₁-K---₊--^v---_τ×---₂--K-₎---ⁿ----^×-----^τ---²- s+
K-----_p-(--τ-×_{--1---K-+---v--τ--}×_-2---)K-----_{n = 0.}
The natural frequency ω_n is
defined as follows:
ω_n = -K----^p-₍--_τ-^×--₁---K-₊---^v--_τ--^×-₂---₎^K-----n.
and the damping value ζ is defined as
follows:
ζ = -2---1_ω-----_n- × -1----⁺-----K-----^p--₍--^×τ---₁-K---₊--^v---_τ×---₂--K-₎---ⁿ----^×-----^τ---²-
In Fig.33 the output frequency response to
a step of input frequency is shown.
The overshoot and settling time
percentages are now used to determine
ω_n. From Fig.33 it can be seen that the
damping ratio ζ = 0.45 will produce an
overshoot of less than 20% and settle to
within 5% at ω_nt = 5. The required settling
time is 1 ms.
This results in:
ω_n = ^5-t- = _{0----.--05---0---1--} = 5 × 10³ r/s.
Rewriting the equation for natural
frequency results in:
(τ₁ + τ₂) = ^K-----p----^×----_ω-K----_n2v----^×----K-----ⁿ .
The maximum overshoot occurs at N_max.:
(τ₁ + τ₂) = ^{0---5-.--40----0×}---0--2-₂----^××-----13---0-0---⁶- = 0.0011 s.
When C2 = 470 nF, then
R4 = --⁽--^τ---¹---K-⁺---_p--^τ--_×-²---⁾-K---^×-_v----_×^2-----K×----_n-^ω---_×-ⁿ----C^×-----2^ζ----^–-----1- = 315 Ω
now R3 can be calculated:
R3 = _C--^τ---¹_2-- – R4 = 2 kΩ.
1997 Nov 25
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