The principle of frequency conversion by the Analog multiplier
Following description is how the frequency conversion occurs by the Analog multiplier based on Math.
sin(A±B) = sinA*cosB ± cosA*sinB …(1)
cos(A+B) = cosA*cosB - sinA*sinB …(2)’
cos(A-B) = cosA*cosB + sinA*sinB …(2)’’
(1), (2)’ and (2)’’ are known as formulas of trìgonométric(al) functions.
From (1), re-write (1) to,
sin(A + B) = sinA*cosB + cosA*sinB …(1)’
sin(A - B) = sinA*cosB - cosA*sinB …(1)’’
(1)’ + (1)’’ then,
sin(A+B)+ sin(A-B) = 2*sinA*cosB …(3)
From (3) then,
sinA*cosB = (1/2)*(sin(A+B)+ sin(A-B)) …(4)
Define A and B as follows,
A ≡ ωc*t, B ≡ ωs*t …(5)
and re-write ωc and ωs to
ωc=2*π*fc, ωs=2*π*fs …(6)
then (4) becomes
sin(ωc*t)*cos(ωs*t) = (1/2)*(sin(ωc*t+ωs*t)+ sin(ωc*t-ωs*t))
=(1/2)*( sin((ωc+ωs)*t)+ sin((ωc-ωs)*t) ) …(6)
Define amplitude of sin to Vc and cos to Vs here.
Then (6)*Vc*Vs becomes
{Vc*sin(ωc*t)} * {Vs*cos(ωs*t)} = (Vc*Vs/2)*(sin((ωc+ωs)*t)+ sin((ωc-ωs)*t)) …(7)
Re-write (7) by definition of angular frequency by (6).
{Vc*sin(2*π*fc*t)} * {Vs*cos(2*π*fs*t)}
= (Vc*Vs/2)*{sin(2*π*(fc+fs)*t)+ sin(2*π*(fc-fs)*t)} …(8)
This means multiplying sine wave (frequency=fc, amplitude=Vc)
and cos wave (frequency=fs, amplitude=Vs)
generates
the frequency (fc+fs) sine wave, (fc-fs) sin wave,
and their amplitudes become (Vc*Vs/2).
Add (2)’ and (2)’’ then get
cos(A+B) + cos(A-B) = 2*cosA*cosB
then,
cosA*cosB = (1/2)*(cos(A+B)+cos(A-B)) … (9)
from the definition of (5), (9) becomes
cos(ωc*t)*cos(ωs*t) = (1/2)*(cos(ωc*t+ωs*t)+cos(ωc*t-ωs*t))
= (1/2)*( cos((ωc+ωs)*t) +cos((ωc-ωs)*t)) ) …(10)
Apply (9)*Vc*Vs, then (9) becomes
{Vc*cos(ωc*t)}*{Vs*cos(ωs*t)} = (Vc*Vs/2)*( cos((ωc+ωs)*t) +cos((ωc-ωs)*t)) ) …(11)
Re-write (11) by definition of angular frequency by (6).
{Vc*cos(2*π*fc*t)}*{Vs*cos(2*π*fs*t)}
= (Vc*Vs/2)*( cos(2*π*(fc+fs)*t)) +cos(2*π*(fc-fs)*t)) ) …(12)
This means multiplying cos wave (frequency=ωc, amplitude=Vc)
and cos wave (frequency=fs, amplitude=Vs)
generates
the frequency (fc+fs) cos wave, (fc-fs) cos wave,
and their amplitudes become (Vc*Vs/2).
Reference and related documents :
[1] Analog multiplier / RF mixer reference
乗算器(ミキサー、周波数変換回路) 参考文献
[2] Deference between heterodyne phenominon and beat phenominon
日本国内における (周波数変換/ヘテロダイン現象)と (ビート現象) 教育内容の誤り
(C) Noboru, Ji1NZL Jun.21, 2016
Rev. 0.1; Corrected some mistypes on the Figure. Jul. 5, 2016
Rev. 0.2; Corrected some mistypes on the Figure. Sep.19, 2016
Rev. 1.0; Added related documents and reference. Sep.26, 2016
Return to INDEX
Rev. 0.1; Corrected some mistypes on the Figure. Jul. 5, 2016
Rev. 0.2; Corrected some mistypes on the Figure. Sep.19, 2016
Rev. 1.0; Added related documents and reference. Sep.26, 2016
Return to INDEX
0 件のコメント:
コメントを投稿
現在コメント機能に不具合が出ています。お手数ですみません。
メッセージは、メールでお送り願います。