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# Equation for Square law circuit modulator

For DSBFC AM (double side band full carrier Amplitude modulation )
The message signal m(t) must be multiplied by carrier maybe $$Ac*cos(\omega_c(t))$$
(For modulation)

This modulation is done in reality by using the non-linear characteristics of the diode,where

$$i(t)=av + bv^2$$
v=applied voltage to diode

Here we apply$$v =Vc(t)+Vm(t)$$
$$i(t)=a(Vc+Vm)+b(Vc+Vm)^2$$
$$i(t)= a*Vc +b*Vm +b*Vc^2+bVm^2 +2*Vc*Vm$$

Here $$Vc=Ac*cos(\omega_c(t))$$

Thus $$i(t)=a*Ac*cos(\omega_c(t))+b*m(t)+b*(Ac*cos(\omega_c(t)))^2 +b*m(t)^2 +2*m(t)*Ac*cos(\omega_c(t))$$

After simplification

$$i(t)=a*Ac*cos(\omega_c(t))+b*m(t)+\frac{b*Ac}{2} +\frac{b*Ac*cos(2\omega_c(t))}{2} +b*m(t)^2 +2*m(t)*Ac*cos(\omega_c(t))$$

In frequency domain I can clearly understand following components
$$f(message frequency),f_c,f_c+f,f_c-f,2*fc$$

but my book tells me there are addition components at $$2f_m,3f_c$$

Can someone help me understand where is it?
Thank you