# Chapter 4: Angle Modulation Techniques¶

## Example 4.1, page no. 71¶

In :

# Variable Declaration
fm1    = 500               # Audio Frequency (Hz)
Vm1    = 2.4               # AF Voltage (V)
del_f1 = 4.8*pow(10,3)     # Deviation (Hz)
fm2    = 500               # Audio Frequency (Hz)
Vm2    = 7.2               # AF Voltage (V)
fm3    = 200               # Audio Frequency (Hz)
Vm3    = 10                # AF Voltage (V)

# Calculation
import math	               # Math Library
kf     = del_f1/Vm1        # Proportionality Constant
mf1    = del_f1/fm1        # Modulation Index
del_f2 = kf*Vm2            # Deviation (Hz)
mf2    = del_f2/fm2        # Modulation Index
del_f3 = kf*Vm3            # Deviation (Hz)
mf3    = del_f3/fm3        # Modulation Index

# Result

print "CASE 1 : Modulation Index, mf1 =",round(mf1,1)
print "         Deviation,     del_f1 =",round(del_f1/pow(10,3),1),"kHz"
print "CASE 2 : Modulation Index, mf2 =",round(mf2,1)
print "         Deviation,     del_f2 =",round(del_f2/pow(10,3),1),"kHz"
print "CASE 3 : Modulation Index, mf3 =",round(mf3)
print "         Deviation,     del_f3 =",round(del_f3/pow(10,3),1),"kHz"

CASE 1 : Modulation Index, mf1 = 9.6
Deviation,     del_f1 = 4.8 kHz
CASE 2 : Modulation Index, mf2 = 28.8
Deviation,     del_f2 = 14.4 kHz
CASE 3 : Modulation Index, mf3 = 100.0
Deviation,     del_f3 = 20.0 kHz


## Example 4.2, page no. 71¶

In :
# Variable Declaration
# GIVEN EXPRESSION : v = 12 sin(6 X 10^(8)t + 5 cos(1250t))
omega1 = 6.00*pow(10,8)         # Angular Velocity (rad/s)
omega2 = 1250                   # Angular Velocity (rad/s)
mf     = 5                      # Modulation Index
A      = 12                     # Amplitude (V)
R      = 10                     # Resistance (Ohms)

# Calculation
import math	                    # Math Library
fc    = omega1/(2*math.pi)      # Carrier frequency (Hz)
fm    = omega2/(2*math.pi)      # Modulating frequency (Hz)
del_f = mf*fm                   # Maximum deviation (Hz)
P     = pow(A/math.sqrt(2),2)/R # Power dissipation (w)

# Result
print "Carrier frequency, fc =",round(fc/pow(10,6),1)," MHz"
print "Modulating frequency, fm =",round(fm)," Hz"
print "Modulation Index, mf =",round(mf)
print "Maximum deviation, del_f =",round(del_f),"Hz"
print "Power dissipation, P =",round(P,1),"W"

Carrier frequency, fc = 95.5  MHz
Modulating frequency, fm = 199.0  Hz
Modulation Index, mf = 5.0
Maximum deviation, del_f = 995.0 Hz
Power dissipation, P = 7.2 W


## Example 4.3, page no. 73¶

In :
# Variable Declaration
fm1    = 500               # Audio Frequency (Hz)
Vm1    = 2.4               # AF Voltage (V)
del_p1 = 4.8               # Deviation (kHz)
fm2    = 500               # Audio Frequency (Hz)
Vm2    = 7.2               # AF Voltage (V)
fm3    = 200               # Audio Frequency (Hz)
Vm3    = 10                # AF Voltage (V)

# Calculation
import math                # Math Library
kp     = del_p1/Vm1        # Proportionality Constant
mp1    = del_p1            # Modulation Index
del_p2 = kp*Vm2            # Deviation (kHz)
mp2    = del_p2            # Modulation Index
del_p3 = kp*Vm3            # Deviation (kHz)
mp3    = del_p3            # Modulation Index

# Result
print "CASE 1 : Modulation Index, mp1 =",round(mp1,1)
print "         Deviation,     del_p1 =",round(del_p1,1),"kHz"
print "CASE 2 : Modulation Index, mp2 =",round(mp2,1)
print "         Deviation,     del_p2 =",round(del_p2,1),"kHz"
print "CASE 3 : Modulation Index, mp3 =",round(mp3)
print "         Deviation,     del_p3 =",round(del_p3,1),"kHz"

CASE 1 : Modulation Index, mp1 = 4.8
Deviation,     del_p1 = 4.8 kHz
CASE 2 : Modulation Index, mp2 = 14.4
Deviation,     del_p2 = 14.4 kHz
CASE 3 : Modulation Index, mp3 = 20.0
Deviation,     del_p3 = 20.0 kHz


## Example 4.4, page no. 74¶

In :
# Variable Declaration
# GIVEN EXPRESSION : v = 12 sin(6 X 10^(8)t + 5 cos(1250t))
omega1 = 6*pow(10,8)          # Angular Velocity (rad/s)
omega2 = 1250                 # Angular Velocity (rad/s)
mp     = 5                    # Modulation Index
A      = 12                   # Amplitude (V)

# Calculation
import math	                  # Math Library
fc     = omega1/(2*math.pi)   # Carrier frequency (Hz)
fm     = omega2/(2*math.pi)   # Modulating frequency (Hz)
del_p  = mp                   # Maximum Deviation (kHz)

# Result
print "Carrier frequency, fc =",round(fc/pow(10,6),1)," MHz"
print "Modulating frequency, fm =",round(fm)," Hz"
print "Modulation Index, mp =",round(mp),"radians"
print "Maximum deviation, del_p =",round(del_p),"kHz"

Carrier frequency, fc = 95.5  MHz
Modulating frequency, fm = 199.0  Hz
Modulation Index, mp = 5.0 radians
Maximum deviation, del_p = 5.0 kHz


## Example 4.5, page no. 75¶

In :
# Variable Declaration
# GIVEN EXPRESSION FM: v=A sin(omega_c*t + mf cos(omega_m*t))
# GIVEN EXPRESSION PM: v=A sin(omega_c*t + mp cos(omega_m*t))
A       = 4                  # Carrier Voltage (V)
del_f   = 10.00*pow(10,3)    # Maximum Frequency Deviation (Hz)
del_p   = 25                 # Maximum Phase Deviation (Hz)
f_c     = 25.00*pow(10,6)    # Carrier Frequency (Hz)
f_m1    = 400                # Modulating Frequency 1 (Hz)
f_m2    = 2000               # Modulating Frequency 2 (Hz)

# Calculation
import math                  # Math Library
omega_c = 2*math.pi*f_c      # Angular Velocity of carrier (rad/s)
omega_m = 2*math.pi*f_m1     # Angular Velocity of Modulating Wave (rad/s)
mf1     = del_f/f_m1         # Modulation Index for FM
mf2     = del_f/f_m2         # Modulation Index for FM
mp      = del_p              # Modulation Index for PM

# Result
print "(a)For FM Case 1, v =",round(A),"sin(",round(omega_c/pow(10,8),2),"* 10^(8) * t +",round(mf1),"cos",round(omega_m),"* t )"
print "(b)For PM Case 1, v =",round(A),"sin(",round(omega_c/pow(10,8),2),"* 10^(8) * t +",round(mp),"cos",round(omega_m),"* t )"
print "(c)For FM Case 2, v =",round(A),"sin(",round(omega_c/pow(10,8),2),"* 10^(8) * t +",round(mf2),"cos",round(omega_m),"* t )"
print "(d)For PM Case 2, v =",round(A),"sin(",round(omega_c/pow(10,8),2),"* 10^(8) * t +",round(mp),"cos",round(omega_m),"* t )"

(a)For FM Case 1, v = 4.0 sin( 1.57 * 10^(8) * t + 25.0 cos 2513.0 * t )
(b)For PM Case 1, v = 4.0 sin( 1.57 * 10^(8) * t + 25.0 cos 2513.0 * t )
(c)For FM Case 2, v = 4.0 sin( 1.57 * 10^(8) * t + 5.0 cos 2513.0 * t )
(d)For PM Case 2, v = 4.0 sin( 1.57 * 10^(8) * t + 25.0 cos 2513.0 * t )


## Example 4.6, page no. 79¶

In :
# Variable Declaration
del1  = 10.00*pow(10,3)    # Maximum Deviation (Hz)
fm    = 2.00*pow(10,3)     # Modulating frequency (Hz)
H     = 8                  # Highest Needed Sideband from Table 4.1

# Calculation
import math                # Math Library
mf    = del1/fm            # Modulation Index
delta = fm*H*2             # Bandwidth required for the FM signal (Hz)

# Result
print "Bandwidth required for the FM signal, delta =",round(delta/pow(10,3)),"kHz"

Bandwidth required for the FM signal, delta = 32.0 kHz


## Example 4.7, page no. 88¶

In :
# Variable Declaration
gm   = 12.00* pow(10,-3) # Transconductance (Siemens)
f    = 5.00*pow(10,6)    # Frequency (Hz)
n    = 9                 # Constant, from X_GS = (1/9)X_GD

# Calculation
import math # Math Library
XCeq = n/gm # Capacitive Reactance of the FET (Ohms)

# Result
print "Capacitive reactance of the FET, XCeq =",round(XCeq),"Ohms"

Capacitive reactance of the FET, XCeq = 750.0 Ohms


## Example 4.8, page no. 89¶

In :
# Variable Declaration
gm = 9.00*pow(10,-3)   # Transconductance (Siemens)
f  = 50.00*pow(10,6)   # Frequency (Hz)
n  = 8                 # Constant, from R_GS = (1/8)XC_GD
C        = 50.00*pow(10,-12)    # Capacitance (F)

# Calculation
import math                             # Math Library
Cn       = 0                            # Minimum Equivalent Capacitance of FET (F)
Cx       = gm/(2*math.pi*f*n)           # Maximum Equivalent Capacitance of FET (F)
fx_by_fn = math.sqrt(1+Cx/C)            # Maximum to Minimum Frequency Ratio
delta    = (fx_by_fn-1)*f/(fx_by_fn+1)  # Total frequency variation of FET (Hz)

# Result
print "Total frequency variation of FET =",round(2*delta/pow(10,6),2),"MHz"

Total frequency variation of FET = 1.73 MHz


## Example 4.9, page no. 90¶

In :
# Variable Declaration
gm_max = 830.00*pow(10,-6)   # Max. Transconductance (Siemens)
gm_min = 320.00*pow(10,-6)   # Min. Transconductance (Siemens)
f      = 88.00*pow(10,6)     # Frequency (Hz)
n      = 10                  # Constant, from R_GS = (1/10)XC_GD
delta  = 75*pow(10,3)        # Maximum Deviation (Hz)
Vgs1   = -2                  # Gate Source Voltage (V)
Vgs2   = -0.5                # Gate Source Voltage (V)

# Calculation
import math                  # Math Library
Vm_rms = -(Vgs1-Vgs2)/(2*math.sqrt(2))  # RMS value of required voltage modulating voltage (V)
Cn     = gm_min/(2*math.pi*f*n)         # Minimum Equivalent Capacitance of FET (F)
Cx     = Cn*gm_max/gm_min               # Maximum Equivalent Capacitance of FET (F)
C      = (Cx-Cn)*f/(4*delta)-Cn         # Capacitance (F)
L      = 1/(4*pow(math.pi*f,2)*C)       # Inductance (H)

# Result
print "(a) RMS value of required modulating voltage, Vm_rms =",round(Vm_rms,2),"V"
print "(b) Capacitance, C =",round(C/pow(10,-12)),"pF"
print "    Inductance,  L =",round(L/pow(10,-6),3),"uH"

(a) RMS value of required modulating voltage, Vm_rms = 0.53 V
(b) Capacitance, C = 27.0 pF
Inductance,  L = 0.121 uH