Unlike rectangular pulses, the power relations in PSM are more complex due to the slope of the transmitted pulses. Here we will consider both trapezoidal and triangular PSM signals.
3.4.1
trapezoidal PSM
Figure 3. 8 shows a trapezoidal PSM pulse which can be represented as;
v(t) = At_
tr 0 < t< tr
A tr < t < T (3.25)
0 T <t < T
where tr is the rise time of the PSM pulse, in the absence of the slope modulation tr = tro. The total energy in a pulse can be obtained from equation 3.26;
E = A \ T - ^ t r) (3.26)
The instantaneous power delivered to 1 Q resistor can be shown to be;
p = j - [ T ~
f'r]
(3-27)
and the average transmitted power is;
Amp. (V) A T tr Time Amp. (V) (a) A Time 0
Fig. 3.8 PSM pulses waveforms: (a) trapezoidal and (b) triangular.
where < > denotes time averaging for the enclosed quantity. Equation 3.28 is a general equation which is true both when slope modulation is present or absent, and is plotted in Fig. 3.9 as a function of the normalised average rise time ( <t,>/T0 ). It is clear that the average power decreases when the average rise time increases. This is because the shape of the pulse becomes closer to a triangular. In the absence o f slope modulation, i.e. <tr> = tro, equation 3.28 is reduced to;
(3.29) o ^
In the case of pulse modulation, the value of tr , as given in equation 3.13, is t
tr = --- --- , but, tr will have discrete values depending on the phase relation
1+'”v»(0
between the sampling pulse and the modulating signal voltage. Therefore, equation 3.28 can be rewritten as;
0.8 T/T =0.9 0.6 < O CL = 0.7 0.4 =0.5 0.2 0.25 0.50 0.75 1.00 <t > / Tr o
Fig. 3.9 Normalised average power of trapezoidal PSM as a function of the normalised rise time.
p °
t
r[1
37;
(3'30)
where tm are the instantaneous discrete values of tr and N is an integer. Substituting equation 3.13 into 3.30 the average power can be written in terms of the
modulating signal values as;
where i„ is the sampling time at which the discrete values of the rise time exists. When N is a large integer, which is true when the sampling ratio is relatively high, the
summation in equation 3.31 can be approximated to;
+ A r-s.oo7’. J0 1+ «*’»,(/) (3.32)
where AT is the time interval (in case of periodic signal AT represents the period). For a unity amplitude sinusoidal signal it can be shown that [76, 77];
Finally, substituting equations 3.33 and 3.32 into 3.31 the average transmitted power can be given as;
Substituting equation 3.19 into 3.34, the average power at the maximum modulation index mc can be evaluated as
The above equation represents the upper limit beyond which the PSM waveform will be distorted. Equations 3.34 and 3.35 have been plotted against modulation index for different values of T/T0 and tro/T0, see Figs. 3.10 and 3.11. It can be seen that the power decreases as the modulation index increases. This is because that as the modulation index increases, the average rise time increases, as explained in the last
(3.33)
(3.34)
p o c
= 4 “7’n-fo-»»c)5(»»c)]
0.9 0.8 = 0.2 = 0.3 0.7
<
O = 0.4 CL t_ CD 0.6 = 0.5 Ql 0.5 m = m Distorted waveform 0.4 0.3 0.2 0.4 1.0 0 0.2 0.6 0.8 Modulation index (m)Fig. 3.10 Normalised average power of trapezoidal PSM signal (T/TQ = 0.9)
0.5 m = m 0.4 = 0.2 o = 0.3 0.3 CL ■o = 0.4 Distorted waveform 0.2 0 0.2 0.4 0.6 0.8 1.0 Modulation index (m)
3.4.2
triangular PSM
This is illustrated in Fig. 3.8.b and may be expressed as;
A t! tr
v(/) = « (3.36)
0 tr < t <Ta
Comparing equation 3.36 with equation 3.25, it is clear that triangular PSM pulses can be considered as trapezoidal PSM with T= tr . Therefore, the average transmitted power can be obtained as;
Equation 3.37 shows the relation between the average rise time and the transmitted power. Unlike the trapezoidal PSM, the average power of triangular PSM increases with increasing average rise time. Following the same procedure used in deriving the power relations of the trapezoidal PSM, the average power per second for the triangular waveform can be found as;
The average PSM waveform power when the modulation index is maximum can be obtained by substituting equation 3.20 into 3.38. this gives;
(3.37)
(3.38)
Equations 3.38 and 3.39 have been plotted as a function of the modulation index m for different values of tro/ro> and shown in Fig 3.12. Unlike the trapezoidal PSM, the average transmitted power of the triangular PSM increases as the modulation index increases. This is because the average rise time increases as the modulation index increases, and consequently the average power increases, as it can seen from equation
A comparison between the power relations of triangular PSM and trapezoidal PSM is shown in Fig. 3.13 which demonstrates results as expected. With small values of tn, triangular PSM needs less power than trapezoidal for the same value o f modulation index. However, as T/T0 decreases, the trapezoidal PSM approaches the triangular behaviour. It can also be seen that the maximum modulation indices in both types of PSM are different, triangular PSM shows wider range for less transmitted power.