This note refers to the article “Achieve MPPT control without power calculation “(Electronic Design, November 05 2009, p. 53, http://electronicdesign.com/article/power/achieve_mppt_control_without_power_calculation.aspx) giving some mathematical justification about the multiplier disuse and the use of a synchronous modulation-demodulation to implement a MPPT

Romeo, Giovanni

Earth-prints Repository

  
 
A more rigorous description of the multiplierless MPPT 
principle. 
Giovanni Romeo, INGV, Roma 
 
 
 
 
Let’s assume the simple circuit described in the article (idea for design #2394) as our environment: 
 
 
 
power input power output
tra
ns
fe
r c
on
tro
l
Vi
Vg
10v
+
-
power source Vo
Rint 5
Rload
2
0
[0..1]
source
0
n
power transfer
 
 
 
There is some minor difference: there is no the MPPT block,  and the load is only a resistor.  
 
 
As written in the article, the power transfer block  (regardless to its practical implementation) is a 
kind of ‘dc transformer’ where 
 
 
 
outputinput WW =                                                                         (1) 
 
nVV io ⋅=              where                                        (2) 10 ≤≤ n
 
so the power transfer acts like a transformer, where  
n
II io
1⋅=           (3) 
 
 
The power supplied to the power transfer may be written as a function of  n. 
From (2) and (3) the impedance at the power transfer input is: 2n
RZ loadi = the power dissipated by Zi 
is 
                                   
 
)(nW
                                                                                                            0 0.5 1
0
2
4
6
 
2
2
int2
2
n
R
V
R
n
R
n
R
W
load
g
load
load
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⋅
+
=
 n
                                            
 
 
The plot on the right is obtained using the variable values shown in the block diagram. 
There is an n that maximizes the power. Its value can be found by equating the derivative of the 
power to 0: 
 
( )int2int
2
2 )2(
RnR
RRnnRV
dn
dW
load
load
loadg ⋅+
−⋅⋅⋅−⋅=  
 
Solving the equation (and skipping trivial and negative results) we have the best value of n: 
 
intR
Rn load=  
 
0 0.5 1
0
2
4
)
This was calculated considering the generator side. Let’s take a look on the load side, measuring the 
voltage: 
 
 
n
R
n
R
n
R
VV
load
load
go ⋅+
=
int2
2
                                           
Vo
 
 n 
 
 
As before, the plot on the right is obtained using the variable values shown on the block diagram. 
 
If we follow the same process, i.e. we equate the output voltage derivative to 0, we obtain 
 
0
)( 2int
2
int
2
0 =+
−⋅⋅−=
load
load
gload RRn
RRnVR
dn
dV  
the positive result of this equation is still   
intR
Rn load=  
which demonstrates the possibility of using the voltage output as MPPT control parameter.  
 
 
 
MPPT with a synchronous modulator. 
 
The MPPT goal is to find the maximum of  the power output (W) vs. the control signal (n),   W(n). 
An conventional feedback cannot be used  because W(n) is not a generally growing curve.  But its 
derivative can be used.    dW(n)/dn  crosses 0  when  W(n)  has a  maximum (fig. 1).  
                                                                                         
 
0 0.5 1
10
0
10
20
)(nW
dn
ndW )(
 
 
 
 
 
 
 
 
                     
 
Fig. 1  Although it contains the information about the maximum power position, the power curve w(n) (right) cannot be 
directly used to drive a feedback loop to seek the maximum. Its derivative (left) can. Despite its funny shape,  it is linear 
around the 0 crossing (when the power has a maximum), and  has a single 0-crossing point in the interval of existence 
of n. 
 
A synchronous modulation demodulation process is represented in fig.2 
 
 
 
 
Fig.2  Synchronous modulation-demodulation scheme. The transfer block  f  is fed with a modulation signal, x, and an 
external signal, n (x frequency>>n frequency) . The controlled switch feeds the integrator with f(n+x) when x>0 and 
with –f(n+x) when x<0.   
 
 
integrator
n
f(n+x)
x
-1
out
 
1generator
n 0n 
0.5 1
0
2
4
6
The schematic in fig.2 may be expressed by the equation: 1
 
∫
Δ
Δ−
+=Δ
x
x
dx
x
xxnfxnout )(),(  
 
We may demonstrate that:2
 
( ) ny
x
x
x dy
ydfdx
x
xxnf
x =
Δ
Δ−→Δ
=+Δ ∫ )()(1lim 20                                   (4) 
 
 
This shows that the circuit in fig.2 computes the derivative of the curve    f(y), y=n .   
If we use this strategy on the curve W(n) we may close the MPPT loop on the power derivative, so 
obtaining a stable maximum  power finder.  Note that the schematic in fig.2 is identical to the 
schematic shown in the article (fig. 3), where the   multiplier inside the MPPT does the same job of 
the switch (assuming a symmetrical square wave generator). 
 
n
power input power output
tra
ns
fe
r c
on
tro
l
0
V1
10v
+
-
power transfer
0
f(n+x)
V2
2v
+
-
MPPT
V3
+-
1
(n+x)
Rload
2
power source
x
Rint 5
. ∫
Δ
Δ−
+
x
x
dx
x
xxnf )(
 
Fig 3. The schematic in the idea for design #2394, identifying some signal paths.  The feedback loop is closed by 
feeding n with ∫
Δ
Δ−
+
x
x
dx
x
xxnf )(  
 
 
 
 
                                                 
1 From an electronic designer point of view this may appear unusual, since there is no time reference. That’s true. The 
equation exists without the time (we want to compute the   f(n)  derivative despite the physical nature of the variable) 
but, in a real circuit, time is usually the integration variable. Defining x as a  time function (i.e. x(t)=at)  the integration 
operation becomes a time integration. Obviously, to implement a real circuit x(t) must be periodic an the integration 
operation has to be extent over the period.  
 
 
                                                 
2 Demonstration: 
 
∫ ∫∫
Δ
Δ−
Δ
Δ−
+−+=+
x
x
x
x
dxxnfdxxnfdx
x
xxnf
0
0
)()()(                                           assuming:        
xnu
xf
dx
xgd
+=
= )())((
 
 
 
)(2)()()()()()( ngxngxngugugduufduuf n
xn
xn
n
xn
n
n
xn
−Δ−+Δ+=−=− Δ−Δ+
Δ+
Δ−
∫ ∫
( )
 
 
 
from (4), and using De L'Hôpital's rule  
 
 
( ) ( )( ) dn
ndf
x
xnfxnf
x
xd
d
ngxngxng
xd
d
ngxngxng
x xxx
)(
2
)()(lim
)(2)()(
lim)(2)()(1lim
020
20
=Δ
Δ−−Δ+=
ΔΔ
−Δ−+Δ+Δ=−Δ−+Δ+Δ →Δ→Δ→Δ
 


A more rigorous description of the multiplierless MPPT principle.

http://www.earth-prints.org/bitstream/2122/5494/1/A%20more%20rigorous%20description%20of%20the%20multiplierless%20MPPT%20principle.pdf

A more rigorous description of the multiplierless MPPT principle.

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