IEEE Power Electronics Magazine - September 2017 - 44
We can now assemble all of these results to form the final
transfer function characterizing the Figure 3 circuit
R
R
G ^ s h = R +2R R + R +3 R | | R
2
1
4
3
1
2
1 + srC C 1
#
1 + s 6rC + ^ R 4 + R 1 | | R 2h | | R 3@ C 1
s
1 + ~z
= G0
s .
1 + ~p
R
G ^ s h = R +2R
2
1
(9)
This is what is called a low-entropy expression in which
you can immediately distinguish a quasi-static gain, G 0; a
R1
R4
Vout
rC
+
Vin
R2
pole, ~ p; and a zero, ~ z . A high-entropy expression would
be obtained by applying the brute-force approach to the
original circuit when considering an impedance divider,
for instance,
R3
FIG 5 You open the capacitor in dc and calculate the transfer
function of this simple resistive arrangement.
1
R 3 | | a rC + sC k
1
.
1
R 3 < a rC + sC k + R 4 + R 1 < R 2
1
Not only could you make mistakes in deriving the
expression, but formatting the result in something like in
(9) would require more energy. Also, note that in this particular example, we did not write a single line of algebra when
writing (9). Should we later identify a mistake, it would be
easy to come back to one of the individual drawings and fix
it separately. The correction in (9) would then be simple.
Try to run the same correction in (10), and you will probably
start from scratch.
FACTs Applied to a Second-Order System
FACTs work equally well for nth-order passive or active circuits. You determine the order of a circuit by counting the
number of energy-storing elements whose state variables
are independent. If we consider a second-order system, H,
featuring a finite quasi-static gain, H 0, its transfer function
can be expressed the following way:
H^ s h = H 0
R1
Iout(sz) = 0
rC
+
Vin(sz)
I1 Vout(sz) = 0
R4
R2
R3
I1
1
sC1
Z1(s)
1 + a1 s + a2 s2
.
1 + b1 s + b2 s2
As H 0 carries the unit of the transfer function, the
ratio made of N over D is unitless. This implies that
the unit for a 1 and b 1 is time, s. You sum up the circuit
time constants determined when the response is nulled
for a 1 and when the excitation is zeroed for b 1 . For the
second-order coefficients, a 2 or b 2, the dimension is time
squared, s , and you combine time constants in a product.
However, in this time constants product, you reuse one of
the time constants already determined for a 1 or b 1, while
the second time constant determination requires a different notation
x 12 or x 21 .
L1
rC
rL
D
Voltage
Mode
Vin
Zout(s)
Rload
C2
rL
rC
Iout(s)
s=0
L1
Vout(s)
rL
R?
C2
Rload
C2
Rload
Small-Signal Model
L1
rC
FIG 7 The determination of the CCM-operated buck converter output impedance is a good example of how FACTs simplify analyses.
44
IEEE POWER ELECTRONICS MAGAZINE
(11)
2
FIG 6 In this transformed circuit, when the series connection
of rC and C1 becomes a transformed short circuit, the response
disappears, and no current flows in R3.
+
(10)
z September 2017
(12)
In this definition, you set the energy-storing element
whose label appears in the "exponent" in its high-frequency
state (a capacitor is replaced by a short circuit, while an
inductor would be replaced by an open circuit), and you
determine the resistance seen from the second element
terminals when it is temporarily removed from the circuit
(subscripted reference). You carry this exercise for a nulled
output when a 2 must be obtained and when the excitation
is reduced to zero for b 2 . Of course, when inspection works,
it is always the fastest and most efficient way to obtain N.
It may be a bit mysterious at first sight but nothing insurmountable, as will be demonstrated.
Figure 7 depicts a classical second-order filter involved
in the determination of the output impedance of a
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