Appendix C
MiddlebrookÕs Extra Element Theorem
C.1
Basic Result
C.2
Derivation
C.3
Discussion
C.4
Examples
C.4.1
C.4.2
C.4.3
C.4.4
Fundamentals of Power Electronics
A simple transfer function
An unmodeled element
Addition of an input filter to a converter
Dependence of transistor current on load in a
resonant inverter
1
Appendix C: MiddlebrookÕs Extra Element Theorem
C.1 Basic Result
Object: find how addition of an element changes a transfer function G(s)
Original conditions:
vin(s) +
–
Addition of element Z(s):
Transfer function
G(s) Z(s) → ∞
Transfer function
G(s)
Linear circuit
Linear circuit
Input
Output
+
vin(s) +
–
vout (s)
Input
Port
Port
{
–
Output
Z(s)
Open-circuit
vout(s)
= G(s)
vin(s)
Fundamentals of Power Electronics
vout(s)
= G(s)
vin(s)
Z(s) → ∞
2
+
vout (s)
–
Z N (s)
Z(s)
Z (s)
1+ D
Z(s)
1+
Z(s) → ∞
Appendix C: MiddlebrookÕs Extra Element Theorem
Dual Form of Basic Result
When the added impedance replaces a short circuit:
Original conditions:
vin(s) +
–
Addition of element Z(s):
Transfer function
G(s) Z(s) → 0
Transfer function
G(s)
Linear circuit
Linear circuit
Input
Output
vin(s) +
–
vout (s)
Input
–
+
vout (s)
–
Z(s)
Short-circuit
vout(s)
= G(s)
vin(s)
Fundamentals of Power Electronics
Output
Port
{
Port
+
3
Z(s)
Z N (s)
Z(s)
1+
Z D(s)
1+
Z(s) → 0
Appendix C: MiddlebrookÕs Extra Element Theorem
Comparison of forms
The two forms of the extra element theorem:
vout(s)
= G(s)
vin(s)
Z N (s)
Z(s)
Z (s)
1+ D
Z(s)
1+
Z(s) → ∞
vout(s)
= G(s)
vin(s)
Z(s)
Z N (s)
Z(s)
1+
Z D(s)
1+
Z(s) → 0
These equations describe the same transfer function, referenced to
the limiting cases of Z = 0 and Z = ∞. Upon equating them, one
obtains the reciprocity relationship:
G(s)
G(s)
Z(s) → ∞
=
Z(s) → 0
Z D(s)
Z N (s)
The quantities ZN and ZD are the same in both forms.
Fundamentals of Power Electronics
4
Appendix C: MiddlebrookÕs Extra Element Theorem
Finding ZD
Linear circuit
vin(s) = 0
Input
+
Output
vout (s)
{
–
Port
Short-circuit
+ v(s) –
Z D(s) =
v(s)
i(s)
v in(s) = 0
i(s)
ZD is the driving-point impedance (i.e., the Thevenin-equivalent
impedance) at the port where the new element is connected. Formally,
it is found by setting independent sources to zero, and injecting a
current i(s) at the port. ZD(s) is the ratio of v(s) to i(s).
Fundamentals of Power Electronics
5
Appendix C: MiddlebrookÕs Extra Element Theorem
Finding ZN
Linear circuit
vin(s) +
–
Input
+
Output
vout (s) o 0
–
Port
+ v(s) –
Z N (s) =
v(s)
i(s)
v out(s) o 0
i(s)
ZN is the impedance seen at the port when the output is nulled. In the
presence of the input vin(s), a current i(s) is injected at the port. This
current is adjusted such that the output vout(s) is nulled to zero. Under
these conditions, ZN(s) is the ratio of v(s) to i(s). Note: nulling is not the
same as shorting.
Fundamentals of Power Electronics
6
Appendix C: MiddlebrookÕs Extra Element Theorem
C.2 Derivation
Original system:
With extra element:
Linear network
Linear network
u(s)
u(s)
y(s)
Input
Input
Output
i(s) + v(s) –
{
i(s) + v(s) –
Z(s)
Open-circuit
y(s)
u(s)
G(s) =
i(s) = 0
[The input and output need not be
voltages, and are denoted here by
the general names u(s) and y(s)]
Fundamentals of Power Electronics
Output
Port
Port
Gold (s) =
y(s)
7
y(s)
u(s)
v(s) = – i(s)Z(s)
Appendix C: MiddlebrookÕs Extra Element Theorem
Current injection at the port
There are now two independent inputs:
Linear network
u(s)
u(s)
y(s)
Input
Output
Port
and
i(s)
The dependent quantities y(s) and v(s)
can be expressed as functions of the
independent inputs using superposition:
+ v(s) –
y(s) = Gold (s)u(s) + Gi(s)i(s)
v(s) = Gv (s)u(s) + Z D(s)i(s)
i(s)
with:
Gold(s) =
Gi(s) =
y(s)
u(s)
y(s)
i(s)
Z D(s) =
i(s) = 0
u(s) = 0
Fundamentals of Power Electronics
Gv(s) =
v(s)
i(s)
u(s) = 0
v(s)
u(s)
i(s) = 0
8
Appendix C: MiddlebrookÕs Extra Element Theorem
Solution for G(s)
Now eliminate v(s) and i(s) from equations of previous slide, and
solve from transfer function G(s):
G(s) =
G (s)Gi(s)
y(s)
= Gold(s) – v
u(s)
Z(s) + Z D(s)
Gold(s) and ZD(s) are found using definitions on previous slide.
We could stop at this point, and use the above equation to evaluate
G(s). The quantities Gi(s) and Gv(s) would be evaluated using the
definitions on the previous slide. However, it is preferable to eliminate
Gi(s) and Gv(s), and instead express G(s) in terms of impedances
measured at the given port. This can be accomplished with an
alternate thought experiment involving null double injection.
Fundamentals of Power Electronics
9
Appendix C: MiddlebrookÕs Extra Element Theorem
Null double injection
In the presence of the input u(s),
inject current i(s) at the port.
Adjust i(s) in the special way that
causes the output y(s) to be
nulled to zero. Under these
conditions, the impedance ZN(s) is
defined as:
v(s)
Z N (s) =
i(s) y(s) o 0
Linear network
u(s)
y(s)
Input
Output
Port
+ v(s) –
i(s)
Nulling: Note that
y(s) = Gold (s)u(s) + Gi(s)i(s)
Therefore, the value of i(s) that
achieves the null condition y(s) o 0 is
given by
So the output is nulled when
i(s) is chosen to satisfy
G (s)
u(s) y(s) o 0 = – i
i(s) y(s) o 0
Gold (s)
Gold (s)u(s) + Gi(s)i(s) o 0
Fundamentals of Power Electronics
10
Appendix C: MiddlebrookÕs Extra Element Theorem
Expression for ZN(s)
Now substitute result from previous slide,
G (s)
u(s) y(s) o 0 = – i
i(s)
Gold (s)
into previous expression for output voltage
y(s) o 0
v(s) = Gv (s)u(s) + Z D(s)i(s)
The result is:
v(s)
y(s) o 0
= Gv(s) u(s)
= –
y(s) o 0
+ Z D(s) i(s)
Gv(s)Gi(s)
+ Z D(s) i(s)
Gold(s)
y(s) o 0
y(s) o 0
Use definition of ZN(s):
v(s)
= Z N (s) i(s)
y(s) o 0
Hence:
Z N (s) = Z D(s) –
Fundamentals of Power Electronics
= –
y(s) o 0
Gv(s)Gi(s)
+ Z D(s) i(s)
Gold(s)
y(s) o 0
Gv(s)Gi(s)
Gold(s)
11
Appendix C: MiddlebrookÕs Extra Element Theorem
Expression for G(s)
Now, eliminate Gi(s) and Gv(s) from expression for G(s), using ZN
result:
G(s) = Gold(s) –
Simplify:
Z D(s) – Z N (s)
G (s)
Z(s) + Z D(s) old
Z N (s)
Z(s)
G(s) = Gold(s)
Z (s)
1+ D
Z(s)
1+
Or,
Z N (s)
Z(s)
Z (s)
1+ D
Z(s)
1+
G(s) = G(s)
Fundamentals of Power Electronics
Z(s) → ∞
12
(Desired result)
Appendix C: MiddlebrookÕs Extra Element Theorem
Example:
A simple transfer function
R1
R3
+
v1(s)
+
–
• Find
G(s) =
v2(s)
v1(s)
R2
v2(s)
R4
C
–
Fundamentals of Power Electronics
13
• Express result in
factored pole-zero
form.
Appendix C: MiddlebrookÕs Extra Element Theorem
C.4.3 Addition of an Input Filter to a Converter
H(s)
Addition of an
input filter changes
the small-signal
transfer functions
of a converter
vg
+
–
Input
filter
Converter
Zo(s)
v
Zi(s)
d
T(s)
Controller
Control-to-output transfer function Gvd(s):
v(s)
Gvd (s) =
d (s)
Converter
Zo(s)
v g(s) = 0
Set vg = 0. Input filter effectively becomes
an impedance Zo(s), added to the converter
power input port.
Fundamentals of Power Electronics
14
v
Gvd(s)
d
Appendix C: MiddlebrookÕs Extra Element Theorem
Application of Extra Element Theorem
Converter
Zo(s)
v
With no input filter, the following
“original” transfer function is obtained:
Gvd (s)
Gvd(s)
Z o(s) = 0
d
In the presence of the input filter, the control-to-output transfer function
can be expressed as:
Gvd (s) = Gvd (s)
Fundamentals of Power Electronics
Z o(s) = 0
15
1+
Z o(s)
Z N (s)
1+
Z o(s)
Z D(s)
Appendix C: MiddlebrookÕs Extra Element Theorem
ZN and ZD
H(s)
vg
+
–
Input
filter
Converter
Zo(s)
v
Zi(s)
Z D(s) = Z i(s)
d(s) = 0
Z N (s) = Z i(s)
v(s) o 0
T(s)
d
Controller
The input filter does not significantly change the control-to-output
transfer function when
Z o < Z N , and
Zo < ZD
Results for basic converters are listed in Table 10.1
Fundamentals of Power Electronics
16
Appendix C: MiddlebrookÕs Extra Element Theorem