# Converter Transfer Functions by alicejenny

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```									              Chapter 8. Converter Transfer Functions

8.1. Review of Bode plots
8.1.1.   Single pole response
8.1.2.   Single zero response
8.1.3.   Right half-plane zero
8.1.4.   Frequency inversion
8.1.5.   Combinations
8.1.6.   Double pole response: resonance
8.1.7.   The low-Q approximation
8.1.8.   Approximate roots of an arbitrary-degree polynomial

8.2. Analysis of converter transfer functions
8.2.1. Example: transfer functions of the buck-boost converter
8.2.2. Transfer functions of some basic CCM converters
8.2.3. Physical origins of the right half-plane zero in converters

Fundamentals of Power Electronics               1             Chapter 8: Converter Transfer Functions
Converter Transfer Functions

8.3. Graphical construction of converter transfer
functions
8.3.1.   Series impedances: addition of asymptotes
8.3.2.   Parallel impedances: inverse addition of asymptotes
8.3.3.   Another example
8.3.4.   Voltage divider transfer functions: division of asymptotes

8.4. Measurement of ac transfer functions and
impedances
8.5. Summary of key points

Fundamentals of Power Electronics                2             Chapter 8: Converter Transfer Functions
The Engineering Design Process

1. Specifications and other design goals are defined.
2. A circuit is proposed. This is a creative process that draws on the
physical insight and experience of the engineer.
3. The circuit is modeled. The converter power stage is modeled as
described in Chapter 7. Components and other portions of the system
are modeled as appropriate, often with vendor-supplied data.
4. Design-oriented analysis of the circuit is performed. This involves
development of equations that allow element values to be chosen such
that specifications and design goals are met. In addition, it may be
necessary for the engineer to gain additional understanding and
physical insight into the circuit behavior, so that the design can be
improved by adding elements to the circuit or by changing circuit
connections.
5. Model verification. Predictions of the model are compared to a
laboratory prototype, under nominal operating conditions. The model is
refined as necessary, so that the model predictions agree with
laboratory measurements.

Fundamentals of Power Electronics        3             Chapter 8: Converter Transfer Functions
Design Process

6. Worst-case analysis (or other reliability and production yield
analysis) of the circuit is performed. This involves quantitative
evaluation of the model performance, to judge whether
specifications are met under all conditions. Computer
simulation is well-suited to this task.
7. Iteration. The above steps are repeated to improve the design
until the worst-case behavior meets specifications, or until the
reliability and production yield are acceptably high.

This Chapter: steps 4, 5, and 6

Fundamentals of Power Electronics         4         Chapter 8: Converter Transfer Functions
Buck-boost converter model
From Chapter 7

L
1:D                                 D' : 1                                 Output

+
–
Line                                                                                                           +
input                                    i(s)
(Vg – V) d(s)
vg(s)   +   Zin(s)       I d(s)                                                I d(s)            C            v(s) R   Zout(s)
–

–

d(s) Control input

v(s)                                                          v(s)
Gvg(s) =                                                Gvd(s) =
vg(s)                                                         d (s)
d(s) = 0                                                      v g(s) = 0

Fundamentals of Power Electronics                          5                      Chapter 8: Converter Transfer Functions
Bode plot of control-to-output transfer function
with analytical expressions for important features

80 dBV
|| Gvd ||              || Gvd ||                                                                              ∠ Gvd
60 dBV
Gd0 = V                             Q = D'R       C
DD'                                         L
D'                                Vg
20 dBV                          2π LC                            ω 2D'LC        DVg
0˚            10 -1/2Q f0                                   ω(D') 3RC
0 dBV                                             fz                                              0˚
∠ Gvd            fz /10                   2
–20 dBV                                           2πDL                                            –90˚
(RHP)
–40 dBV                                                                                           –180˚
10 1/2Q f0                          10fz
–270˚
–270˚
10 Hz            100 Hz            1 kHz              10 kHz          100 kHz           1 MHz
f

Fundamentals of Power Electronics                              6                  Chapter 8: Converter Transfer Functions
Design-oriented analysis

How to approach a real (and hence, complicated) system
Problems:
Complicated derivations
Long equations
Algebra mistakes
Design objectives:
Obtain physical insight which leads engineer to synthesis of a good design
Obtain simple equations that can be inverted, so that element values can
be chosen to obtain desired behavior. Equations that cannot be inverted
are useless for design!
Design-oriented analysis is a structured approach to analysis, which attempts to
avoid the above problems

Fundamentals of Power Electronics          7             Chapter 8: Converter Transfer Functions
Some elements of design-oriented analysis,
discussed in this chapter

• Writing transfer functions in normalized form, to directly expose salient
features
• Obtaining simple analytical expressions for asymptotes, corner
frequencies, and other salient features, allows element values to be
selected such that a given desired behavior is obtained
• Use of inverted poles and zeroes, to refer transfer function gains to the
most important asymptote
• Analytical approximation of roots of high-order polynomials
• Graphical construction of Bode plots of transfer functions and
polynomials, to
avoid algebra mistakes
approximate transfer functions
obtain insight into origins of salient features

Fundamentals of Power Electronics           8             Chapter 8: Converter Transfer Functions
8.1. Review of Bode plots

Decibels                            Table 8.1. Expressing magnitudes in decibels

G   dB
= 20 log 10 G             Actual magnitude          Magnitude in dB

1/2                      – 6dB
Decibels of quantities having                   1                       0 dB
units (impedance example):
normalize before taking log                     2                       6 dB
5 = 10/2           20 dB – 6 dB = 14 dB
Z
Z        = 20 log 10                      10                       20dB
dB                 Rbase
1000 = 103             3 ⋅ 20dB = 60 dB

5Ω is equivalent to 14dB with respect to a base impedance of Rbase =
1Ω, also known as 14dBΩ.
60dBµA is a current 60dB greater than a base current of 1µA, or 1mA.

Fundamentals of Power Electronics           9                   Chapter 8: Converter Transfer Functions
Bode plot of fn

Bode plots are effectively log-log plots, which cause functions which
vary as fn to become linear plots. Given:
f n
G =
f0                    60dB                              2
Magnitude in dB is                                 40dB                                                f0
n                                                                                            f
f                       f                                            =
2
G        = 20 log 10            = 20n log 10        20dB                                n                                f0
n=
1
0dB
n=
–20dB
n                   f0
• Magnitude is 1, or 0dB, at                                                                        =
–2
–40dB
frequency f = f0
–2
f
f0
–60dB
0.1f0           f0                        10f0
f
log scale

Fundamentals of Power Electronics                            10              Chapter 8: Converter Transfer Functions
8.1.1. Single pole response

Simple R-C example                        Transfer function is
R                                              1
v2(s)
+                   G(s) =       = sC
v1(s)   1 +R
sC
v1(s)    +              C    v2(s)
–                                Express as rational fraction:

G(s) =      1
–
1 + sRC

This coincides with the normalized
form
G(s) =    1
1+ ωs
0

with            ω0 = 1
RC

Fundamentals of Power Electronics           11              Chapter 8: Converter Transfer Functions
G(jω) and || G(jω) ||

Let s = jω:
ω
1– j ω                               Im(G(jω))
G( jω) =   1      =        0
ω         ω 2                                                              G(jω)
1+ j ω     1+ ω
0        0

|
)|
(jω
Magnitude is

G
||
2                   2
G( jω) =        Re (G( jω)) + Im (G( jω))                               ∠G(jω)
=       1
ω 2                                                     Re(G(jω))
1+ ω
0

Magnitude in dB:

G( jω)      = – 20 log 10           ω
1+ ω
2
dB
dB                           0

Fundamentals of Power Electronics                       12           Chapter 8: Converter Transfer Functions
Asymptotic behavior: low frequency

For small frequency,                                                    1
G( jω) =
ω << ω0 and f << f0 :                                                     ω      2
1+ ω
0
ω
ω0 << 1
|| G(jω) ||dB

Then || G(jω) ||                        0dB
0dB
becomes                                                                                                –1
–20dB
f
G( jω) ≈ 1 = 1                                                                            f0
–40dB
Or, in dB,
–60dB
G( jω)      ≈ 0dB                              0.1f0             f0            10f0
dB                                                                                     f

This is the low-frequency
asymptote of || G(jω) ||

Fundamentals of Power Electronics                    13             Chapter 8: Converter Transfer Functions
Asymptotic behavior: high frequency

For high frequency,                                                                   1
G( jω) =
ω >> ω0 and f >> f0 :                                                                   ω      2
1+ ω
0
ω
ω0 >> 1                              || G(jω) ||dB

ω       2     ω         2                                0dB
1+ ω           ≈ ω                               0dB
0             0
–1
f
Then || G(jω) ||                                   –20dB
f0
–40dB
–1
1                f
G( jω) ≈                         =              –60dB
ω      2         f0
ω0                                            0.1f0             f0            10f0
f

The high-frequency asymptote of || G(jω) || varies as f-1.
Hence, n = -1, and a straight-line asymptote having a
slope of -20dB/decade is obtained. The asymptote has
a value of 1 at f = f0 .
Fundamentals of Power Electronics                                  14             Chapter 8: Converter Transfer Functions
Deviation of exact curve near f = f0

Evaluate exact magnitude:
at f = f0:
G( jω0) =             1           = 1
ω      2
2
1 + ω0
0

ω     2
G( jω0)   dB
= – 20 log 10       1 + ω0        ≈ – 3 dB
0

at f = 0.5f0 and 2f0 :
Similar arguments show that the exact curve lies 1dB below
the asymptotes.

Fundamentals of Power Electronics                        15              Chapter 8: Converter Transfer Functions
Summary: magnitude

|| G(jω) ||dB

0dB
1dB                        3dB
0.5f0                1dB
f0
–10dB
2f0

–30dB
f
Fundamentals of Power Electronics            16           Chapter 8: Converter Transfer Functions
Phase of G(jω)

Im(G(jω))
G(jω)                                 ω
1– j ω
G( jω) =   1      =        0
ω
1+ j ω     1+ ωω 2
|
)|
0        0
(jω
G
||

∠G(jω)
ω
Re(G(jω))          ∠G( jω) = – tan – 1
ω0

–1
Im G( jω)
∠G( jω) = tan
Re G( jω)

Fundamentals of Power Electronics            17         Chapter 8: Converter Transfer Functions
Phase of G(jω)

ω
0˚              0˚ asymptote                                     ∠G( jω) = – tan – 1
∠G(jω)                                                                                            ω0
-15˚

-30˚                                                                      ω       ∠G(jω)

-45˚                        -45˚
0          0˚
f0
-60˚
ω   0
–45˚
-75˚
–90˚ asymptote
-90˚                                                                      ∞         –90˚
0.01f0        0.1f0            f0        10f0          100f0

f

Fundamentals of Power Electronics                   18               Chapter 8: Converter Transfer Functions
Phase asymptotes

Low frequency: 0˚
High frequency: –90˚
Low- and high-frequency asymptotes do not intersect
Hence, need a midfrequency asymptote

Try a midfrequency asymptote having slope identical to actual slope at
the corner frequency f0. One can show that the asymptotes then
intersect at the break frequencies

fa = f0 e – π / 2 ≈ f0 / 4.81
fb = f0 e π / 2 ≈ 4.81 f0

Fundamentals of Power Electronics                         19        Chapter 8: Converter Transfer Functions
Phase asymptotes

fa = f0 / 4.81
0˚
∠G(jω)
-15˚

-30˚
–π/2
fa = f0 e        ≈ f0 / 4.81
fb = f0 e π / 2 ≈ 4.81 f0               -45˚                          -45˚
f0
-60˚

-75˚

-90˚
0.01f0        0.1f0               f0     fb = 4.81 f0           100f0

f

Fundamentals of Power Electronics                          20                     Chapter 8: Converter Transfer Functions
Phase asymptotes: a simpler choice

fa = f0 / 10
0˚
∠G(jω)
-15˚

-30˚
fa = f0 / 10
-45˚                        -45˚
fb = 10 f0
f0
-60˚

-75˚

-90˚
0.01f0          0.1f0          f0         fb = 10 f0           100f0

f

Fundamentals of Power Electronics                        21            Chapter 8: Converter Transfer Functions
Summary: Bode plot of real pole

0dB                                                                  1
|| G(jω) ||dB                                                                 G(s) =
1dB                              3dB
1+ ωs
0.5f0                                                      0
1dB
f0

2f0

0˚         f0 / 10
∠G(jω)                                  5.7˚

-45˚
f0

-90˚
5.7˚
10 f0

Fundamentals of Power Electronics                         22                  Chapter 8: Converter Transfer Functions
8.1.2. Single zero response

Normalized form:
s
G(s) = 1 + ω
0

Magnitude:
G( jω) =          ω
1+ ω
2

0

Use arguments similar to those used for the simple pole, to derive
asymptotes:
0dB at low frequency, ω << ω0
+20dB/decade slope at high frequency, ω >> ω0
Phase:
ω
∠G( jω) = tan – 1
ω0
—with the exception of a missing minus sign, same as simple pole

Fundamentals of Power Electronics                23      Chapter 8: Converter Transfer Functions
Summary: Bode plot, real zero

s
G(s) = 1 + ω

2f0

f0
0.5f0                    1dB
0dB    1dB                                3dB
|| G(jω) ||dB

10 f0      +90˚
5.7˚

f0
45˚

∠G(jω)         0˚
5.7˚
f0 / 10

Fundamentals of Power Electronics                           24                   Chapter 8: Converter Transfer Functions
8.1.3. Right half-plane zero

Normalized form:
s
G(s) = 1 – ω
0

Magnitude:
G( jω) =          ω
1+ ω
2

0

—same as conventional (left half-plane) zero. Hence, magnitude
asymptotes are identical to those of LHP zero.
Phase:
ω
∠G( jω) = – tan – 1
ω0
—same as real pole.
The RHP zero exhibits the magnitude asymptotes of the LHP zero,
and the phase asymptotes of the pole

Fundamentals of Power Electronics                 25   Chapter 8: Converter Transfer Functions
Summary: Bode plot, RHP zero

s
G(s) = 1 – ω
0

2f0

f0
0.5f0                     1dB
0dB    1dB                                 3dB
|| G(jω) ||dB

0˚         f0 / 10
∠G(jω)                                  5.7˚

-45˚
f0

-90˚
5.7˚
10 f0

Fundamentals of Power Electronics                          26                   Chapter 8: Converter Transfer Functions
8.1.4. Frequency inversion

Reversal of frequency axis. A useful form when describing mid- or
high-frequency flat asymptotes. Normalized form, inverted pole:
G(s) =     1
ω
1 + s0

An algebraically equivalent form:
s
ω0
G(s) =
s
1+ ω
0

The inverted-pole format emphasizes the high-frequency gain.

Fundamentals of Power Electronics            27         Chapter 8: Converter Transfer Functions
Asymptotes, inverted pole

G(s) =        1                                                                0dB
ω                          3dB                           1dB
1 + s0                                                2f0
1dB
f0

0.5f0

|| G(jω) ||dB

+90˚       f0 / 10
∠G(jω)                                    5.7˚

+45˚
f0

0˚
5.7˚
10 f0

Fundamentals of Power Electronics                             28                 Chapter 8: Converter Transfer Functions
Inverted zero

Normalized form, inverted zero:
ω
G(s) = 1 + s0

An algebraically equivalent form:

1+ ωs
0
G(s) =   s
ω0

Again, the inverted-zero format emphasizes the high-frequency gain.

Fundamentals of Power Electronics            29        Chapter 8: Converter Transfer Functions
Asymptotes, inverted zero

ω
G(s) = 1 + s0
|| G(jω) ||dB

0.5f0

f0
1dB                     2f0
3dB                             1dB        0dB

10 f0       0˚
5.7˚

f0
–45˚

∠G(jω)          –90˚                    5.7˚
f0 / 10

Fundamentals of Power Electronics                           30                 Chapter 8: Converter Transfer Functions
8.1.5. Combinations

Suppose that we have constructed the Bode diagrams of two
complex-values functions of frequency, G1(ω) and G2(ω). It is desired
to construct the Bode diagram of the product, G3(ω) = G1(ω) G2(ω).
Express the complex-valued functions in polar form:
G1(ω) = R1(ω) e jθ 1(ω)
G2(ω) = R2(ω) e jθ 2(ω)
G3(ω) = R3(ω) e jθ 3(ω)

The product G3(ω) can then be written
G3(ω) = G1(ω) G2(ω) = R1(ω) e jθ 1(ω) R2(ω) e jθ 2(ω)

G3(ω) = R1(ω) R2(ω) e j(θ 1(ω) + θ 2(ω))

Fundamentals of Power Electronics                     31             Chapter 8: Converter Transfer Functions
Combinations

G3(ω) = R1(ω) R2(ω) e j(θ 1(ω) + θ 2(ω))

The composite phase is
θ 3(ω) = θ 1(ω) + θ 2(ω)

The composite magnitude is

R3(ω) = R1(ω) R2(ω)

R3(ω)   dB
= R1(ω)   dB
+ R2(ω)    dB

Composite phase is sum of individual phases.
Composite magnitude, when expressed in dB, is sum of individual
magnitudes.

Fundamentals of Power Electronics                      32        Chapter 8: Converter Transfer Functions
G0
Example 1: G(s) =                                 s            s
1+ ω         1+ ω
1            2

with G0 = 40 ⇒ 32 dB, f1 = ω1/2π = 100 Hz, f2 = ω2/2π = 2 kHz

40 dB        G0 = 40 ⇒ 32 dB
|| G ||                                                                                     ∠G
|| G ||                     f1      –20 dB/decade
20 dB
100 Hz
0 dB
0 dB
f2
–20 dB                                                                             0˚
∠G         f1/10                f2/10
–40 dB             10 Hz               200 Hz                                     –45˚

–60 dB                                                                            –90˚
10f1            20 kHz        –135˚
–180˚
1 Hz            10 Hz         100 Hz          1 kHz       10 kHz        100 kHz
f
Fundamentals of Power Electronics                             33                Chapter 8: Converter Transfer Functions
Example 2

Determine the transfer function A(s) corresponding to the following
asymptotes:

f2              || A∞ ||dB
|| A ||
f1
|| A0 ||dB                         +20 dB/dec

10f1    f2 /10

∠A                        +45˚/dec          –90˚             –45˚/dec
0˚                                                                  0˚
f1 /10                                  10f2

Fundamentals of Power Electronics                 34               Chapter 8: Converter Transfer Functions
Example 2, continued

One solution:                                    s
1+ ω
1
A(s) = A 0
s
1+ ω
2

Analytical expressions for asymptotes:
For f < f1                1+➚
s
ω
= A0 1 = A0
1
A0
1+➚
s
ω
1
2
s = jω

For f1 < f < f2
s
➚+ ω
1   s
1                    ω1       s = jω        ω      f
A0                         = A0                     = A0 ω = A0
1+➚
ω
s                              1                  1     f1
2
s = jω

Fundamentals of Power Electronics                        35                        Chapter 8: Converter Transfer Functions
Example 2, continued

For f > f2
s
➚+ ω
1  s
1                   ω1   s = jω         ω        f
A0                          = A0   s             = A 0 ω2 = A 0 2
➚+ ω
1  s
2                   ω2   s = jω
1      f1
s = jω

So the high-frequency asymptote is
f
A∞ = A0 2
f1
Another way to express A(s): use inverted poles and zeroes, and
express A(s) directly in terms of A∞
ω
1 + s1
A(s) = A ∞
ω
1 + s2

Fundamentals of Power Electronics                         36                   Chapter 8: Converter Transfer Functions

Example                                              L
v2(s)        1
G(s) =          =                                                                    +
v1(s) 1 + s L + s 2LC
R
v1(s)   +                C              R        v2(s)
Second-order denominator, of               –
the form
–
G(s) =        1
1 + a 1s + a 2s 2              Two-pole low-pass filter example

with a1 = L/R and a2 = LC

How should we construct the Bode diagram?

Fundamentals of Power Electronics        37            Chapter 8: Converter Transfer Functions
Approach 1: factor denominator

G(s) =          1
1 + a 1s + a 2s 2
We might factor the denominator using the quadratic formula, then
construct Bode diagram as the combination of two real poles:
1                                  a1                  4a 2
G(s) =                                with        s1 = –      1–           1– 2
s
1– s             s
1– s                           2a 2                 a1
1             2

a1                    4a 2
s2 = –        1+         1–
2a 2                  a21

• If 4a2 ≤ a12, then the roots s1 and s2 are real. We can construct Bode
diagram as the combination of two real poles.
• If 4a2 > a12, then the roots are complex. In Section 8.1.1, the
assumption was made that ω0 is real; hence, the results of that
section cannot be applied and we need to do some additional work.

Fundamentals of Power Electronics                      38              Chapter 8: Converter Transfer Functions
Approach 2: Define a standard normalized form

G(s) =           1                   or           G(s) =             1
s   s
1 + 2ζ ω + ω
2                                    s
1+ s + ω
2

0   0                                      Qω0    0

• When the coefficients of s are real and positive, then the parameters ζ,
ω0, and Q are also real and positive
• The parameters ζ, ω0, and Q are found by equating the coefficients of s
• The parameter ω0 is the angular corner frequency, and we can define f0
= ω0/2π
• The parameter ζ is called the damping factor. ζ controls the shape of the
exact curve in the vicinity of f = f0. The roots are complex when ζ < 1.
• In the alternative form, the parameter Q is called the quality factor. Q
also controls the shape of the exact curve in the vicinity of f = f0. The
roots are complex when Q > 0.5.

Fundamentals of Power Electronics           39             Chapter 8: Converter Transfer Functions
The Q-factor
In a second-order system, ζ and Q are related according to
Q= 1
2ζ

Q is a measure of the dissipation in the system. A more general
definition of Q, for sinusoidal excitation of a passive element or system
is
(peak stored energy)
Q = 2π
(energy dissipated per cycle)

For a second-order passive system, the two equations above are
equivalent. We will see that Q has a simple interpretation in the Bode
diagrams of second-order transfer functions.

Fundamentals of Power Electronics           40            Chapter 8: Converter Transfer Functions
Analytical expressions for f0 and Q

Two-pole low-pass filter               v2(s)
example: we found that        G(s) =         =      1
v1(s) 1 + s L + s 2LC
R

Equate coefficients of like                   1
powers of s with the
G(s) =
s
1+ s + ω
2
standard form                            Qω0    0

Result:                          ω0    1
f0 =    =
2π 2π LC
Q=R C
L

Fundamentals of Power Electronics       41            Chapter 8: Converter Transfer Functions

In the form          G(s) =           1
s
1+ s + ω
2

Qω0    0

let s = jω and find magnitude:             G( jω) =                     1
ω      2 2        ω
+ 12 ω
2
1– ω
0            Q    0

Asymptotes are                    || G(jω) ||dB
0 dB
0 dB
G →1            for ω << ω0
–2
–2                     –20 dB                                       f
f                                                                         f0
G →                   for ω >> ω0
f0                           –40 dB
–60 dB
0.1f0           f0            10f0
f

Fundamentals of Power Electronics                    42               Chapter 8: Converter Transfer Functions
Deviation of exact curve from magnitude asymptotes

G( jω) =                 1
ω        2 2        ω
+ 12 ω
2
1– ω
0              Q    0

At ω = ω0, the exact magnitude is
G( jω0) = Q         or, in dB:              G( jω0)   dB
= Q    dB

The exact curve has magnitude
|| G ||
Q at f = f0. The deviation of the
exact curve from the                            0 dB                                    | Q |dB
asymptotes is | Q |dB
f0

Fundamentals of Power Electronics              43                  Chapter 8: Converter Transfer Functions
Two-pole response: exact curves

0°
Q=∞                                                                  Q=∞
Q=5                                                                  Q = 10
10dB                                                                                          Q =5
Q=2                                                                   Q=2
Q=1                                                                        Q=1
-45°
Q = 0.7
Q = 0.7
Q = 0.5

0dB                                                               Q = 0.2
Q = 0.1

Q = 0.5                         ∠G    -90°
|| G ||dB

-10dB
Q = 0.2
-135°

Q = 0.1

-20dB
0.3      0.5        0.7     1      2   3
-180°
f / f0                         0.1                  1                      10
f / f0

Fundamentals of Power Electronics                          44                  Chapter 8: Converter Transfer Functions
8.1.7. The low-Q approximation

Given a second-order denominator polynomial, of the form

G(s) =          1                 or                          1
1 + a 1s + a 2s 2                     G(s) =
s
1+ s + ω
2

Qω0    0

When the roots are real, i.e., when Q < 0.5, then we can factor the
denominator, and construct the Bode diagram using the asymptotes
for real poles. We would then use the following normalized form:

G(s) =              1
s
1+ ω            s
1+ ω
1            2

This is a particularly desirable approach when Q << 0.5, i.e., when the
corner frequencies ω1 and ω2 are well separated.

Fundamentals of Power Electronics                        45           Chapter 8: Converter Transfer Functions
An example

A problem with this procedure is the complexity of the quadratic
formula used to find the corner frequencies.
R-L-C network example:                                      L
+
v2(s)        1
G(s) =         =                      v1(s)   +             C            R      v2(s)
v1(s) 1 + s L + s 2LC                –
R
–

Use quadratic formula to factor denominator. Corner frequencies are:
2
L/R±     L / R – 4 LC
ω1 , ω2 =
2 LC

Fundamentals of Power Electronics              46               Chapter 8: Converter Transfer Functions
Factoring the denominator

2
L/R±    L / R – 4 LC
ω1 , ω2 =
2 LC

This complicated expression yields little insight into how the corner
frequencies ω1 and ω2 depend on R, L, and C.
When the corner frequencies are well separated in value, it can be
shown that they are given by the much simpler (approximate)
expressions
ω1 ≈ R ,     ω2 ≈ 1
L           RC

ω1 is then independent of C, and ω2 is independent of L.
These simpler expressions can be derived via the Low-Q Approximation.

Fundamentals of Power Electronics                  47          Chapter 8: Converter Transfer Functions
Derivation of the Low-Q Approximation

Given
G(s) =           1
s
1+ s + ω
2

Qω0    0

Use quadratic formula to express corner frequencies ω1 and ω2 in
terms of Q and ω0 as:

ω 1–         1 – 4Q 2                    ω 1+    1 – 4Q 2
ω1 = 0                                   ω2 = 0
Q            2                           Q       2

Fundamentals of Power Electronics                  48            Chapter 8: Converter Transfer Functions
Corner frequency ω2

ω 1+         1 – 4Q 2                    1
ω2 = 0
Q            2              F(Q)
0.75
can be written in the form
ω0                                   0.5
ω2 =      F(Q)
Q

where                                      0.25

F(Q) = 1 1 +        1 – 4Q 2                 0
2                                          0   0.1         0.2       0.3       0.4       0.5
For small Q, F(Q) tends to 1.                                               Q
We then obtain
For Q < 0.3, the approximation F(Q) = 1 is
ω0
ω2 ≈         for Q << 1           within 10% of the exact value.
Q              2

Fundamentals of Power Electronics                49                   Chapter 8: Converter Transfer Functions
Corner frequency ω1

ω 1–         1 – 4Q 2                    1
ω1 = 0
Q            2              F(Q)
0.75
can be written in the form
Q ω0                                 0.5
ω1 =
F(Q)
where                                      0.25

F(Q) = 1 1 +        1 – 4Q 2                 0
2                                          0   0.1         0.2       0.3       0.4       0.5
For small Q, F(Q) tends to 1.                                               Q
We then obtain
For Q < 0.3, the approximation F(Q) = 1 is
ω1 ≈ Q ω0       for Q << 1        within 10% of the exact value.
2

Fundamentals of Power Electronics                50                   Chapter 8: Converter Transfer Functions
The Low-Q Approximation

|| G ||dB                      Q f0
f1 =
F(Q)
f0          f0F(Q)
0dB                      ≈ Q f0
f2 =
Q
f0
≈
Q

Fundamentals of Power Electronics               51           Chapter 8: Converter Transfer Functions
R-L-C Example

For the previous example:

v (s)                          ω0    1
G(s) = 2 =         1               f0 = =
2π 2π LC
v1(s) 1 + s L + s 2LC
R                Q=R C
L

Use of the Low-Q Approximation leads to

ω1 ≈ Q ω0 = R   C 1 =R
L LC L
ω
ω2 ≈ 0 = 1       1   = 1
Q      LC R   C  RC
L

Fundamentals of Power Electronics           52          Chapter 8: Converter Transfer Functions
8.1.8. Approximate Roots of an
Arbitrary-Degree Polynomial

Generalize the low-Q approximation to obtain approximate
factorization of the nth-order polynomial

P(s) = 1 + a 1 s + a 2 s 2 +    + an s n

It is desired to factor this polynomial in the form
P(s) = 1 + τ 1 s 1 + τ 2 s         1 + τn s

When the roots are real and well separated in value, then approximate
analytical expressions for the time constants τ1, τ2, ... τn can be found,
that typically are simple functions of the circuit element values.
Objective:        find a general method for deriving such expressions.
Include the case of complex root pairs.

Fundamentals of Power Electronics                     53                 Chapter 8: Converter Transfer Functions
Derivation of method

Multiply out factored form of polynomial, then equate to original form
(equate like powers of s):
a1 = τ1 + τ2 +       + τn
a2 = τ1 τ2 +        + τn + τ2 τ3 +   + τn +
a 3 = τ1τ2 τ3 +      + τn + τ2τ3 τ4 +   + τn +

a n = τ1τ2τ3   τn

• Exact system of equations relating roots to original coefficients
• Exact general solution is hopeless
• Under what conditions can solution for time constants be easily
approximated?

Fundamentals of Power Electronics                    54               Chapter 8: Converter Transfer Functions
Approximation of time constants
when roots are real and well separated

a1 = τ1 + τ2 +       + τn
System of equations:                 a2 = τ1 τ2 +        + τn + τ2 τ3 +        + τn +
(from previous slide)                a 3 = τ1τ2 τ3 +      + τn + τ2τ3 τ4 +          + τn +

a n = τ1τ2τ3   τn

Suppose that roots are real and well-separated, and are arranged in
decreasing order of magnitude:
τ 1 >> τ 2 >>    >> τ n

Then the first term of each equation is dominant
⇒ Neglect second and following terms in each equation above

Fundamentals of Power Electronics                   55                   Chapter 8: Converter Transfer Functions
Approximation of time constants
when roots are real and well separated

System of equations:                   Solve for the time
(only first term in each
constants:
equation is included)                      τ1 ≈ a1
a1 ≈ τ1                                    a2
a 2 ≈ τ 1τ 2                          τ2 ≈
a1
a 3 ≈ τ 1τ 2τ 3                            a
τ3 ≈ 3
a2
a n = τ 1τ 2τ 3   τn

an
τn ≈
an – 1

Fundamentals of Power Electronics       56        Chapter 8: Converter Transfer Functions
Result
when roots are real and well separated

If the following inequalities are satisfied
a2    a                  an
a 1 >>         >> 3 >>         >>
a1    a2                an – 1

Then the polynomial P(s) has the following approximate factorization
a2            a3                   an
P(s) ≈ 1 + a 1 s 1 +      s     1+      s          1+           s
a1            a2                  an – 1
•     If the an coefficients are simple analytical functions of the element
values L, C, etc., then the roots are similar simple analytical
functions of L, C, etc.
•     Numerical values are used to justify the approximation, but
analytical expressions for the roots are obtained

Fundamentals of Power Electronics             57                Chapter 8: Converter Transfer Functions
When two roots are not well separated
then leave their terms in quadratic form

Suppose inequality k is not satisfied:
a2                     ak                    ak + 1                 an
a 1 >>        >>          >>                    ✖
>>             >>       >>
a1                    ak – 1                  ak                   an – 1
↑
not
satisfied
Then leave the terms corresponding to roots k and (k + 1) in quadratic
form, as follows:
a2                     ak       a                       an
P(s) ≈ 1 + a 1 s 1 +             s            1+           s + k + 1 s2       1+            s
a1                    ak – 1    ak – 1                 an – 1
This approximation is accurate provided
a2                  ak       a a             a                             an
a 1 >>       >>        >>           >> k – 22 k + 1 >> k + 2 >>               >>
a1                 ak – 1      ak – 1        ak + 1                       an – 1

Fundamentals of Power Electronics                         58               Chapter 8: Converter Transfer Functions
When the first inequality is violated
A special case for quadratic roots

When inequality 1 is not satisfied:
a2    a3                  an
a1      ✖
>>          >>    >>           >>
a1    a2                 an – 1
↑
not
satisfied
Then leave the first two roots in quadratic form, as follows:
a3                    an
P(s) ≈ 1 + a 1s + a 2s 2 1 +          s        1+              s
a2                   an – 1
This approximation is justified provided
a2
2           a    a                            an
>> a 1 >> 3 >> 4 >>                   >>
a3           a2   a3                          an – 1

Fundamentals of Power Electronics              59                 Chapter 8: Converter Transfer Functions
Other cases

•     When several isolated inequalities are violated
—Leave the corresponding roots in quadratic form
—See next two slides

•     When several adjacent inequalities are violated
—Then the corresponding roots are close in value
—Must use cubic or higher-order roots

Fundamentals of Power Electronics            60           Chapter 8: Converter Transfer Functions

In the case when inequality k is not satisfied:

a2                    ak      a                      an
a1 >          >             >          ≥ k+1 >           >
a1                   ak – 1     ak                  an – 1

Then leave the corresponding roots in quadratic form:

a2              ak       a                       an
P(s) ≈ 1 + a 1 s           1+      s      1+          s + k + 1 s2        1+           s
a1             ak – 1    ak – 1                 an – 1

This approximation is accurate provided that
a2                       ak      a    a    a                              an
a1 >          >             >             > k–2 k+1 > k+2 >                    >
a1                      ak – 1     a2 – 1
k      ak + 1                        an – 1

(derivation is similar to the case of well-separated roots)

Fundamentals of Power Electronics                    61              Chapter 8: Converter Transfer Functions
When the first inequality is not satisfied

The formulas of the previous slide require a special form for the case when
the first inequality is not satisfied:
a2   a                 an
a1 ≥         > 3 >          >
a1   a2               an – 1

We should then use the following form:

a3            an
P(s) ≈ 1 + a 1s +   a 2s 2   1+    s      1+        s
a2           an – 1

The conditions for validity of this approximation are:

a2        a   a                               an
2
> a1 > 3 > 4 >                    >
a3        a2  a3                             an – 1

Fundamentals of Power Electronics                62                Chapter 8: Converter Transfer Functions
Example
Damped input EMI filter

L1

ig                                         ic
L2         R                               Converter
vg   +                                        C
–

L + L2
i g(s)             1+s 1
G(s) =        =                    R
i c(s)        L + L2                 L L C
1+s 1     + s 2 L 1C + s 3 1 2
R                     R

Fundamentals of Power Electronics                 63          Chapter 8: Converter Transfer Functions
Example
Approximate factorization of a third-order denominator

The filter transfer function from the previous slide is
L + L2
i g(s)            1+s 1
G(s) =        =                  R
i c(s)       L1 + L2   2 L C + s 3 L 1 L 2C
1+s         +s 1
R                      R
—contains a third-order denominator, with the following coefficients:

L1 + L2
a1 =
R
a 2 = L 1C
LLC
a3 = 1 2
R

Fundamentals of Power Electronics              64            Chapter 8: Converter Transfer Functions
Real roots case
Factorization as three real roots:
L1 + L2                 L1            L2
1+s                1 + sRC              1+s
R                  L1 + L2         R
This approximate analytical factorization is justified provided
L1 + L2         L1      L
>> RC         >> 2
R          L1 + L2    R
Note that these inequalities cannot be satisfied unless L1 >> L2. The
above inequalities can then be further simplified to
L1         L
>> RC >> 2
R           R
And the factored polynomial reduces to             • Illustrates in a simple
way how the roots
L1                    L2
1+s           1 + sRC 1 + s                 depend on the
R                     R             element values

Fundamentals of Power Electronics                65         Chapter 8: Converter Transfer Functions
When the second inequality is violated

L1 + L2         L1                     L2
>> RC                ✖
>>
R          L1 + L2                  R
↑
not
satisfied
Then leave the second and third roots in quadratic form:
a2    a3 2
P(s) = 1 + a 1s 1 + a s + a s
1     1

which is

L1 + L2               L1
1+s               1 + sRC           + s 2 L 1||L 2 C
R                L1 + L2

Fundamentals of Power Electronics               66             Chapter 8: Converter Transfer Functions
Validity of the approximation

This is valid provided
L1 + L2         L1        L 1||L 2                (use a0 = 1)
>> RC         >>           RC
R          L1 + L2    L1 + L2

These inequalities are equivalent to
L1
L 1 >> L 2, and          >> RC
R
It is no longer required that RC >> L2/R
The polynomial can therefore be written in the simplified form

L1
1+s            1 + sRC + s 2L 2C
R

Fundamentals of Power Electronics               67       Chapter 8: Converter Transfer Functions
When the first inequality is violated

L1 + L2                     L1       L2
✖
>>       RC         >>
R                      L1 + L2    R
↑
not
satisfied
Then leave the first and second roots in quadratic form:
a
P(s) = 1 + a 1s + a 2s 2 1 + a 3 s
2

which is

L1 + L2                       L2
1+s            + s 2L 1C     1+s
R                          R

Fundamentals of Power Electronics                68        Chapter 8: Converter Transfer Functions
Validity of the approximation

This is valid provided
L 1RC   L + L2   L
>> 1     >> 2
L2       R      R

These inequalities are equivalent to
L2
L 1 >> L 2, and RC >>
R
It is no longer required that L1/R >> RC
The polynomial can therefore be written in the simplified form
L1                    L2
1+s       + s 2 L 1C   1+s
R                     R

Fundamentals of Power Electronics                69      Chapter 8: Converter Transfer Functions
8.2. Analysis of converter transfer functions

8.2.1. Example: transfer functions of the buck-boost converter
8.2.2. Transfer functions of some basic CCM converters
8.2.3. Physical origins of the right half-plane zero in converters

Fundamentals of Power Electronics         70          Chapter 8: Converter Transfer Functions
8.2.1. Example: transfer functions of the
buck-boost converter

Small-signal ac model of the buck-boost converter, derived in Chapter 7:

L
1:D                               D' : 1

+
–
+
i(s)
(Vg – V) d (s)
vg (s)    +   I d (s)                                                             I d (s)
–                                                                                    C v(s)      R

–

Fundamentals of Power Electronics                    71                    Chapter 8: Converter Transfer Functions
Definition of transfer functions

The converter contains two inputs, d(s) and vg(s) and one output, v(s)
Hence, the ac output voltage variations can be expressed as the
superposition of terms arising from the two inputs:

v(s) = Gvd(s) d(s) + Gvg(s) vg(s)

The control-to-output and line-to-output transfer functions can be
defined as
v(s)                              v(s)
Gvd(s) =                        and Gvg(s) =
d(s)                              vg(s)   d(s) = 0
vg(s) = 0

Fundamentals of Power Electronics                            72              Chapter 8: Converter Transfer Functions
Derivation of
line-to-output transfer function Gvg(s)

Set d sources to                      1:D            D' : 1
zero:                                                                         +
L

vg (s)   +
–                                        C v(s)       R

–

Push elements through                                                       +
L
transformers to output                                  D' 2
side:                                 vg(s) – D    +
C v(s)        R
D'   –

–

Fundamentals of Power Electronics                73         Chapter 8: Converter Transfer Functions
Derivation of transfer functions

+
Use voltage divider formula                                                   L
to solve for transfer function:                                               D' 2
+
vg(s) – D
D'   –                       C v(s)       R
R || 1
v(s)                           sC
Gvg(s) =                    =– D                                                                  –
vg(s)               D' sL           1
d(s) = 0          2 + R || sC
D'

Expand parallel combination and express as a rational fraction:
R
1 + sRC
Gvg(s) = – D
D' sL         R
2 + 1 + sRC
D'
We aren’t done yet! Need to
write in normalized form, where
= – D            R
the coefficient of s0 is 1, and
D'      sL + s 2RLC
R+ 2
D'      D' 2     then identify salient features

Fundamentals of Power Electronics                     74                Chapter 8: Converter Transfer Functions
Derivation of transfer functions

Divide numerator and denominator by R. Result: the line-to-output
transfer function is
v(s)
Gvg(s) =                = – D            1
vg(s) d(s) = 0     D' 1 + s L + s 2 LC
D' 2 R   D' 2

which is of the following standard form:

Gvg(s) = Gg0        1
s
1+ s + ω
2

Qω0    0

Fundamentals of Power Electronics                75     Chapter 8: Converter Transfer Functions
Salient features of the line-to-output transfer function

Equate standard form to derived transfer function, to determine
expressions for the salient features:

Gg0 = – D
D'
1 = LC
ω 2 D' 2         ω0 = D'
0                   LC

1 = L                      C
Qω0 D' 2R        Q = D'R
L

Fundamentals of Power Electronics          76           Chapter 8: Converter Transfer Functions
Derivation of
control-to-output transfer function Gvd(s)
L
D' : 1

+
–
In small-signal model,                                                                               +
(Vg – V) d (s)
set vg source to zero:
I d (s)      C v(s)          R

–

+
L
Push all elements to            Vg – V         –           D' 2
output side of                         d (s)   +                  I d(s)          C          v(s)       R
D'
transformer:
–

There are two d sources. One way to solve the model is to use superposition,
expressing the output v as a sum of terms arising from the two sources.

Fundamentals of Power Electronics                  77                     Chapter 8: Converter Transfer Functions
Superposition

With the voltage                                               +                                   R || 1
source only:                              L                                 v(s)
= –
Vg – V            sC
Vg – V            –     D' 2                              d (s)       D'     sL + R || 1
d (s)                      C         v(s)     R                                 sC
D' 2
+
D'

–

With the current                                              +
source alone:                                                               v(s)
= I sL2 || R || 1
L                                    d(s)     D'          sC
I d (s)                     C         v(s)     R
D' 2

–

Vg – V             R || 1
Total:                                             sC
Gvd(s) = –                                + I sL2 || R || 1
D'            sL + R || 1        D'          sC
D' 2      sC
Fundamentals of Power Electronics                       78              Chapter 8: Converter Transfer Functions
Control-to-output transfer function

Express in normalized form:

1–s      LI
v(s)                     Vg – V            Vg – V
Gvd(s) =                      = –
d(s)                      D' 2           L + s 2 LC
vg(s) = 0                  1+s
D' 2 R   D' 2

This is of the following standard form:

s
1– ω
z
Gvd(s) = Gd0
s
1+ s + ω
2

Qω0    0

Fundamentals of Power Electronics                          79            Chapter 8: Converter Transfer Functions
Salient features of control-to-output transfer function

Vg – V   Vg
Gd0 = –        =– 2 = V
D'     D'  DD'

Vg – V D' R
ωz =         =         (RHP)
LI     DL

ω0 = D'
LC

Q = D'R       C
L

— Simplified using the dc relations:   V = – D Vg
D'
I=– V
D' R

Fundamentals of Power Electronics            80          Chapter 8: Converter Transfer Functions
Plug in numerical values

Suppose we are given the             Then the salient features
following numerical values:          have the following numerical
values:

D = 0.6                    Gg0 = D = 1.5 ⇒ 3.5 dB
R = 10Ω                            D'
Vg = 30V                            V
Gd0 =       = 187.5 V ⇒ 45.5 dBV
L = 160µH                          DD'
ω
C = 160µF                    f0 = 0 = D' = 400 Hz
2π 2π LC
Q = D'R C = 4 ⇒ 12 dB
L
ωz D' 2R
fz =    =       = 2.65 kHz
2π 2πDL

Fundamentals of Power Electronics        81          Chapter 8: Converter Transfer Functions
Bode plot: control-to-output transfer function

80 dBV        || Gvd ||
|| Gvd ||                                                                                                         ∠ Gvd
60 dBV    Gd0 = 187 V
⇒ 45.5 dBV                           Q = 4 ⇒ 12 dB
400 Hz
20 dBV
10 -1/2Q f0
0˚              300 Hz
0 dBV                                                   fz                                            0˚
∠ Gvd               fz /10                 2.6 kHz
–20 dBV                      260 Hz                                                                   –90˚

–40 dBV                                                                                               –180˚
1/2Q                        10fz
10    f0
533 Hz                          26 kHz             –270˚
–270˚
10 Hz               100 Hz                 1 kHz            10 kHz        100 kHz           1 MHz

f

Fundamentals of Power Electronics                                   82                 Chapter 8: Converter Transfer Functions
Bode plot: line-to-output transfer function

20 dB
|| Gvg ||            Gg0 = 1.5                                                            ∠ Gvg
⇒ 3.5 dB                       Q = 4 ⇒ 12 dB
0 dB
|| Gvg ||              f0
–20 dB

–40 dB
10 –1/2Q 0 f0
0˚         300 Hz
–60 dB                                                                          0˚
∠ Gvg
–80 dB                                                                         –90˚

–180˚
–180˚
1/2Q 0
10     f0
533 Hz
–270˚
10 Hz            100 Hz              1 kHz          10 kHz          100 kHz

f
Fundamentals of Power Electronics                      83                    Chapter 8: Converter Transfer Functions
8.2.2. Transfer functions of
some basic CCM converters

Table 8.2. S alient features of the small-signal CCM transfer functions of some basic dc-dc converters

Converter             Gg0                 Gd0             ω0               Q                    ωz
V                  1
buck               D               D                                 R C                    ∞
LC                L
1               V                  D'
D'R C                  D' 2R
boost              D'               D'                 LC                 L                  L
D
– D'              V                  D'
D'R C                  D' 2 R
buck-boost                           D D'
2
LC                 L                 DL

where the transfer functions are written in the standard forms
s
1– ω
z
Gvg(s) = Gg0            1
Gvd(s) = Gd0                                                                s           2
s           2                                 1+ s + ω
1+ s + ω                                               Qω0    0
Qω0    0

Fundamentals of Power Electronics                     84                Chapter 8: Converter Transfer Functions
8.2.3. Physical origins of the right half-plane zero

s
G(s) = 1 – ω
1
0

uin(s)                         +      uout(s)
–

s
ωz
• phase reversal at
high frequency
• transient response:
output initially tends
in wrong direction

Fundamentals of Power Electronics            85        Chapter 8: Converter Transfer Functions
Two converters whose CCM control-to-output
transfer functions exhibit RHP zeroes

iD   Ts
= d' iL   Ts

L                     2     iD(t)
+
Boost                                                        iL(t)
1
vg   +                                               C           R       v
–

–

iD(t)

Buck-boost                                               1           2                           +

iL(t)
vg    +                                    C            R       v
–                     L

–

Fundamentals of Power Electronics                            86                  Chapter 8: Converter Transfer Functions
Waveforms, step increase in duty cycle

iL(t)

iD   Ts
= d' iL   Ts

• Increasing d(t)                                                                         t
causes the average             iD(t)
diode current to                                      〈iD(t)〉T
s
initially decrease
• As inductor current
increases to its new                                                                    t
equilibrium value,           | v(t) |
average diode
current eventually
increases
t
d = 0.4   d = 0.6

Fundamentals of Power Electronics                       87                      Chapter 8: Converter Transfer Functions
Impedance graph paper

80dBΩ                                                                          100          10kΩ
pF

60dBΩ       10H
1nF          1kΩ

40dBΩ       1H
10n          100Ω
F
m    H
20dBΩ       100
100          10Ω
nF
H
0dBΩ       10m                                                                             1Ω
1µF
H
–20dBΩ       1m                                                                              100mΩ
10µ
F
µH
–40dBΩ       100                                                                             10mΩ
100
100                                       µF
1F             mF        nH   10m          1m
H              1µH            100         F 10nH       F 1nH
–60dBΩ       10µ
1mΩ
10Hz                100Hz         1kHz            10kHz        100kHz                1MHz

Fundamentals of Power Electronics                     88            Chapter 8: Converter Transfer Functions
Transfer functions predicted by canonical model

He(s)
e(s) d(s)
+         1 : M(D)
–

+                             +
Le

+               j(s) d(s)
Zin                                     Zout
vg(s)                                       ve(s)              C           v(s)       R
–

–                               –

{
{
Z1                Z2

Fundamentals of Power Electronics             89                Chapter 8: Converter Transfer Functions
Output impedance Zout: set sources to zero

Zout
Le {           C               R

Z1
{     Z2

Zout = Z1 || Z2

Fundamentals of Power Electronics               90         Chapter 8: Converter Transfer Functions
Graphical construction of output impedance

1                          || Z1 || = ωLe
ωC
R
Q = R / R0
R0
f0

|| Zout ||

Fundamentals of Power Electronics     91           Chapter 8: Converter Transfer Functions
Graphical construction of
filter effective transfer function

ωL e                             Q = R / R0
=1
ωL e

f0                1 /ωC               1
=
ωL e           ω 2L eC
Z out
He =
Z1

Fundamentals of Power Electronics        92          Chapter 8: Converter Transfer Functions
Boost and buck-boost converters: Le = L / D’ 2

1                                                                 ωL
ωC                                             increasing
D               D' 2
R
Q = R / R0
R0
f0

|| Zout ||

Fundamentals of Power Electronics     93            Chapter 8: Converter Transfer Functions
8.4. Measurement of ac transfer functions
and impedances

Network Analyzer
Injection source          Measured inputs         Data
vz           vz                                                  Data bus
magnitude   frequency                         vy                    to computer
17.3 dB
vx

vz               vx           vy
output           input        input
+    –       +   –     vy
– 134.7˚
+
–

vx

Fundamentals of Power Electronics                     94         Chapter 8: Converter Transfer Functions
Swept sinusoidal measurements

• Injection source produces sinusoid vz of controllable amplitude and
frequency
• Signal inputs vx and vy perform function of narrowband tracking
voltmeter:
Component of input at injection source frequency is measured
Narrowband function is essential: switching harmonics and other
noise components are removed
• Network analyzer measures
vy                vy
vx
and    ∠v     x

Fundamentals of Power Electronics           95           Chapter 8: Converter Transfer Functions
Measurement of an ac transfer function

Network Analyzer
Injection source              Measured inputs                  Data                          • Potentiometer
vz           vz                                                            Data bus
magnitude   frequency                                      vy
vx
–4.7 dB     to computer        establishes correct
vz                   vx            vy                                                    quiescent operating
output               input
+     –
input
+   –              vy
– 162.8˚
point
+
–

vx

• Injection sinusoid
coupled to device
DC                                                                                vy(s)
blocking                                                                                     = G(s)         input via dc blocking
capacitor                                                                               vx(s)                capacitor
VCC                                                                                                    • Actual device input
and output voltages
DC                                                                                                             are measured as vx
bias
output
input

G(s)
• Dynamics of blocking
capacitor are irrelevant
Device
under test

Fundamentals of Power Electronics                                                           96                  Chapter 8: Converter Transfer Functions
Measurement of an output impedance

v(s)
Z(s) =
i(s)
VCC                                                                                           Zs
Device

{
under test                                               DC blocking
DC                                                                      i out            capacitor R
source
bias
output
input

G(s)                    Zout                    probe                         +   vz
–

voltage
probe
vy(s)
Z out(s) =
i out(s)   amplifier
ac input
=0                    + – + –
vy  vx

Fundamentals of Power Electronics                                97                  Chapter 8: Converter Transfer Functions
Measurement of output impedance

• Treat output impedance as transfer function from output current to
output voltage:
v(s)                     vy(s)
Z(s) =              Z out(s) =
i(s)                    i out(s) amplifier
=0
ac input

• Potentiometer at device input port establishes correct quiescent
operating point
• Current probe produces voltage proportional to current; this voltage
is connected to network analyzer channel vx
• Network analyzer result must be multiplied by appropriate factor, to
account for scale factors of current and voltage probes

Fundamentals of Power Electronics          98            Chapter 8: Converter Transfer Functions
Measurement of small impedances

injection              Network Analyzer
Grounding problems                     Impedance                          source
cause measurement                       under test                        return                   Injection source
connection
i out                           Rsource
to fail:
Injection current can                 Z(s)                                           Zrz               +
–   vz
i out k i out

{
two paths. Injection
current which returns                            (1 – k) i out
Measured
via voltage probe ground                                                                                inputs
voltage
induces voltage drop in                          probe                                                      +
vx
voltage probe, corrupting the               voltage                                                         –
probe
measurement. Network                        return                         Zprobe                           +
analyzer measures                           connection

Z + (1 – k) Z probe = Z + Z probe || Z rz                        +  {              –
(1 – k) i out Z probe
– vy

For an accurate measurement, require
Z >>        Z probe || Z rz
Fundamentals of Power Electronics                          99                       Chapter 8: Converter Transfer Functions

injection                          Network Analyzer
Impedance                                     source
Injection                   under test                                   return                               Injection source
connection
current must                                       i out                                      1:n       Rsource
now return
+
entirely                      Z(s)                                                    Zrz                         –   vz
i out

{
through
transformer.                                       0
inputs
voltage is                               voltage
induced in                               probe                                                                         +
vx
–
voltage probe                        voltage
probe
ground                               return                                  Zprobe                                    +
connection
{
connection                                                                                                             – vy
+                   –
0V

Fundamentals of Power Electronics                                        100                  Chapter 8: Converter Transfer Functions
8.5. Summary of key points

1. The magnitude Bode diagrams of functions which vary as (f / f0)n
have slopes equal to 20n dB per decade, and pass through 0dB at
f = f0.
2. It is good practice to express transfer functions in normalized pole-
zero form; this form directly exposes expressions for the salient
features of the response, i.e., the corner frequencies, reference
gain, etc.
3. The right half-plane zero exhibits the magnitude response of the
left half-plane zero, but the phase response of the pole.
4. Poles and zeroes can be expressed in frequency-inverted form,
when it is desirable to refer the gain to a high-frequency asymptote.

Fundamentals of Power Electronics           101           Chapter 8: Converter Transfer Functions
Summary of key points

5. A two-pole response can be written in the standard normalized
form of Eq. (8-53). When Q > 0.5, the poles are complex
conjugates. The magnitude response then exhibits peaking in the
vicinity of the corner frequency, with an exact value of Q at f = f0.
High Q also causes the phase to change sharply near the corner
frequency.
6. When the Q is less than 0.5, the two pole response can be plotted
as two real poles. The low- Q approximation predicts that the two
poles occur at frequencies f0 / Q and Qf0. These frequencies are
within 10% of the exact values for Q ≤ 0.3.
7. The low- Q approximation can be extended to find approximate
roots of an arbitrary degree polynomial. Approximate analytical
expressions for the salient features can be derived. Numerical
values are used to justify the approximations.

Fundamentals of Power Electronics            102           Chapter 8: Converter Transfer Functions
Summary of key points

8. Salient features of the transfer functions of the buck, boost, and buck-
boost converters are tabulated in section 8.2.2. The line-to-output
transfer functions of these converters contain two poles. Their control-
to-output transfer functions contain two poles, and may additionally
contain a right half-pland zero.
9. Approximate magnitude asymptotes of impedances and transfer
functions can be easily derived by graphical construction. This
approach is a useful supplement to conventional analysis, because it
yields physical insight into the circuit behavior, and because it
exposes suitable approximations. Several examples, including the
impedances of basic series and parallel resonant circuits and the
transfer function He(s) of the boost and buck-boost converters, are
worked in section 8.3.
10. Measurement of transfer functions and impedances using a network
analyzer is discussed in section 8.4. Careful attention to ground
connections is important when measuring small impedances.
Fundamentals of Power Electronics           103           Chapter 8: Converter Transfer Functions

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