ECE317 : Feedback and Controlweb.cecs.pdx.edu/~tymerski/ece317/ECE317 L8 System... · 2019. 4....
Transcript of ECE317 : Feedback and Controlweb.cecs.pdx.edu/~tymerski/ece317/ECE317 L8 System... · 2019. 4....
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ECE317 : Feedback and Control
Lecture System Modelling: dc-dc converter
Dr. Richard Tymerski
Dept. of Electrical and Computer Engineering
Portland State University
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Course roadmap
2
Laplace transform
Transfer function
Block Diagram
Linearization
Models for systems
• electrical
• mechanical
• example system
Modeling Analysis Design
Stability
• Pole locations
• Routh-Hurwitz
Time response
• Transient
• Steady state (error)
Frequency response
• Bode plot
Design specs
Frequency domain
Bode plot
Compensation
Design examples
Matlab & PECS simulations & laboratories
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Filtering a Square Wave
• Given a square wave input with frequency :
• into a low pass filter with corner frequency :
• If what is v (the voltage across R)?
𝑓𝑠𝑞
𝑓𝑐
𝑓𝐶 <<< 𝑓𝑠𝑞
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Consider the spectrum
Fourier Series:
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Spectrum cont’d.
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Spectrum cont’d.
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Spectrum cont’d.
• DC (zero freq.) component is ½ the peak• All even harmonics are 0• Therefore only odd harmonics are present
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Spectrum cont’d. • Amplitude spectrum (for 10 V pk-pk square wave):
• Zero freq. amplitude:
• 1st harmonic peak amplitude: • 2nd harmonic peak amplitude: • 3rd harmonic peak amplitude:
2 ×1
𝜋= 6.366 V
2 ×1
3𝜋= 2.122 V
0 V
5 V
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Filtering the square wave• Low pass filter magnitude response:• Filter component values:• Filter corner frequency:• Square wave:
• Attenuation at 1st harmonic (40 kHz) is -71 dB• Attenuation at 3rd harmonic (120 kHz) is -90 dB
𝐿 = 560 𝜇𝐻, 𝐶 = 100 𝜇𝐹, 𝑅 = 5 𝛺
𝑓𝐶 = 673 𝐻𝑧
𝑓𝑠𝑞: = 40 𝑘𝐻𝑧
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Filtering the square wave, cont’d
• First harmonic filtered peak amplitude:• -71 dB attenuation attenuation by a factor:
10−
71
20 = 0.000283
• peak-peak amplitude:
6.366 × 0.000283 = 1.8 mV
2 × 1.8 = 3.6 mV
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PECS simulation:
• Peak-peak output ripple: 3.5 mV
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Filtering the square wave, cont’d
• PECS simulation result:Peak-peak output ripple: 3.5 mV
• Fourier analysis result:First harmonic only filtered pk-pk amplitude: 3.6 mV
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Dc-to-dc Buck Converter• Convert a DC voltage to a lower DC voltage level at high efficiency• Chop up the input voltage (Vg) and then filter it• Inclusion of a single-pole, double-throw switch produces a
rectangular waveform to the output filter
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Dc-to-dc Buck Converter, cont’d• Three sections
1. Input voltage, vg2. Single-pole, double-throw switch3. Low pass filter
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Average Switch Modelling, cont’d
𝑣𝑠 =1
𝑇𝑠
0
𝑇𝑠
𝑣𝑠 (𝑡)𝑑𝑡 = 𝑑 ⋅ 𝑣𝑔
x Denotes the average of the variable x.
Note: d and vg are permitted to vary.
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Average Switch Modelling, cont’d
x
Notation:
Let x be a variable of interest, it may represent quantities such as duty ratio, , input voltage, , input voltage to the low pass filter, , and output current, .
𝑑 𝑣𝑔
𝑣𝑠 𝑖𝑜
Due to the switching action the system is time varying. To simplify the analysis we will look at the average behavior of the system and produce a time invariant model of the system.
Therefore we are interested in looking at average signals denoted by . We have just derived the relationship: 𝑣𝑠 = 𝑑 ⋅ 𝑣𝑔
However, to simplify the notation we will drop the angular bracket notation with the understanding that henceforth x now represents an averaged variable. With this simplified notation, the above equation becomes:
𝑣𝑠 = 𝑑 ⋅ 𝑣𝑔
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Average Switch Modelling, cont’d
We will use a caret ‘^’ to indicate small deviations so that represents a small deviation away from the steady state operating point , so that:
𝑣𝑠 = 𝑑 ⋅ 𝑣𝑔
Note that this formula represents the product of two variables. Consequently it is nonlinear. The above is a nonlinear model.
To derive a linear model we will need to linearize around a steady state operating point under the assumption of small-signal deviations from this operating point.
𝑥 = 𝑋 + 𝑥
𝑥
𝑋 𝑥 = 𝑥 − 𝑋 or
This states that an averaged variable 𝑥 in general consists of a steady state average 𝑋 plus a small signal deviation 𝑥 . This relationship will be used next to produce a linear model.
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𝑣𝑠 = 𝑑 ⋅ 𝑣𝑔
𝑑 = 𝐷 + 𝑑
𝑣𝑔 = 𝑉𝑔 + 𝑣𝑔
𝑣𝑠 << 𝑉𝑠𝑣𝑠 = 𝑉𝑠 + 𝑣𝑠
𝑑 << 𝐷
𝑣𝑔 << 𝑉𝑔
𝑤ℎ𝑒𝑟𝑒
𝑤ℎ𝑒𝑟𝑒
𝑤ℎ𝑒𝑟𝑒
𝐿𝑒𝑡:
⇒𝑉𝑠 + 𝑣𝑠 = 𝐷 + 𝑑 ⋅ 𝑉𝑔 + 𝑣𝑔
= 𝐷𝑉𝑔 + 𝐷 𝑣𝑔 + 𝑉𝑔 𝑑 + 𝑑 𝑣𝑔
Average Switch Modelling, cont’d
First order termsDC term(zero order term)
Nonlinear term(higher order terms)
Small signal assumption
• The product of the two small-signal variations will be small and so will be neglected. This leaves only the linear terms. In this way we have linearized the model.
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Average Switch Modelling, cont’d
Equating zero order terms on both sides results in the DC model:
𝑉𝑠 = 𝐷𝑉𝑔
Since the low pass filter has a DC gain of unity (neglecting losses) the DC converter output voltage 𝑉 is given by:
𝑉 = 𝑉𝑠
𝑉 = 𝐷𝑉𝑔
⇒
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Average Switch Modelling, cont’d
Equating first order terms on both sides results in the small-signal model:
𝑣𝑠 = 𝐷 𝑣𝑔 + 𝑉𝑔 𝑑
⇒ 𝑣𝑠
𝑣𝑔= 𝐷, ( 𝑑 = 0)
𝑎𝑛𝑑
𝑣𝑠
𝑑= 𝑉𝑔, 𝑣𝑔 = 0
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𝐺𝑣𝑔(𝑠) 𝑣
𝑣𝑔
System Transfer Functions
≜1) Transfer Function 𝐺𝑣𝑔 𝑠 :
This transfer function quantifies how variations in the input voltage 𝑣𝑔
propagate to appear as variations in the output voltage 𝑣. input 𝑣𝑔 represents a disturbance signal
2) Transfer Function 𝐺𝑣𝑑 𝑠 :𝐺𝑣𝑑 𝑠
𝑣
𝑑≜
This transfer function quantifies how variations in the duty ratio 𝑑propagate to appear as variations in the output voltage 𝑣.
duty ratio 𝑑 represents the control signal
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𝐺𝑣𝑔(𝑠) 𝑣
𝑣𝑔=
𝑣𝑠
𝑣𝑔∙
𝑣
𝑣𝑠= 𝐷 ∙ 𝐺𝐿𝑃𝐹 (𝑠)
𝐺𝑣𝑑 𝑠 𝑣
𝑑=
𝑣𝑠
𝑑∙
𝑣
𝑣𝑠= 𝑉𝑔 ∙ 𝐺𝐿𝑃𝐹 (𝑠)
≜
≜
System Transfer Functions
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Output section of the Buck converter is a low pass filter:
𝐺𝐿𝑃𝐹(𝑠) = 𝑣
𝑣𝑠=
1
1 +𝑠
𝜔0𝑄+
𝑠𝜔0
2
𝑤ℎ𝑒𝑟𝑒
𝑄 =𝐿𝐶
𝑟𝐿𝐶+𝐿
𝑅
, 𝜔0 =1
𝐿𝐶
System Transfer Functions, cont’d
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System Transfer Functions, cont’d
The effect of output current variations, 𝑖𝑜, causing output voltage variations, 𝑣, is quantified by transfer function −𝑍𝑜𝑢𝑡.
Input 𝑖𝑜represents a disturbance signal
𝑍𝑜𝑢𝑡 = 𝑣
− 𝑖𝑜
𝑣
𝑖𝑜(𝑠) = −𝑍𝑜𝑢𝑡
⇒
3) Transfer Function 𝑣
𝑖𝑜𝑠 :
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System Transfer Functions, cont’dfirst subinterval, 𝐷𝑇𝑠: second subinterval, 𝐷′𝑇𝑠:
𝑍𝑜𝑢𝑡 = 𝑠𝐿 + 𝑟𝐿 ∥1
𝑠𝐶∥ 𝑅
−𝑍𝑜𝑢𝑡 = −𝑟𝐿 1+
𝑠𝐿
𝑟𝐿
1+𝑠
𝜔0𝑄+
𝑠
𝜔0
2
⇒
𝑤ℎ𝑒𝑟𝑒
𝑄 =𝐿𝐶
𝑟𝐿𝐶+𝐿
𝑅
, 𝜔0 =1
𝐿𝐶
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System Transfer Functions, cont’d
Addition of the Pulse Width Modulator:
Control-to-output transfer function: 𝑣
𝑣𝑐= 𝐺𝑣𝑑 ∙ 𝐺𝑃𝑊𝑀 =
𝑣
𝑑∙
𝑑
𝑣𝑐
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System Transfer Functions, cont’d
𝐺𝑃𝑊𝑀 ≜ 𝑑
𝑣𝑐Pulse Width Modulator (PWM) ‘describing function’:
The modulator consists of a comparator driven by a sawtooth waveform, 𝑣𝑠𝑎𝑤, at one input and the control signal, 𝑣𝑐, at the other. This produces a rectangular waveform at the output with a duty ratio, d.
Sawtooth waveform
has a peak-to-peak
amplitude of 𝑉𝑀.
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System Transfer Functions, cont’d
𝐺𝑃𝑊𝑀 ≜ 𝑑
𝑣𝑐=
1
𝑉𝑀
• Through a Fourier analysis of the waveform produced by the PWM the ‘describing function’ can be derived.
• A describing function gives the frequency response of the fundamental component in the output spectrum.
• The result is given by:
𝑉𝑀 is the peak-to-peak amplitude of sawtooth waveform.
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𝐺𝑣𝑑 ≜ 𝑣
𝑑
𝑉𝑔
Δ(s)
𝐺𝑣𝑔 ≜ 𝑣
𝑣𝑔
𝐷
Δ(s)
−𝑍𝑜𝑢𝑡≜ 𝑣
𝑖𝑜 −𝑟𝐿 1 +
𝑠𝐿𝑟𝐿
Δ(s)
𝐺𝑃𝑊𝑀 ≜ 𝑑
𝑣𝑐
1
V𝑀
System Transfer Functions, cont’d
Summary of Transfer Functions of the
buck converter and modulator
Δ s = 1 +𝑠
𝜔0𝑄+
𝑠
𝜔0
2, 𝑄 =
𝐿𝐶
𝑟𝐿𝐶+𝐿
𝑅
, 𝜔0 =1
𝐿𝐶
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System Transfer Functions, cont’d
Open Loop transfer functions:
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Closed loop system
• The open loop system is closed in a loop through a voltage divider and feedback compensator
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Closed loop system: block diagram representation
𝐻 𝑠 – feedback gain
𝑅𝑏
𝑅𝑎 + 𝑅𝑏
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Summary• Buck dc-to-dc voltage converter
• Basic operation of chopping up input voltage and then filtering • Used Fourier analysis to determine unfiltered residuals• The system has one output, the output voltage, 𝑣• Identified inputs to the system:
𝑑 – duty ratio, the control input 𝑣𝑔 – input voltage, a disturbance input 𝑖𝑜 - output current, a disturbance input
• Determined three transfer functions for the converter:𝐺𝑣𝑑 𝑠 , 𝐺𝑣𝑔 𝑠 and 𝑍𝑜𝑢𝑡 𝑠 .
• Discussed the ‘describing function’ analysis to model the PWM• Through a process of averaging and linearization a linear time
invariant (small-signal) model of the system was obtained
• Next, stability analysis
33