EE464-Static Power Conversion-II

class: center, middle

# EE-464 STATIC POWER CONVERSION-II

# Controller Design in Power Electronics

## Ozan Keysan

## [keysan.me](http://keysan.me)

### Office: C-113 <span class="meta">•</span> Tel: 210 7586

---
# Control in Power Electronics

## Control of a Wind Turbine

---

## Detailed Control of a Wind Turbine

---

# Control in Power Electronics

## Most DC/DC converters controlled by analog controllers:

- ### Micro-controllers are not fast enough (both for computing and sampling) at high switching frequencies

- ### Cheap (just an IC and a few passive elements)

- ### Could be integrated to with drive circuit ([LM1771](http://www.ti.com/product/LM1771/datasheet/abstract#snvs44619))

---
# Control in Power Electronics

## Control with a microcontroller
---
# Control in Power Electronics

## Control with an error amplifier
---
## Buck Converter Controller

<img src="./images/ee464/buck_controller.png" alt="Drawing" style="width: 800px;">
---
## Buck Converter Controller

<img src="./images/ee464/buck_control_blocks.png" alt="Drawing" style="width: 800px;">
---
# Control Loop Stability

- ## Small steady-state error (i.e. gain at low frequencies should be large)

- ## No resonance: (i.e. gain at switching frequency should be small)

- ## Enough [phase-margin](https://www.youtube.com/watch?v=ThoA4amCAX4): ( usually at least 45 degree phase margin is aimed for stability)

---

# Phase Margin

## Difference to -180 degrees when the gain is unity (0dB)

---
# Phase Margin

---
# Small Signal Analysis
--

### Small Signal Model of a Transistor (EE311)

---
# Small Signal Analysis for the Buck Converter

---
# Small Signal Analysis
--

## For a parameter, x:
--

- ## \\(x\\): total quantity
--

- ## \\(X\\): steady-state (DC) component
--

- ## \\(\tilde{x}\\): AC term (small-signal variation)

## \\(x = X + \tilde{x}\\)

---
# Small Signal Analysis
--

## For the buck converter

## \\(v_o = V_o + \tilde{v}_o\\)

## \\(d = D + \tilde{d}\\)

##  \\(i_L = I_L + \tilde{i}_L\\)

## \\(v_s = V_s + \tilde{v}_s\\)

---
# Small Signal Analysis

## Average Model of the buck converter

<img src="./images/ee464/buck_average.png" alt="Drawing" style="width: 700px;">
---
## Average Model of the buck converter

---
## Average Model of the buck converter

---
## Average Model of the buck converter

### Averaged model is useful for the controller design
---

<img src="./images/ee464/average_models.png" alt="Drawing" style="width: 500px;">
--

#### (a) Averaged model for switch and diode; (b) Buck equivalent; (c) Boost equivalent; (d) Buck-boost equivalent; (e) Ćuk equivalent
---
# Small Signal Analysis for the Buck Converter

## Let's derive the small signal model for voltage
--

### \\(v_x = v_s d \\)
--
 \\( = (V_s + \tilde{v}_s ) (D + \tilde{d} ) \\)
--

### \\(v_x = V_s D + \tilde{v}_s D + V_s \tilde{d} + \tilde{v}_s \tilde{d} \\) 
--

## ignoring the last term
--

### \\(v_x \approx V_s D + \tilde{v}_s D + V_s \tilde{d} = v_s D + V_s \tilde{d} \\)

---
# Small Signal Analysis for the Buck Converter

## Let's repeat for current

### \\(i_s = i_L d = (I_L + \tilde{i}_L)(D +\tilde{d})\\)

### \\(\approx i_L D + I_L \tilde{d}\\)
---

## Exercise Assignment:

### Power Electronics, Hart, Section 7.13
###  Buck Converter Small Signal Model

---
# Transfer Functions

## RC Filter

---

## RC Filter [Bode Plot](https://blanco.io/education/principles-of-ee-2/how-to-bode-plot/)

### [Bode Plotter](https://lpsa.swarthmore.edu/Bode/BodeWhat.html) ,[Wolfram Alpha Plotter](https://www.wolframalpha.com/input/?i=transfer+function+(1)%2F(3*(s%5E2%2B2*s%2B1))
---
## Transfer Functions

## Let's do for the LCR part of the converter

### Representation in the s-domain

---
## Transfer Functions

## Let's do for the LCR part of the converter

### \\(\dfrac{v_o(s)}{v_x(s)}=\dfrac{1}{LC(s^2 +(1/RC)s+1/LC)}\\)

### \\(v_x(s)= V_s d(s)\\)

### Transfer function in terms of d(s)

### \\(\dfrac{v_o(s)}{d(s)}=\dfrac{V_s}{LC(s^2 +(1/RC)s+1/LC)}\\)

---
# Realistic RLC

## Non-ideal elements can effect stability

- ## Resistance of the inductor

- ## ESR of capacitor (series resistance)

---
## Let's repeat the case with non-ideal capacitor

## Capacitor with series resistance
---
## Let's repeat the case with non-ideal capacitor

#### \\(\dfrac{v_o(s)}{d(s)} = \dfrac{V_s}{LC}\dfrac{1 + s r_C R}{s^2(1+r_C/R) +s(1/RC+r_C/L)+1/LC)}\\)
--

### Can be simplified by assuming \\(r_C << R\\)

### \\(\dfrac{V_s}{LC}\dfrac{1 + s r_C R}{s^2 +s(1/RC+r_C/L)+1/LC)}\\)

### Notice the extra zero introduced by ESR!
---
### PWM Block Transfer Function
--

## \\(d=\dfrac{v_c}{V_p}\\)

### for a saw-tooth PWM generator with Vp peak voltage
--

### Transfer function

## \\(\dfrac{d(s)}{v_c(s)}=\dfrac{1}{V_p}\\)

---
## PWM Block Transfer Function

## Be careful with high-frequency control bandwidth
--

<--

## What about in switching components?

## Problem:

- ## Multi-mode systems (topology changes with switching)

- ## Different transfer function for on-off states

- ## Can use non-linear controller, or multiple linear controllers (but difficult to implement)

## What about in switching components?

## Solution:

- ## Convert multi-mode to single-mode system

- ## Linearizing the system with averaging wrt duty cycle

- ## Use a linear controller with required characteristics

## Details in the textbook (Mohan)

# Case Study (Mohan 10-1)

## Find the transfer function of forward converter

## Note the state variables

# Case Study (Mohan 10-1)

## Switch ON

# Case Study (Mohan 10-1)

## Switch OFF

<img src="./images/ee464/forward_controller_off.png" alt="Drawing" style="width: 700px;">
# Case Study (Mohan 10-1)

## Steady State Transfer Function
--

### \\(\dfrac{V_o}{V_d} = -C A^{-1}B\\)
--

### \\(\dfrac{V_o}{V_d} = D \dfrac{R+ r_C}{R+(r_C + r_L)}\\)
--

### If parasitic resistances are small
--

### \\(\dfrac{V_o}{V_d} \approx D \\)

# Case Study (Mohan 10-1)

## AC Transfer Function
--

### \\(T_p(s)=\dfrac{\tilde{v}_o(s)}{\tilde{d}(s)}\\)
--

### \\(=V_d \dfrac{1 + s r_C C}{LC [s^2 + s (1/RC + (r_C + r_L)/L)+ 1/LC]}\\)
--

## Remember this equation?
--

### \\(s^2 + 2 \xi \omega_0 s + \omega_0^2\\)

# Case Study (Mohan 10-1)

## AC Transfer Function

### \\( s^2 + 2 \xi \omega_0 s + \omega_0^2\\)
--

### \\(\omega_0 = \dfrac{1}{\sqrt{LC}}\\)
--

### \\(\xi = \dfrac{1/RC + (r_C + r_L)/L}{2\omega_0}\\)
-->
---
# Case Study (Mohan 10-1)

## AC Transfer Function for forward converter

### \\(T_p(s)=V_d \dfrac{\omega_0^2}{\omega_z} \dfrac{s + \omega_z}{s^2 + 2 \xi \omega_0 s + \omega_0^2}\\)

### where \\(\omega_z = \dfrac{1}{r_C C}\\)
---
# Example (Mohan 10-1)

### Put the parameters into the equation

#### \\(V_d = 8 V\\)
#### \\(V_o = 5 V\\)
#### \\(r_L = 20 m\Omega\\)
#### \\(L = 5 \mu H\\)
#### \\(r_C = 10 m\Omega\\)
#### \\(C = 2 mF\\)
#### \\(R = 200 m\Omega\\)
#### \\(f_s = 200 kHz\\)

---
# Example (Mohan 10-1)

## Bode Plot (Gain)

---
# Example (Mohan 10-1)

## Bode Plot (Phase)

<img src="./images/ee464/forward_bode_phase.png" alt="Drawing" style="width: 650px;">
---
# Flyback Converter
--

#### Equation 10-86

### \\(T_p(s)=\dfrac{\tilde{v}_o(s)}{\tilde{d}(s)}\\)
--

### \\(T\_p(s)=V\_d f(D)  \dfrac{(1 + s/\omega\_{z1})(1 - s/\omega\_{z2})}{as^2 + bs + c}\\)

---
# Flyback Converter
--

## Bode Plot (Gain)

---
# Flyback Converter
--

## Bode Plot (Phase)

<img src="./images/ee464/flyback_bode_phase.png" alt="Drawing" style="width: 650px;">
---
## A few readings for controller design
--

- ### [Control Design of a Boost Converter Using Frequency Response Data](https://www.mathworks.com/help/slcontrol/examples/control-design-of-a-boost-converter-using-frequency-response-data.html)
- ### [PID Control Tuning for Buck Converter](https://www.mathworks.com/videos/pid-controller-tuning-for-a-buck-converter-1504291092156.html)
- ### [Design digital controllers for power electronics using simulation](https://www.mathworks.com/discovery/power-electronics-simulation.html)
- ### [Bode Response of Simulink Model](https://www.mathworks.com/help/slcontrol/gs/bode-response-of-simulink-model.html)
- ### [How to Run an AC Sweep with PSIM?](https://www.youtube.com/watch?v=nbFo-NMeY_w)
- ### [Peak Current Control with PSIM](https://powersimtech.com/drive/uploads/2016/03/Tutorial-Peak-current-mode-control.pdf)
- ### [Plexim-Frequency Analysis of Buck Converter ](https://www.plexim.com/support/application-examples/187) 
---
# Controller Design
--

## Generalized Compensated Error Amplifier
--

---
## Types of Error Amplifier
--

### Common Ones:

- ### Type-1

- ### Type-2

- ### Type-3

---
# Type-1 Error Amplifier

## Simple Integrator

## Has one pole at the origin

---
# Type-1 Error Amplifier

---
## Type-2 Error Amplifier (Most Common Type)
--

### Has two poles: at origin and one at zero-pole pair
### 90 degrees phase boost can be obtained due to single zero
---

## Type-2 Error Amplifier (Most Common Type)
--

### Note the phase boost

---

## Type-3 Error Amplifier
--

### has two zeros can can boost up to 180 degrees
---
# A Few Examples

- ## [TL494](https://pdf.direnc.net/upload/tl494-datasheet.pdf),  pg. 7, 15
- ## [LM5015](http://www.ti.com/lit/ds/snvs538c/snvs538c.pdf), Fig. 12, 15

---

## Putting all Together
--

## A controller just increases the gain (Proportional)
--

### Increasing gain usually reduces phase margin (and reduces stability)

---

## Putting all Together
--

## A proper controller  (adjust gain and phase margin)
--

---

## More Information

---
## More Information

- ### Fundamentals of Power Electronics, Erickson

- ### [Phase Margin, Crossover Frequency, and Stability](https://www.eeweb.com/profile/cody-miller/articles/bode-plot-phase-margin-crossover-frequency-and-stability)

- ### [Loop Stability Analysis of Voltage Mode Buck Regulator](http://www.ti.com/lit/an/slva301/slva301.pdf)

- ### [DC-DC Converters Feedback and Control](http://www.onsemi.com/pub/Collateral/TND352-D.PDF)

- ### [Modeling and Loop Compensation Design](http://cds.linear.com/docs/en/application-note/AN149fa.pdf)

- ### [Compensator Design Procedure](https://www.infineon.com/dgdl/an-1162.pdf?fileId=5546d462533600a40153559a8e17111a)

---
## You can download this presentation from: [keysan.me/ee464](http://keysan.me/ee464)

---
## Saved for further reference
## Ready?
--

## Down the rabbit hole

---
## Linearization with State-Space Averaging
--

- ## Represent everything in matrix form
--

- ## Inductor current, and capacitor voltage as state variables
--

- ## Obtain two states (for switch on and switch off)
--

- ## Find the weighted average
---

## Linearization with State-Space Averaging
--

## Obtain two states (for switch on and switch off)
--

## \\(\dot{x} = A_1 x + B_1 v_d\\)   (for switch on, dTs)
--

## \\(\dot{x} = A_2 x + B_2 v_d\\)   (for switch off, (1-d)Ts)

### where, A1 and A2 are state matrices

### B1 and B2 are vectors

---
# Example:

## Mohan 10.1

---

## Linearization with State-Space Averaging
--

## Find weighted average
--

## \\(A = d A_1 + (1-d) A_2\\)

## \\(B = d B_1 + (1-d) B_2\\) 
--

## \\(\dot{x} = A x + B v_d\\)   (for switch off, (1-d)Ts)
---

## Similar calculations for the output voltage

## \\(v_o = C_1 x \\)   (for switch on, dTs)

## \\(v_o = C_2 x \\)   (for switch off, (1-d)Ts)

## where C1 and C2 are transposed vectors

---

## Similar calculations for the output voltage

## \\(v_o = C x \\)

## \\(C = d C_1 + (1-d) C_2\\)  
## where C1 and C2 are transposed vectors
---
### Use Small signal model 
Equations 10.46-10.52
--

###\\(x = X + \tilde{x}\\)
--

### \\(\dot{\tilde{x}}= AX + B V_d + A\tilde{x}\\)
### \\( \quad \quad + [(A_1-A_2)X + (B_1 - B_2)V_d]\tilde{d}\\)

--
### In the steady state:

## \\(\dot{X}\\)=0

Use derivations from eq.10.53-10.59
---
# Steady State DC Voltage Transfer Function
--

## \\(\dfrac{V_o}{V_d} = -C A^{-1}B\\)

---
# Small Signal Model to Get AC Transfer Function
--

### \\(T_p(s)=\dfrac{\tilde{v}_o(s)}{\tilde{d}(s)}\\)

### \\(=C[sI-A]^{-1}[(A_1-A_2)X + (B_1-B_2)V_d] \\)
### \\(\quad + (C_1-C_2)X)]\\)
---
# Example (Mohan 10-1)

## Find the transfer function of forward converter
--

## Note the state variables
---
# Example (Mohan 10-1)

## Switch ON
--

---
# Example (Mohan 10-1)

## Switch OFF
--

<img src="./images/ee464/forward_controller_off.png" alt="Drawing" style="width: 700px;">
---
# Example (Mohan 10-1)

## Steady State Transfer Function
--

### \\(\dfrac{V_o}{V_d} = -C A^{-1}B\\)
--

### \\(\dfrac{V_o}{V_d} = D \dfrac{R+ r_C}{R+(r_C + r_L)}\\)
--

### If parasitic resistances are small
--

### \\(\dfrac{V_o}{V_d} \approx D \\)

---
# Example (Mohan 10-1)

## AC Transfer Function
--

### \\(T_p(s)=\dfrac{\tilde{v}_o(s)}{\tilde{d}(s)}\\)
--

#### \\(T_p(s)=V_d \dfrac{1 + s r_C C}{LC [s^2 + s (1/RC + (r_C + r_L)/L)+ 1/LC]}\\)
--

## Remember this equation?
--

### \\(s^2 + 2 \xi \omega_0 s + \omega_0^2\\)

---
# Example (Mohan 10-1)

## AC Transfer Function

### \\( s^2 + 2 \xi \omega_0 s + \omega_0^2\\)
--

### \\(\omega_0 = \dfrac{1}{\sqrt{LC}}\\)
--

### \\(\xi = \dfrac{1/RC + (r_C + r_L)/L}{2\omega_0}\\)

---
# Example (Mohan 10-1)

## AC Transfer Function Becomes

### \\(T_p(s)=V_d \dfrac{\omega_0^2}{\omega_z} \dfrac{s + \omega_z}{s^2 + 2 \xi \omega_0 s + \omega_0^2}\\)

### where \\(\omega_z = \dfrac{1}{r_C C}\\)
---
# Example (Mohan 10-1)

### Put the parameters into the equation

#### \\(V_d = 8 V\\)
#### \\(V_o = 5 V\\)
#### \\(r_L = 20 m\Omega\\)
#### \\(L = 5 \mu H\\)
#### \\(r_C = 10 m\Omega\\)
#### \\(C = 2 mF\\)
#### \\(R = 200 m\Omega\\)
#### \\(f_s = 200 kHz\\)

---
# Example (Mohan 10-1)

## Bode Plot (Gain)

---
# Example (Mohan 10-1)

## Bode Plot (Phase)

<img src="./images/ee464/forward_bode_phase.png" alt="Drawing" style="width: 650px;">
---
# Flyback Converter
--

#### Equation 10-86

### \\(T_p(s)=\dfrac{\tilde{v}_o(s)}{\tilde{d}(s)}\\)
--

### \\(T\_p(s)=V\_d f(D)  \dfrac{(1 + s/\omega\_{z1})(1 - s/\omega\_{z2})}{as^2 + bs + c}\\)

---
# Flyback Converter
--

## Bode Plot (Gain)

---
# Flyback Converter
--

## Bode Plot (Phase)