ECE1750, Spring 2017 Week 11 Power Electronics Control 1
Power Electronic Circuits Control In most power electronic applications we need to control some variable, such as the put voltage of a dc-dc converter, automatically so we can compensate for potential deviations. When we control a given variable so that it reaches a given reference value automatically, it is said that we are regulating such variable. Some reasons for regulating a variable, such as the put voltage of a dc-dc converter, are: Deviations from ideal (expected) behavior to real behavior (e.g. account for losses). Deviations in input variables (e.g. different input voltage from the one expected).
Consider a buck converter: Buck converter modeling (review) State variables (those associated to energy storage): Capacitor voltage Inductor current 3
Buck converter modeling (review) In continuous conduction we have two states: State 1, main switch is ON State 0, main switch is OFF dil() t 1 dil() t 1 in vc () t vc () t dt L dt L dvc() t 1 vc() t il() t dv C() t 1 () () v C t il t dt C dt C 4
Buck converter modeling (review) In continuous conduction we have two states: State 1, main switch is ON State 0, main switch is OFF qt ( ) 1 qt ( ) 0 dil() t 1 in vc () t dil() t 1 v () dt L C t dt L dv C() t 1 C() il() t v t dt C dv C() t 1 C() il() t v t dt C dil() t 1 qt () in vc () t dt L dv C () t 1 C () il() t v t dt C Switched model of a dc-dc buck converter q(t) is the switching function 5
Switching Function ealization MOSFETS are usually controlled with a pulse-width modulation (PWM) strategy. In essence, the PWM process involves comparing a sawtooth (or any other periodic function with linear transitions) with a voltage level. If the voltage level is constant, it will tend to produce a constant (dc) put. Sawtooth or triangle adjustable analog input (duty cycle control) ---- vcont(t) Linear portions of sawtooth allows to directly translate voltage levels into time intervals (on-times) 12 Gate Driver (e.g. TC1427) Comparator (e.g., LM393) 6
PWM Switching Function realization D = 8/12 = 075 0.75 Gate Driver (e.g. TC1427) Comparator (e.g., LM393) 7
PWM Switching Function realization D = 6/12 = 05 0. Gate Driver (e.g. TC1427) Comparator (e.g., LM393) 8
PWM Switching Function realization D = 4/12 = 1/3 Where is coming from? What if it can be adjusted so the put t voltage remains constant regardless of potential deviations? Gate Driver (e.g. TC1427) Comparator (e.g., LM393) 9
Buck converter modeling Average model Let s go back to the switched model of a buck converter: dil() t 1 qt () in vc () t dt L dv C() t 1 C() il() t v t dt C Let s apply the average operator on both sides of both equations 1 T T 1 T di ( ) 1 T L t 1 dt q () t in vc () t dt T0 dt T0 L 1 T () 1 T dvc t 1 vc() t dt il() t dt T0 dt T0 C 0 dt 10
Buck converter modeling Average model 1 T di ( ) 1 T L t 1 dt q() t in vc () t dt T0 dt T0 L T T 1 dv C() t 1 1 v C () t dt il() t dt T 0 dt T 0 C 1 Now, let s change the order of some operators (this can be done due to linearity of integrals and differentials) T d il() t dt T 0 1 1 T 1 T in qtdt () vc () tdt 0 dt L T T 0 1 T () 1 T d vc t dt () 0 1 1 T vc t dt T 0 il() t dt T dt C 0 T 11
Buck converter modeling Average model 1 T d il() t dt T 0 1 1 T 1 T in 0 0 qtdt () vc () tdt dt L T T 1 T 1 T d v () C t dt () 0 1 1 T vc t dt T 0 il() t dt T dt C T 0 Let s define the following average variables: 1 T 1 T v i L() t C() t vc() t dt, il () t dt T 0 T 0 Also, 1 T dt () qtdt () T 0 d il() t 1 dt () in vc () t dt L d v () 1 () C t C il() t v t dt C 12
Buck converter modeling Average model Comparison between average and switched model: d il() t 1 dt () in vc () t dt L d v C() t 1 C() il() t v t dt C dil() t 1 qt () in vc () t dt L dv C() t 1 C() il() t v t dt C Application to the previously modeled and simulated buck converter: vc () t vc () t il () t il () t Notice that the average operator works like a low pass filter by filtering all of the ripple caused by switching. 13
PI Controller for DC-DC Buck Converter Output oltage Open Loop, DC-DC Converter Process vcont PWM mod. and MOSFET driver DC-DC conv. q(t) has a fixed duty cycle D D in d i () t 1 in C dt L d vc() t 1 vc() t il() t dt C L q(t) dt () v () t Issue: is not controlled automatically ti 14
et () v () t set error e(t) PI Controller q(t) has a variable v cont duty cycle d(t) set + PI controller PWM mod. and MOSFET driver DC-DC conv. 1 vcont () t d () t Pet () etdt () Pet () i etdt () T i The switching function has a duty cycle that changes from cycle to cycle until the converter reaches steady state t operation. Proportional term: Immediate correction but steady state error (v cont equals zero when there is no error (that is when set = )). Integral term: Gradual correction Consider the integral as a continuous sum (iemman s sum) 15 Thank you to the sum action, v cont is not zero when the e = 0 Has some memory
E.g. Buck converter in = 24 = 16 (goal) L = 200 uh, C = 500 uf, = 2 Ohm d il() t 1 dt () in vc () t dt L d v C() t 1 C() il() t v t dt C 1 dt () et P () etdt () T i 16
in = 24 = 16 (goal) L = 200 uh, C = 500 uf, = 2 Ohm E.g. Buck converter 1 dt () etdt () T i = 40, p = 0 A large value for i yields too many transient oscillations i i il vc d e 17
in = 24 = 16 (goal) E.g. Buck converter 1 dt () etdt () T L = 200 uh, C = 500 uf, = 2 Ohm i A moderate value for i regulates with t overshoot i = 10, p = 0 or too many oscillations but it takes some time to converge il vc d e 18
E.g. Buck converter in = 24 = 16 (goal) L = 200 uh, C = 500 uf, = 2 Ohm Al large value for p yields fast convergence but there is a i = 0, p = 1 steady state error d e() t P i il vc = 15 d e 19
E.g. Buck converter in = 24 = 16 (goal) dt () et () L = 200 uh, C = 500 uf, = 2 Ohm A small value for p yields slower convergence and there i = 0, p = 0.1 is still a steady state error P i il vc = 11 d e 20
in = 24 = 16 (goal) E.g. Buck converter 1 dt () et P () etdt () T L = 200 uh, C = 500 uf, = 2 Ohm API controller achieves faster convergence than an integral i = 10, p = 0.1 controller with the steady state error of a proportional controller i il vc d e 21
Euler method to numerically solve differential equations Euler (forward) method algorithm: Step 0) We have the value of x(tn) from the previous iteration or from the initial conditions. where t t t n1 Step 1) For time tn we calculate n f ( xt ( ), t) Step 2) For time tn+1 we calculate n n x( t t) xt ( ) xt ( ) f( xt ( ), t) t n n1 n n n Now x(tn+1) is the input for the next iteration. To start the new iteration go to Step #1. 22
Euler method applied to buck converters Euler (forward) method is an algorithm to solve differential equations based on: d il() t 1 dt () in vc () t dt L d vc() t 1 vc() t il () t dt C 1 i L ( tn 1 ) i L ( tn ) d ( tn ) in v C ( tn ) t L 1 vc( tn) vc( tn1) vc( tn) il( tn) t C 23
Euler method applied to buck converters Euler solver for PI controller 1 il( tn1) il( tn) d( tn) in vc( tn) t L 1 vc( tn) vc( tn1) vc( tn) il( tn) t C kn1 dt () et () etdt () d ( t n1) Pet ( n1) i et ( k) t P i k0 et ( ) v ( t ) n1 set C n1 Consider a Buck converter with in = 100, set=40, i = 15, =2 ohms, L=300 μh, C=200 μf (horizontal axis is time in seconds). v () C t i () L t 24
Euler solver for PI controller Integral controller simulation dt () et P () i etdt () kn1 dt ( ) et ( ) et ( ) t n1 P n1 i k k 00 et ( ) v ( t ) n1 set C n1 Consider a Buck converter with in = 100, set=40, i = 15, =2 ohms, L=300 μh, C=200 μf (horizontal axis is time in seconds). v () C t d(%) et () <vc(t)> > set=40 so e(t)<0 and d decreases 25
E.g. Buck converter in = 24 = 16 (goal) L = 200 uh, C = 500 uf, = 2 Ohm i = 10, p = 0.1 In steady state d(t)=d=2/3 26
E.g. Buck converter in = 30 = 16 (goal) L = 200 uh, C = 500 uf, = 2 Ohm i = 10, p = 0.1 In steady state d(t)=d=0.53 27
E.g. Buck converter in = 20 = 16 (goal) L = 200 uh, C = 500 uf, = 2 Ohm In steady state d(t)=d=0.8 i = 10, p = 0.1 28
Op Amps + I I + + Assumptions for ideal op amp = (+ ), large (hundreds of thousands, or one million) I+ = I = 0 oltages are with respect to power supply ground (not shown) Output current is not limited 29
Example 1. Buffer Amplifier (converts high h impedance signal to low impedance signal) in + ( + ) = ( in ) in (1 ) in in 1 is large i in 30
Example 2. Inverting Amplifier (used for proportional control signal) in in f ( 0 ), so. CL at the node is 0. in + in f Eliminating yields in in f 0, so 1 in 1 f 1 f in in. For large, then f in in, so f in. in 31
Example 3. Inverting Difference ( d f i l) (used for error signal) b ) ( + a b 2 ) (, so b 2. b CL at the node is 0 a, so 0 a, yielding 2 a. 2 Eliminating yields a b so a b or 1 a b t 2 2, so 2 2, or 2 2 1. For large then 32 For large, then b a
Example 4. Inverting Sum (used to sum proportional and integral control signals) a b + (0 ), so CL at the node is. a b 0, so 3. a b Substitutingfor 3 yields 3 a b,so 1 a b. Thus, for large,, g, a b 33
in i Example 5. Inverting Integrator (used for integral control signal) C i ~ ~ Using phasor analysis, (0 ), so ~ ~. CL at the node is + ~ ~ ~ ~ in 0. i 1 jc ~ ~ Eliminating ~ in ~ ~ yields j C 0. Gathering terms yields i ~ ~ 1 1 in ~ ~ j C 1 i, or 1 1 jic 1 i in For large, the ~ ~ ~ ~ expression reduces to j ic in, so in (thus, negative integrator action). j C For a given frequency and fixed C, increasing i i reduces the magnitude of ~. 34
Op Amp Implementation of PI Controller α set Signal flow error p + + Buffers (Gain = 1) + Difference (Gain = 1) 15kΩ i + Proportional (Gain = p) Ci + Inverting Integrator (Time Constant = Ti) (see slide #14) PWM mod. and MOSFET driver + cont Summer (Gain = 1) 35
Power Electronic Circuits Control Buck converters are relatively simple to control because a PI controller applied to an average model yields a linear system. d i () t 1 1 ( ( ) ) ( ( ) ) ( ) L P set vc t set vc t dt in vc t dt L T i d vc() t 1 vc() t il() t dt C However, for other converters, PI controllers applied to an average model still yields a non-linear system: d il() t 1 1 1 P( set vc () t ) ( set vc () t ) dt in 1 P ( set vc () t ) ( set vc () t ) dt vc () t dt L T i T i d v () 1 1 C t v P( set vc( t) ) ( set vc( t) ) dt il( t) C dt C T () t i Moreover, boost and other converters are what are called non-minimal phase converters in which direct regulation of the put voltage likely yields an unstable controller. For boost converters, the variable that can be controlled directly is the inductor current.