Karadeniz Technical University Department of Electrical and Electronics Engineering 61080 Trabzon, Turkey Chapter 3-2- 1 Modelling and Representation of Physical Systems 3.1. Electrical Systems Bu ders notları sadece bu dersi alan öğrencilerin kullanımına açık olup, üçüncü sahıslara verilmesi, herhangi bir yöntemle çoğaltılıp başka yerlerde kullanılması, yayınlanması Prof. Dr. İsmail H. ALTAŞ ın yazılı iznine tabidir. Aksi durumlarda yasal işlem yapılacaktır. Chapter 3-2- 2
Model representation Integro-differential equations Simulation diagrams (detailed models) Block diagrams (Combined models) Signal flow graghs State-Space equations Transfer functions Chapter 3-2- 3 Model representation input cause block system BLOCK DIAGRAM output effect It helps in the realization, synthesis or fabrication of systems A simulation of the system may be realized from the block diagram by interconnecting the basic building blocks. It helps in the analysis and the design of systems as it provides a clear picture of the cause and effect relationship governing the various signals within the system. v Operation and analysis in time domain v Operation and analysis in frequency domain Chapter 3-2- 4
Model representation BLOCK DIAGRAM Laplace transform or Fourier transform or z- transform Transfer Function is a mathematical operator that relates the input and the output. A system element represented by a block may be either a static element or a dynamic element: Chapter 3-2- 5 Electrical Component Models i i v _ v _ voltage/current voltage/charge Inductance v = L di/dt v = L dq 2 /dt 2 Resistance v = R i v = R dq/dt i v _ Capacitance v = 1/C ò i dt v = 1/C q Chapter 3-2- 6
Model representation u a y BLOCK DIAGRAM Linear gain u y u Integral gain f(u) Nonlinear gain y Chapter 3-2- 7 Operation and analysis in time domain Model representation BLOCK DIAGRAM The input and the outputs are functions of time u=sin(t) y y y u u u Integral gain t Linear gain y=au y=3u Nonlinear gain F=f(u) y = u 3u 3 y = u 2 y=sin(u) Chapter 3-2- 8
The integration notation Model representation BLOCK DIAGRAM Initial condition, y(0) integration terms Chapter 3-2- 9 Model representation BLOCK DIAGRAM The integration notation Integrator Integrator An integrator block with an indefinite integral sign is equivalent to a definite integrator block and a summer Chapter 3-2- 10
Model representation Static element NO MEMORY BLOCK DIAGRAM y = 3u, y = u, y=sin(u) 3 The output at the instant t 1 depands on the input at the same instant t 1. Chapter 3-2- 11 Model representation Static element t(s) E (V) u R (V) t1 E1 U R 1 t2 E2 U R 2 t3 E3 U R 3 t4 E4 U R 4 BLOCK DIAGRAM NO MEMORY U R (V) u R (V) E(V) t (s) Chapter 3-2- 12
Model representation Potentiometer Static element Obtain the block representation of a potentiometer shown below Chapter 3-2- 13 Potentiometer Static element case 1: microscopic picture: a detailed block diagram representation. step 1: identify the input and the output : Model representation step 2: usually input is located on the left and output to the right step 3: relate v i to v o using internal variables. v i is input, v o is output and i is an internal variable. Chapter 3-2- 14
Model representation Potentiometer Summing Rectangular Blocks Take off point Detailed Block Representation Chapter 3-2- 15 Model representation Potentiometer Chapter 3-2- 16
case 2: Model representation Potentiometer macroscopic picture: a single block representation relating merely the input and the output The block diagram is thus an interconnection of : a.rectangular gain blocks b.summing junctions and c.takeoff points Chapter 3-2- 17 Model representation Dynamic element HAS MEMORY Chapter 3-2- 18
Model representation Dynamic element HAS MEMORY This is called a low-pass filter. Chapter 3-2- 19 Model representation Dynamic element HAS MEMORY Chapter 3-2- 20
Model representation Dynamic element X low-pass filter. seconds t(s) Chapter 3-2- 21 Model representation Dynamic element t(s) - A low-pass filter. Chapter 3-2- 22
Model representation Dynamic element Solution: This is called a low-pass filter. Simulate in Matlab/Simulink using the numerical data given as; C=1 µf, R=10 kw; V=10 V Chapter 3-2- 23 Model representation Dynamic element HAS MEMORY (0) Chapter 3-2- 24
Model representation Dynamic element HAS MEMORY Chapter 3-2- 25 Model representation Dynamic element X Step input t(s) Chapter 3-2- 26
Model representation HAS MEMORY Dynamic element - t(s) Chapter 3-2- 27 Model representation Dynamic element HAS MEMORY Simulate in Matlab/Simulink using the numerical data given as; L=200 mh, R=10 kw; V=10 V - Chapter 3-2- 28
Model representation Obtain a block diagram of the following system described by an integral equation. where u is the input and y is the output. t(s) Initial conditions are zero, u C (0)=0. Chapter 3-2- 29 Model representation EXAMPLES step 1: usually input is located on the left and output to the right step 2: Rewrite the integral equation with the highest orfer term on the left hand side and the rest on the right hand side Chapter 3-2- 30
Model representation EXAMPLES OR step 3: treat the term, inputs are u(t) and as an output of a summer whose step 4: generate from the output of the summer, y(t) and the terms Chapter 3-2- 31 Model representation EXAMPLES step 5: interconnect the blocks summer For capacitor circuit - pick-off REPLACE: blocks Chapter 3-2- 32
Model representation EXAMPLES Block diagram should not contain differentiator elements. Only integrators should be present. To avoid using differentiators, use integrators to generate various derivatives. Here we need to generate the 3 derivative of y, which is generated as follows: Starting from the highest order (third order) derivative generate the lower order derivatives (second, first and zero order) by cascading integrators. Chapter 3-2- 33 Model representation EXAMPLES Obtain the block diagram representation of the following integro-differential equation Chapter 3-2- 34
Model representation EXAMPLES step 1: express the equation with the term representing the highest order derivative of the output on the left hand side and the rest of the terms on the right hand side step 2: treat the highest order term as an output of a summer with rest of the terms as inputs Chapter 3-2- 35 Step 4: interconnect the blocks Model representation EXAMPLES step 3: from the highest order term, generate the terms and - - Chapter 3-2- 36
Model representation State-Space Model State-Space Model Chapter 3-2- 37 Model representation State-Space Model State vector State variables A: System matrix B: Input matrix C: Output matrix Output variable State-Space Model Chapter 3-2- 38
Model representation Select Kp=20 and Ki=50 to simulate with a PI controller. L=2 H, R=10 Ohm; C=100 F V=50 V Chapter 3-2- 39 Obtain the block diagram for a spacecraft platform : Solution 6 Chapter 3-2- 40
Solution 6 Chapter 3-2- 41 Operational amplifiers Chapter 3-2- 42
Operational amplifiers This image is taken from the internet https://www.st-andrews.ac.uk/~jcgl/scots_guide/experiment/lab/expt6/expt6.html Chapter 3-2- 43 Operational amplifiers The operational amplifiers are used to build analog circuits. OP-AMPs are the key elements used in realization of analog controllers. They are used as: v GAIN: multiplication of a signal by a constant for amplification or attenuation v SUMMER: addition of two or more signals v INTEGRATOR: integration of a signal Chapter 3-2- 44
Inverting operational amplifier circuits inverting configuration Inversion Choose Z f =Z i u -u -1 Chapter 3-2- 45 Inverting operational amplifier circuits Addition: n signals (v i, i=1,2,3,..., n) are added to give an output v 0 Chapter 3-2- 46
Inverting operational amplifier circuits Addition: -1 Chapter 3-2- 47 Inverting operational amplifier circuits Multiplication Multiplying by a constant: Choose Z f =R f and Z i =R i ; that is connected resistors in the feedback path and at the input: Chapter 3-2- 48
Inverting operational amplifier circuits Multiplication Proportional Control -1 Chapter 3-2- 49 Inverting operational amplifier circuits Integration Choose: and That is connect a capacitor in the feedback path and a resistor in the input path. Chapter 3-2- 50
Integration Inverting operational amplifier circuits Integral Control Chapter 3-2- 51 Inverting operational amplifier circuits Derivation Choose: and That is connect a capacitor in the input path and the resistor in the feedback path. Chapter 3-2- 52
Inverting operational amplifier circuits Derivation Derivative Control Chapter 3-2- 53 PID CONTROLLERS PROPORTIONAL INTEGRAL DERIVATIVE Chapter 3-2- 54
CONTROLLERS Power Source R(s) E(s) Controller U(s) Actuator u x The plant Sensor Feedback Controllers Two position (on-off) controllers Proportional controllers (P) Integral controllers (I) Proportional Integral controllers (PI) Proportional Derivative controllers (PD) Proportional Integral Derivative controllers (PID) State-Feedback Controllers Lead / lag compensators Chapter 3-2- 55 CONTROLLERS PID Proportional Integral (PI) Controller Chapter 3-2- 56
CONTROLLERS Proportional Derivative (PD) Controller PID Chapter 3-2- 57 CONTROLLERS Proportional Derivative (PD) Controller PID Chapter 3-2- 58
CONTROLLERS PID Proportional Integral Derivative (PID) Controller Chapter 3-2- 59 Model representation ADDING A CONTROLLER TO A PHYSICAL SYSTEM Chapter 3-2- 60
Model representation ADDING A CONTROLLER TO A PHYSICAL SYSTEM Chapter 3-2- 61 Model representation ADDING A CONTROLLER TO A PHYSICAL SYSTEM Chapter 3-2- 62
ADDING A CONTROLLER TO A PHYSICAL SYSTEM Application of a PI controller in order to control the current i flowing through a series RLC circuit. L=2 H, R=10 Ohm; C=100 F V=50 V Select Kp=20 and Ki=50 to simulate with a PI controller. Chapter 3-2- 63 ADDING A CONTROLLER TO A PHYSICAL SYSTEM PI Controller Chapter 3-2- 64
Model representation Elements of Simulation Diagrams Summer - - - - - Chapter 3-2- 65 Model representation Elements of Simulation Diagrams Pick-off point pick-off point pick-off point Gain B locks a Linear gain Integral gain f(u) Nonlinear gain Chapter 3-2- 66
Model representation Elements of Simulation Diagrams The block diagram is a pictorial representation of the mathematical model of the system. The block diagram is not unique. For a given system one may obtain an infinite number of block diagrams all of which characterize the same mathematical model. An elemental block diagram should not contain differentiator blocks. It may only contain Linear gain, nonlinear gain, integrator and summer blocks. Pick off points Chapter 3-2- 67 Model representation Solved Problems in time domain Chapter 3-2- 68
Model representation Obtain the block diagram representation of the following differential equation step 1: express the equation with the term representing the highest order derivative of the output on the left hand side and the rest of the terms on the right hand side Chapter 3-2- 69 Model representation - - - Chapter 3-2- 70
Model representation illustrate the importance of a block diagram using the following circuits. v is input, i 2 is the output Fig. A v is input, i 2 is the output Fig. B the loop equations for the above RL circuits can be chosen in two ways as shown above. Chapter 3-2- 71 Model representation A) Consider the figure A on the left. The loop equation yields Expressing in the matrix form yields Chapter 3-2- 72
Model representation The matrix is singular. Hence there is redundancy in the set of equations. Adding the equations we get The currents i 1 and i 2 lie on a plane. There is only one differential equation (and not two differential equations) governing the above system. Chapter 3-2- 73 Model representation B) Consider the figure B on the right. Using a different set of loop equations we get Now you can easily draw the block diagram. Chapter 3-2- 74
Model representation Obtain the block diagram of the following circuit. Solution P3 Re-write the equations, with the highest derivative terms of the corresponding loop on the LHS. Chapter 3-2- 75 Model representation Chapter 3-2- 76
Model representation NOTE In the first equation, for the loop 1 highest order derivative term and Lowest order term. In the second equation, for the loop 2 highest order derivative term and and Lowest order terms. Chapter 3-2- 77 Model representation Obtain the mathematical model and simulation diagram Solution P4 Let us use the mesh equation : Chapter 3-2- 78
Model representation Step 2: Express the equations with highest order derivative to the left. Step 3: Since the left hand side of equation (1) and (2) contains more than one derivative term: we will use matrix formulation to obtain only a single derivative term in each equation. Chapter 3-2- 79 Model representation Expressing in the matrix form yields: inverting the matrix yields : Chapter 3-2- 80
Model representation simplifying yields Hence simplifying we get 1 Chapter 3-2- 81 Model representation Simplifying yields Simplifying we get 2 Chapter 3-2- 82
Model representation Chapter 3-2- 83 Model representation Obtain the block diagram of the following Solution P5 Chapter 3-2- 84
Model representation Expressing the differential equations with the highest order derivatives to the left yields The inputs The output internal variables, Chapter 3-2- 85 Model representation Chapter 3-2- 86
Obtain the block diagram representation of the following passive circuit with e i as input and e o as the output in the frequency domain. Chapter 3-2- 87 Taking the Laplace transform yields (assume zero initial condition) or and Chapter 3-2- 88
Chapter 3-2- 89 Now consider the case when the initial conditions are not zero. Taking the Laplace transform yields (assume non-zero initial condition) Solution: or Chapter 3-2- 90
ASIDE Chapter 3-2- 91 Chapter 3-2- 92
Obtain the block diagram of the following circuit. Solution P7 Taking the Laplace transform and assuming zero initial conditions, we get Chapter 3-2- 93 This is called a high-pass filter. Chapter 3-2- 94
Unsolved Examples Chapter 3-2- 95 Model representation UE 1 Obtain the simulation diagram of the following circuit in time domain. Assume that e i (t) as input and e 0 (t) as the output This is called a high pass filter Chapter 3-2- 96
Model representation UE 2 Obtain the block diagram of the following RC circuit. Exercise for students Chapter 3-2- 97 Model representation UE 3 Obtain the simulation diagram of the following circuit in time domain. Assume that e i (t) as input and e 0 (t) as the output This is called a high pass filter Chapter 3-2- 98
Operational amplifiers Additional study topic for the students This section will not be included in exams of this course Chapter 3-2- 99 Operational amplifiers The operational amplifiers are used to build analog circuits. OP-AMPs are the key elements used in realization of analog controllers. They are used as: v GAIN: multiplication of a signal by a constant for amplification or attenuation v SUMMER: addition of two or more signals v INTEGRATOR: integration of a signal Chapter 3-2- 100
Operational amplifiers Most physical systems may be modeled using an ordinary differential equation and the ordinary differential equation can be expressed in terms of four basic operations. v addition. v inversion v gain (multiplication by a constant) v integration. Using the operational amplifiers the ordinary differential equation model can be simulated by realizing these four basic operations. When the operational amplifiers are configured to simulate a mathematical model, the operational amplifier set up is called an analog computer. Chapter 3-2- 101 Operational amplifiers The analog computers The analog computers are used to simulate a real world (continuoustime) systems. They have advantages over the digital computers: the signals need not be sampled and hence can handle very high bandwidth signals; simulation is performed in real time; hardware in the loop (the physical system is simulated on an analog computer while the control or filtering operations are performed using on a digital computer) simulations is performed to provide a physical insight to the actual operation. However in recent times as the digital computers are becoming are becoming more powerful and cheaper, and analog to digital and digital converters are becoming faster many of the continuous time simulations may be performed on a digital computers fairly accurately and in real time. Chapter 3-2- 102
Operational amplifiers The further application of the operational amplifiers include v realization of a circuit with a specified transfer function: op-amplifiers serve as building blocks to realize analog circuits v realization of filters: (e.g. low-pass, high pass, band-pass and notch filters) v realization of a buffer between a high impedance source and a low impedance load. Chapter 3-2- 103 A mathematical model of the operational amplifier In practice, three different models are employed in the analysis and design of the operational amplifiers Rigorous model Approximate model Ideal model Chapter 3-2- 104
A mathematical model of the operational amplifier The rigorous model 15 V Symbol -15 V Equivalent circuit Z inp is the input impedance, Z out is the output impedance, A is the amplifier gain. Chapter 3-2- 105 A mathematical model of the operational amplifier The approximate model Assuming: Z inp = and Z out =0. However, the gain A is finite as in the rigorous model Chapter 3-2- 106
A mathematical model of the operational amplifier Ideal model by assuming Z inp = and Z out = 0 and A=. This implies that v1 v2=0, i inp =0. Chapter 3-2- 107 The operational amplifier circuits: Always used in closed-loop configuration to ensure: very low sensitivity to noise, variations in the supply voltage, variations in the parameters of the transistors etc., improve linearity, increase the input impedance, reduce the output impedance and increase the bandwidth. Chapter 3-2- 108
The operational amplifier circuits: In the open-loop configuration the output will saturate due to ever present noise: The output v 0 will be very large even (v 1 v 2 )is negligible since A. Chapter 3-2- 109 The operational amplifier circuits: The operational amplifier circuit configurations are of two types: 1. non-inverting configuration 2. inverting configuration Non-inverting operational amplifier circuits Non-inverting configuration Voltage follower (For Zi = and Zf =0) Chapter 3-2- 110
The operational amplifier circuits: The gain of the non-inverting amplifier is positive and is always greater than unity, and the input impedance is very large (» ). If we set Z i = and Z f =0 the circuit is reduced to a voltage follower (an unity gain amplifier). The voltage follower is useful as a buffer amplifier or as a impedance converter. As a buffer amplifier the voltage follower isolates one circuit from the loading effect of the following stage. Chapter 3-2- 111 The operational amplifier circuits: For example For D/A converters, a high input impedance load is required for correct operation but do not desire output voltage scaling. The load generally has low input impedance and hence a voltage follower may be used as buffer between the D/A converter and the low input impedance load. It is also useful following the hold circuit in the sample-andhold circuit to prevent capacitor discharge. Chapter 3-2- 112
Inverting operational amplifier circuits: inverting configuration The inverting operational amplifier circuits are used widely to realize addition, multiplication by a constant, inversion, integration. Chapter 3-2- 113 Inverting operational amplifier circuits: Addition: n signals (v i, i=1,2,3,..., n) are added to give an output v 0 Chapter 3-2- 114
Inverting operational amplifier circuits: Multiplication Multiplying by a constant: Choose Z f =R f and Z i =R i ; that is connected resistors in the feedback path and at the input: Chapter 3-2- 115 Inverting operational amplifier circuits: Inversion Choose Z f =Z i Chapter 3-2- 116
Integration Choose: Inverting operational amplifier circuits: and That is connect a capacitor in the feedback path and a resistor in the input path. Chapter 3-2- 117 Inverting operational amplifier circuits: Derivation Choose: and That is connect a capacitor in the input path and the resistor in the feedback path. Chapter 3-2- 118
Inverting operational amplifier circuits: The operational amplifier operates correctly (operates in the linear range) if the input voltages are constrained to some range which is determined by the power supply voltage. For the usual ±15V supply, the input voltage range is If the voltage exceeds ±13V range, then the saturation sets in and the operation amplifier circuits operate incorrectly. The input impedance of the inverting circuit is usually low when compared to the non-inverting circuit( input impedance is infinity). Since the inverting terminal is virtually grounded, the input impedance is Z i. However the output impedance is zero. Chapter 3-2- 119 inverting operational amplifier Node 1 Node 2 V E(s) - A(s)E(s) V 0 Chapter 3-2- 120
inverting operational amplifier Node 1 V E(s) - Node 2 V 0 A(s)E(s) Kirchoff s law: node equations. Choose the nodes as node 1: junction of Z 1 and Z f node 2 : the junction of Z 0, Z f and Z L. Chapter 3-2- 121 inverting operational amplifier Consider the node 1: The expression of E(s) is obtained by considering the input circuit : V 1 (s) Z i (s) E(s) V 0 (s) Z 1 Z i Y 1 Y i Z f Y f K.C.L: 1 Chapter 3-2- 122
inverting operational amplifier Consider the node 2: The expression of V 0 (s) is obtained by considering the input circuit : Z f Y f E Z 0 Z L V 0 Z 0 Z L Y 0 Y L A(s)E(s) - 2 Chapter 3-2- 123 1 = inverting operational amplifier 2 = Chapter 3-2- 124
inverting operational amplifier The block diagram indicate that there is feedback. The feedback block can be eliminated by block reduction Chapter 3-2- 125 inverting operational amplifier Chapter 3-2- 126
inverting operational amplifier - Chapter 3-2- 127 inverting operational amplifier Chapter 3-2- 128
inverting operational amplifier Chapter 3-2- 129 inverting operational amplifier The transfer function: Chapter 3-2- 130
inverting operational amplifier It is instructive to represent the block diagram with standard negative feedback system as follows: - Negative feedback Chapter 3-2- 131 inverting operational amplifier - Further simplification of the feedback loop yields feedback system as follows: case 1: the amplifier gain A is infinitely large Since A is very large AY 0 >> Y f and hence we can approximate the inverting amplifier block further. Chapter 3-2- 132
inverting operational amplifier - If the output impedance Z 0 =0 that is Y 0 = then the block diagram reduces to - Chapter 3-2- 133 inverting operational amplifier Let the input impedance be infinity ( op-amp draws no current), Y i =0 and the output impedance be zero, Y 0 = but the gain A is finite as shown in the figure below. V E(s) - V 0 A(s)E(s) Then the block diagram becomes Chapter 3-2- 134
inverting operational amplifier - Y i =0 Y 0 = - Chapter 3-2- 135 inverting operational amplifier Approximate model of an inverting operational amplifier V 1 e - V 0 An op-amp and its equivalent circuit is given. Obtain a block diagram. W V e - 1 Ae V 0 Chapter 3-2- 136
inverting operational amplifier W V e - 1 Ae V 0 Consider the nodal equation at the junction of the three impedances: the output Chapter 3-2- 137 inverting operational amplifier Consider the output Chapter 3-2- 138
inverting operational amplifier Obtain the block diagram of an operational amplifier shown below. V e - 1 Ae V 0 Assume the input impedance, Z i is infinity the output impedance, Z 0, is zero the amplifier gain, A, is finite ( the gain A is not infinity) Chapter 3-2- 139 inverting operational amplifier Consider the nodal equation at the junction of the two impedances: V e - 1 Ae V 0 the output Chapter 3-2- 140
inverting operational amplifier Chapter 3-2- 141 Ideal model of an operational amplifier Ideal model of an operational amplifier will have inverting operational amplifier The amplification, A, as a constant, i.e. the gain A is independent of the frequency (the bandwidth is infinity), The gain, A is infinitely large, A The input impedance, Z i is infinitely large, Z i The output impedance, Z 0 is negligibly small, Z 0 0 In view of A, v 1 v 2» 0, v 1 and v 2 are at the same potential. If v 2 is grounded (that is v 2 =0), then v 1 is at a virtual ground ( v 1» 0). It is comforting to know that the assumption of an ideal model is valid in many application. Chapter 3-2- 142
Ideal model of an operational amplifier One may obtain the model of the ideal operational amplifier circuit in two ways: 1. Derive from the non-ideal case by imposing the above 3 conditions (A, Z i and Z 0 0) 2. Obtain the model directly 1. Derivation from the non-ideal case inverting operational amplifier The conditions 2 and 3 are equivalent to Y i = 0 and Y 0 =. Since A(s)Y 0 (s) is large, dividing both numerator and denominator by A(s)Y 0 (s) and taking the limit as A(s)Y 0 (s) approaches infinity yields: Chapter 3-2- 143 Ideal model of an operational amplifier 2. Direct derivation inverting operational amplifier V 1 I=0 - E=0 V 0 Ideal operational amplifier, I=0 and E=0. Chapter 3-2- 144
Ideal model of an operational amplifier inverting operational amplifier Consider the ideal operational amplifier circuit. Since the gain, A, is infinite, the voltage across i the input terminals, -, and,, namely E=0. Further since the input impedance, Z i, is infinite, the current input to the op-amp, i=0. Hence applying Kirchhoff s law and using the nodal equation we get Since E=0 we get I= V 1 0 - E=0 V 0 Chapter 3-2- 145 Ideal model of an operational amplifier inverting operational amplifier Comment The operational amplifiers are used to implement summers, constant gain multipliers, integrators and in general filters. To implement a constant gain amplifier Z i = R i and Z f = R f Chapter 3-2- 146
Ideal model of an operational amplifier inverting operational amplifier Comment To implement an integrator Z 1 = R 1 and Z f = 1/C f s To implement a filter Z 1 = R 1 (1/Cs) and Z f = R f (1/C f s) Chapter 3-2- 147 Chapter 3-2- 148