dynamic system simulation. charging capacitor the capacitor is initially uncharged there is no...
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Dynamic system simulation
Charging Capacitor• The capacitor is initially uncharged• There is no current while switch S is open (Fig.b)• If the switch is closed at t= 0 (Fig.c) the
charge begins to flow, setting up a current in the circuit, and the capacitor begins to charge
• Note that during charging, charges do not jump across the capacitor plates because the gap between the plates represents an open circuit
• The charge is transferred between each plate and its connecting wire due to E by the battery
• As the plates become charged, the potential difference across the capacitor increases
• Once the maximum charged is reached, the current in the circuit is zero
Charging Capacitor (2)• Apply Kirchhoff’s loop rule to the
circuit after the switch is closed
• Note that q and I are instantaneous values that depend on time
• At the instant the switch is closed (t = 0) the charge on the capacitor is zero. The initial current
• At this time, the potential difference from the battery terminals appears entirely across the resistor
• When the charge of capacitor is maximum Q, The charge stop flowing and the current stop flowing as well. The V battery appears entirely across the capacitor
Charging Capacitor (3)
• The current is , substitute to voltage equation
• The equation is called Ordinary Differential Equation (ODE)
• How to solve this equation? Solve mean we can express the equation into q(t)=….
Solution of ODE
• Using Deterministic Approach• Using Numerical approach:
1. Euler’s method2. Heun’s method3. Predictor-corrector method4. Runge-kutta method5. Etc.
Deterministic Approach
• The current is , substitute to voltage equation
• Integrating this expression
• we can write this expression as
Deterministic Approach
• If you integrate to obtain the solution, then you use exact/deterministic method.
• However in practical use, we often cannot integrate the function directly.
• The numerical approach is often preferable.
Numerical approach
Numerical approach (2)
Numerical approach (3)
Solution in Matlab
• Using ODE solver (m-file)• Using Simulink
State space of charging capacitor
( ) 1 1( )
( ) [1] ( ) [0]
dq tq t x Ax Bu
dt RC R
q t q t y Cx Du
State space in practical use
• In practical use, the A matrix consists of many states space
• Simulating the power system is just solving the differential equation of system states and (sometimes) algebraic equation related to load flow .
• Normally we use states space in power system simulation such as rotor speed, rotor angle, Flux-linkage change, etc.
( )x t
State space in practical use (2)• Example: state space of synchronous generator with PSS
x Ax
Order greater than 1 (n>1)
• Suppose second order (n=2) equation
• We need to write second order equation into n order first order differential equation
• These equations can be solved simultaneously• Homework 1: how to solve this equation for
a=b=c=1 using Matlab (use function: ode45)? With all initial states are zero
Transformer Simulation
• Equivalent circuit
Transformer Simulation (2)
• Voltage Equation
• The flux linkage per second
• Mutual flux linkage
Transformer Simulation (3)
• The current can be expressed as
• Eqn. 4.29 is now
• Collecting mutual flux linkage
Transformer Simulation (4)
• Define
• Eqn 4.33 can be expressed as
Transformer Simulation (5)
• The flux linkage in integral form
Transformer Simulation (6)
Implementation in Simulink
Homework 2: Build this block in Simulink with all initial values of flux linkage are zero
Rules for student
• Maksimal terlambat 20 min• Tidak boleh titip absen• Tidak boleh menggunakan barang elektronik
kec berhubungan dengan kegiatan perkuliahan
• All materials are posted at http://husniroisali.staff.ugm.ac.id/