Dynamic system simulation Charging Capacitor The capacitor is

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 circ uit, 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 method 2. Heun’s method 3. Predictor-corrector method 4. Runge-kutta method 5. 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

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.

State space in practical use (2) • Example: state space of synchronous generator with PSS

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: ode 45)? 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/