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In this section, you
Apply the phasor simulation method to a simple linear circuit
Learn advantages and limitations of this method
Up to now you have used two methods to simulate electrical circuits:
Simulation with variable time steps using the continuous Simulink^{®} solvers
Simulation with fixed time steps using a discretized system
This section explains how to use a third simulation method, the phasor solution method.
The phasor solution method is mainly used to study electromechanical oscillations of power systems consisting of large generators and motors. An example of this method is the simulation of a multimachine system in Three-Phase Systems and Machines. However, this technique is not restricted to the study of transient stability of machines. It can be applied to any linear system.
If, in a linear circuit, you are interested only in the changes in magnitude and phase of all voltages and currents when switches are closed or opened, you do not need to solve all differential equations (state-space model) resulting from the interaction of R, L, and C elements. You can instead solve a much simpler set of algebraic equations relating the voltage and current phasors. This is what the phasor solution method does. As its name implies, this method computes voltages and currents as phasors. Phasors are complex numbers representing sinusoidal voltages and currents at a particular frequency. They can be expressed either in Cartesian coordinates (real and imaginary) or in polar coordinates (amplitude and phase). As the electrical states are ignored, the phasor solution method does not require a particular solver to solve the electrical part of your system. The simulation is therefore much faster to execute. You must keep in mind, however, that this faster solution technique gives the solution only at one particular frequency.
You now apply the phasor solution method to a simple linear circuit. Open the example named Transient Analysis of a Linear Circuit (power_transientpower_transient).
This circuit is a simplified model of a 60 Hz, 230 kV three-phase power system where only one phase is represented. The equivalent source is modeled by a voltage source (230 kV RMS / sqrt(3) or 132.8 kV RMS, 60 Hz) in series with its internal impedance (Rs Ls). The source feeds an RL load through a 150 km transmission line modeled by a single PI section (RL1 branch and two shunt capacitances, C1 and C2). A circuit breaker is used to switch the load (75 MW, 20 Mvar) at the receiving end of the transmission line. Two measurement blocks are used to monitor the load voltage and current.
The Powergui block at the lower-left corner indicates that the model is continuous. Start the simulation and observe transients in voltage and current waveforms when the load is first switched off at t = 0.0333 s (2 cycles) and switched on again at t = 0.1167 s (7 cycles).
You now simulate the same circuit using the phasor simulation method. This option is accessible through the Powergui block. Open the Powergui, click Configure Parameters, and in the Powergui block parameters dialog box set Simulation type to Phasor. You must also specify the frequency used to solve the algebraic network equations. A default value of 60 Hz should already be entered in the Phasor frequency field. Close the Powergui and notice that the word Phasors now appears on the Powergui icon, indicating that the Powergui now applies this method to simulate your circuit. Before restarting the simulation, you need to specify the appropriate format for the two signals sent to the Scope block.
If you now double-click the Voltage Measurement block or the Current Measurement block, you see that a menu allows you to output phasor signals in four different formats: Complex (default choice), Real-Imag, Magnitude-Angle, or just Magnitude. The Complex format is useful when you want to process complex signals. Note that the oscilloscope does not accept complex signals. Select Magnitude format for both the Line Voltage and the Load Current Measurement blocks. This will allow you to observe the magnitude of the voltage and current phasors.
Restart the simulation. The magnitudes of the 60 Hz voltage and current are now displayed on the scope. Waveforms obtained from the continuous simulation and the phasor simulation are superimposed in this plot.
Waveforms Obtained with the Continuous and Phasor Simulation Methods
Note that with continuous simulation, the opening of the circuit breaker occurs at the next zero crossing of current following the opening order; whereas for the phasor simulation, this opening is instantaneous. This is because there is no concept of zero crossing in the phasor simulation.
The Complex format allows the use of complex operations and processing of phasors without separating real and imaginary parts. Suppose, for example, that you need to compute the power consumption of the load (active power P and reactive power Q). The complex power S is obtained from the voltage and current phasors as
$$\overline{S}=P+jQ=\frac{1}{2}\cdot V\cdot {I}^{\ast}$$
where I* is the conjugate of the current phasor. The 1/2 factor is required to convert magnitudes of voltage and current from peak values to RMS values.
Select the Complex format for both current and voltage and, using blocks from the Simulink Math library, implement the power measurement as shown.
Power Computation Using Complex Voltage and Current
The Complex to Magnitude-Angle blocks are required to convert complex phasors to magnitudes before sending them to the scope.