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

^{®}solversSimulation 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. 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_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. Select the I_load
and V_line signals. From the Simulation Data Inspector, select **Log
Selected Signals**. 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 block and 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 words `Phasor 60 Hz`

now appear
on the Powergui icon, indicating that the Powergui now applies this
method to simulate your circuit. Before restarting the simulation,
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 the **Output Signal** parameter 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
Scope block does not accept complex signals. Select
`Magnitude`

format for both the Line Voltage and the Load
Current Measurement blocks. This allows you to observe the magnitude of the
voltage and current phasors.

Restart the simulation. Open the Simulation Data Inspector. 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`

`S`

$$\overline{S}=P+jQ=\frac{1}{2}\cdot V\cdot {I}^{\ast}$$

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.