In this section, you

Obtain the state-space representation of your model with the

`power_analyze`

commandCompute the steady-state voltages and currents using the graphical user interface of the Powergui block

Analyze an electrical circuit in the frequency domain

The electrical state variables are the Simulink^{®} states
of your diagram associated to the capacitor and inductor devices of
the Simscape™
Power Systems™ blocks. Inductors and capacitors elements
are found in the RLC-branch type blocks such as the Series RLC Branch
block, Three-Phase Parallel RLC Load block, in the transformer models,
in the PI Section Line block, in the snubber devices of the power
electronic devices, etc.

The electrical state variables consist of the inductor currents
and the capacitor voltages. Variable names for Simscape
Power Systems electrical
states contain the name of the block where the inductor or capacitor
is found, preceded by the `Il_`

prefix for inductor
currents or the `Uc_`

prefix for capacitor voltages.

You compute the state-space representation of a model with the `power_analyze`

command.
Enter the following command at the MATLAB^{®} prompt.

[A,B,C,D,x0,electrical_states,inputs,outputs]=power_analyze('power_gui')

The `power_analyze`

command returns the state-space
model of power_gui example model
in the four matrices A, B, C, and D. x0 is the vector of initial conditions
of the electrical states of your circuit. The names of the electrical
state variables, inputs, and outputs are returned in three matrices.

See the `power_analyze`

reference
page for more details on how to use this function.

Open the power_gui example model by typing `power_gui`

at
the command line. In the Powergui block, in the **Tools** tab,
select **Steady-State**. This opens the **Powergui Steady-State Voltages and Currents Tool** window
where the steady-state phasors voltages measured by the voltage measurement
and current measurement blocks of the model are displayed in polar
form.

Each measurement output is identified by a character vector corresponding to the measurement block name. The magnitudes of the phasors correspond to the peak value of the sinusoidal voltages.

From the **Powergui Steady-State Voltages
and Currents Tool** window, you can also display the steady-state
values of the source voltage or the steady-state values of the inductor
currents and capacitor voltages by selecting either the **Sources** or the **States **check
box.

Refer to the section Measuring Voltages and Currents for
more details on the sign conventions used for the voltages and currents
of sources and electrical state variables listed in the **Steady-State Tool **window.

The Fundamental Blocks/Measurements library contains an Impedance Measurement block that measures the impedance between any two nodes of a circuit. In the following two sections, you measure the impedance in the power_gui model by using two methods:

Automatic measurement using the Impedance Measurement block and the Powergui block

Calculation from the state-space model

The process to measure a circuit impedance from the state-space model (which is described in detail in the next section, Obtaining the Impedance vs. Frequency Relation from the State-Space Model) has been automated in a Simscape Power Systems block. In the power_gui example model, two Impedance Measurement blocks of powerlib measure impedance at two points in the model. Delete the Impedance B3 block and reconnect the Impedance B1 as shown.

**Measuring Impedance vs. Frequency with the
Impedance Measurement Block**

In the** 150 km Line** block, set **Number of pi sections**
to 1, and click **OK**. Open the Powergui dialog. In the
**Tools** tab, select **Impedance Measurement**. A new window opens, showing the list of
Impedance Measurement blocks available in the circuit.

Fill in the frequency range by entering `0:2:5000`

(zero
to 5000 Hz by steps of 2 Hz). Select the logarithmic scale to display
Z magnitude.

When the calculation is finished, the window displays the magnitude
and phase as functions of frequency. In the **150 km Line** block
dialog box, set the **Number of pi sections** to
10. In the Powergui Impedance vs Frequency Measurement tool, click
the **Update** button. The block dialog box displays
the frequency response for the new circuit.

The following section assumes that you have Control System Toolbox™ software installed.

To measure the impedance versus frequency at the same node where
the impedance measurement block is connected, you need a current source
providing a second input to the state-space model. Open the Electrical
Sources library and copy the AC Current
Source block into your model. Connect this source, as shown
below. Set the current source magnitude to zero and keep its frequency
at `60 Hz`

.

**AC Current Source at the B2 Node**

Now compute the state-space representation of the model with
the `power_analyze`

command. Enter the following
command at the MATLAB prompt.

sys1 = power_analyze('power_gui','ss')

This command returns a state-space model representing the continuous-time state-space model of your electrical circuit.

In the Laplace domain, the impedance is defined as the transfer function between the current injected by the AC current Source block and the voltage measured by the U2 Voltage Measurement block.

$$Z2\left(s\right)=\frac{U2\left(s\right)}{I2\left(s\right)}$$

You obtain the names of the inputs and outputs of this state-space model as follows.

sys1.InputName ans = 'U_Vs (60Hz)' 'I_AC Current Source' sys1.OutputName ans = 'U_Ub2' 'U_Ub1'

The impedance at the measured node then corresponds to the transfer function between output 1 and input 2 of this state-space model. For the 0–5000 Hz range, it can be calculated and displayed as follows.

freq=0:5000; w=2*pi*freq; bode(sys1(1,2),w);

Repeat the same process to get the frequency response with a
10 section line model. Open the **PI Section Line** dialog
box and change the number of sections from `1`

to `10`

.
To calculate the new frequency response and superimpose it upon the
one obtained with a single line section, enter the following commands.

sys10 = power_analyze('power_gui','ss'); bode(sys1(1,2),sys10(1,2),w);

Open the property editor of the Bode plot and select units for Frequency in Hz using linear scale and Magnitude in absolute using log scale. The resulting plot is shown below.

**Impedance at the Measured Node as Function
of Frequency**

This graph indicates that the frequency range represented by the single line section model is limited and the 10 line section model gives a better approximation for higher frequencies.

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