Vector diagrams. Construction of vector diagrams

The processes occurring in an alternating current electrical circuit with active resistance and reactive inductance can be clearly expressed graphically.

The article will describe what vector diagrams are, where and what they are used for. The timing diagram and its purpose will also be described. At the end, an example of constructing a simple diagram for an electrical circuit with a series connection of elements will be given.

Definition

A vector diagram of currents and voltages is a geometric representation of all processes, quantities and amplitudes of a sinusoidal current. All available quantities are located on the plane in the form of vectors.

The construction of a vector diagram is used in physics and electrical engineering. By creating such a diagram, you can significantly simplify the calculations performed, as well as display the ongoing processes in a clear and accessible form.

The vector diagram method also allows you to see short and phase-to-phase short circuits that occur in an alternating current circuit, as well as calculate possible power losses.

Usually such a diagram is built together with a time diagram. A timing diagram is a graphical representation of the input and output in an electrical circuit. Timing diagrams help determine the time interval between the start, flow and end of a signal. For example, when you press a button, a signal is generated that goes to the receiver and starts the process of its operation.

Timing diagrams are also applicable to a sinusoidal electrical circuit, since this current has an initial reference point (power on) and a time of movement from the current source to the consumer. Such diagrams are a graph that depicts the initial reference point, the time vector and phase angles.

Preparing data for a ray diagram

As mentioned above, this template will have the ability to visually build connections for up to 20 participants (companies, branches, counterparties, etc.). The template worksheet "Data" provides a table for populating incoming values. For example, let’s fill it out for 14 market participants:

On the same sheet we will create an additional table, which is a matrix of connections of all possible participants, generated by the formula:

With the data preparation completed, we move on to processing.

How to calculate the sum of vectors?

Vectors and matrices in a spreadsheet are stored as arrays.

It is known that the sum of vectors is a vector whose coordinates are equal to the sums of the corresponding coordinates of the original vectors:

To calculate the sum of vectors you need to perform the following sequence of actions:

– Enter the values ​​of the numerical elements of each vector into ranges of cells of the same size.

– Select a range of cells for the calculated result of the same dimension as the original vectors.

– Enter the formula for multiplying ranges into the selected range

– = Vector_Address_1 + Vector_Address_2 Address

– Press the key combination [Ctrl] + [Shift] + [Enter].

Example.

Two vectors are given:

It is required to calculate the sum of these vectors.

Solution:

– To cells in the range A2:A4

Let's enter the values ​​of the coordinates of vector a1, and into the cells of the range
C2:C4
- the coordinates of vector a2.

– Select the cells of the range in which the resulting vector C will be calculated ( E2:E4

) and enter the formula into the selected range:

=A2:A4+C2:C4

– Press the key combination [Ctrl] + [Shift] + [Enter]. In cells in the range E2:E4

the corresponding coordinates of the resulting vector will be calculated.

Varieties

Having figured out what a vector diagram is and what it is used for, you need to find out what types of construction exist. They differ in the nature of their construction and type. By nature they are:

1. Accurate. A vector exact diagram is a display of the performed numerical calculation on the appropriate scale. Using it, phase parameters and amplitude values ​​are determined in a strictly geometric way.
2. High quality. Such histograms are constructed to observe the relationship between electrical quantities without using numerical characteristics. This method allows you to experiment with various parameters and simulate processes in electrical circuits.

A vector diagram of currents can be constructed in 2 different ways:

1. Circular. Its principle is a vector that describes a change in characteristics by forming a circle or semicircle on a plane. With this option, the direction of movement is taken into account, taking into account the direction of the vector position.

2. Linear. In such a vector diagram, when the characteristics change, the direction changes strictly rectilinearly.

Both constructions can be used to calculate the characteristics of alternating current in a circuit with resistance and inductance.

Construction

The construction of simple vector diagrams will be discussed in this section. For example, you can take a simple chain with several elements and their values. This scheme involves sequential connection of elements to each other. The circuit consists of an inductor, a capacitor and an active resistance. The parameters of each circuit element are given below.

1. UL 15 volt inductor. The current in the inductive reactance has a phase shift of 90°.
2. A UC capacitor with a voltage of 20 volts and a 90 degree lead.
3. The voltage of the resistor UR is 10 volts, its direction coincides with the current I.
4. The current in circuit I is 3 amperes.

Next, you can make a simple diagram that will help determine the voltage for the entire circuit.

1. Place it on plane I in the form of a horizontal line with a scale of 1 A/cm (the scale can be any, the main thing is to carry out all the elements of the diagram of the same type on the same scale). The current itself is 3 amperes, so its length will be 3 cm.

2. Now we need to plot the vertical vector UL on a scale of 5 V/cm. It displays the voltage of the inductor and is equal to 15 volts. Its length on the plane will also be 3 cm on this scale.

3. Next, you need to graphically designate the voltage vector of the active resistance. Its reference point is located at the end of the vertical vector UL. For the accepted scale of 5 V/cm, it corresponds to a vector 2 cm long. The line must be strictly parallel to the horizontal vector I.

4. Now we need to display the voltage of the capacitor UC on this diagram. Its beginning will be the end point of the vector UR, and the end of this vector will be located below the horizontal vector I. On a scale of 5 V/cm, it corresponds to a vector with a length of 4 cm.

5. To determine the voltage U corresponding to such a communication circuit, you will need to do the following. The beginning of the vector is located at the accepted reference point, and its end will be located at the end point of the vector UC.

Therefore, if there is a circuit with a series connection of elements, then you can always quite simply construct a vector diagram and calculate the total voltage for such a circuit.

Method 2

Construction of vector diagrams taking into account all known values ​​for an alternating current circuit with a series connection of a capacitor, resistor and inductor. With this construction, we also know the voltage of the circuit itself. The circuit consists of:

• Resistor UR;
• Capacitor UC;
• UL coils.

Next you need:

1. The vector UR (resistor) is plotted on the Im plane. Its direction is exactly the same as the current, so it will be a horizontal line.
2. The vector UC (capacitor) is laid down from the reference point. The vector is laid down at an angle of 90 degrees, since it has the previously specified 90° lead.

3. The vector UL (inductor) is plotted from the same reference point. Its value is plotted exactly 90 degrees vertically, since there is a phase shift of 90 degrees.

This diagram can be used to control and calculate the influence of all known parameters of the circuit and elements, as well as their relationship with each other.

1. Show the result of adding the vector UL and UC.

2. As the resistance value increases, the difference between voltage and resistance can be determined using the new vector Um.

3. In addition, you can determine the phase shift angle φ in the circuit.

The main advantage of a vector diagram is the following: simple and quick addition and subtraction of two parameters during the calculation of electrical circuits.

The concept of vectors and vector diagrams also implies the calculation of the power circuit of a three-phase network connected using the star method. It is constructed taking into account 3 deferred vectors from the 0 ordinate axis at once. This construction determines the vector from the current source to the receiver. A vector is constructed with the following values:

1. The real values ​​of quantities are plotted on the OX axis, and imaginary values ​​are plotted on the OY axis.
2. The angular magnitude is denoted as W.
3. The vector Im itself and the phase shift angle φ are also present.

Next you need to do:

1. Select a reference point on the plane.
2. Delay the vector Im from it, taking into account the phase shift angle of 90°.
3. The length of the vector Im is equal to the value of its voltage and is plotted on the selected scale.

In the same way, two more straight lines are superimposed on the plane. The general diagram will show the symmetry of the phases or their shift when a short circuit occurs. Such a diagram can serve as an example for calculating voltage, current or load on each phase with modeling of various parameters.

Algorithm for creating a ray vector diagram in Excel

To simplify our lesson, let's assume that we are talking about relationships not between fourteen as in the graph, but for now only with 4 people named Anton, Alice, Boris and Bella.

Our matrix of the level of relationships and connections between them looks like this:

• 0 means no relationship;
• 1 means weak relationship (for example: Anton and Alice just know each other);
• 2 means a strong relationship (for example, Boris and Alisa are friends).

How can we geometrically model the visualization of this raw data? If we were to draw the relationship between these four people (Anton, Alice, Boris and Bella), it would schematically look like this:

2 criteria that we need to determine:

1. Location of dots (where people's names are printed).
2. Lines (starting and ending points of connecting lines).

Defining and plotting points

First we need to plot our points so that the space between each point is the same. This will create a balanced schedule.

Which geometric figure best satisfies our need for such equal intervals? Of course it's a circle!

You might object that the finished diagram model does not have a circle shape. Yes, really no - that's it. We don't need to draw a circle. We just need to plot the points around it.

So we have 4 stakeholders, we need 4 points:

1. If we have 12 stakeholders, we need 12 points.
2. If we have 20, we need 20 points.

Assuming the origin of our circle is (x, y), the radius is r and theta is 360 divided by the number of points we need. The first point (x1, y1) on the circle will be at this position:

• x1 = x + r * COS (theta);
• y1 = y + r * SIN (theta).

Once all the points are calculated and connected to the XY chart (scatter plot), let's move on.

Drawing lines on a ray diagram

Let's say we have n people in our network. This means that each person can have a maximum of n-1 relationships.

Thus, the total number of possible lines on our graph is n * (n-1) / 2.

We need to divide it by 2, as if A knows B, then B also knows A. But we only need to draw 1 line.

The network graph analysis ray diagram template is configured to work with 20 people. It can be downloaded at the end of the article and used as a ready-made analytical tool for visualizing these connections. This means that the maximum number of rows we can have will be 190.

Each line requires adding a separate series to the chart. This means we need to add 190 series of data for just 20 people. And it only satisfies one line type (dashed or thick). If we want different lines depending on the type of relationship, we need to add another 190 episodes.

It's painful and funny at the same time. Fortunately, there is a way out!

We can use a much smaller number of series and still produce the same graph.

Let's say we have 4 people - A, B, C and D. For the sake of simplicity, let's assume that the coordinates of these 4 people are as follows:

• A – (0.0);
• B – (0.1);
• C – (1,1);
• D – (1.0).

And let's say A has relationships with B, C and D.

This means that we need to draw 3 lines, from A to B, from A to C and A to D.

Now, instead of putting 3 series for the chart, what if we put one long series that looks like this:

(0,0), (0,1), (0,0), (1,1), (0,0), (1,0)

This means we simply draw one long line from A to B, A to C, A to D. Granted, it's not a straight line, but Excel scatter plots can draw any line if you give it a set of coordinates.

See this illustration to understand the technique:

So instead of 190 series of data for the chart, we just need 20 series.

In the last chart we have 40 + 2 + 1 series of data. It's because:

• 20 lines for weak relationships (dashed lines);
• 20 lines for strong relationships (thick lines);
• 1 line for highlighting in blue the weak relationships of the selected participant;
• 1 line for highlighting in green the strong relationships of the selected participant;
• 1 set without lines, but just dots for data labels on the graph.

How to generate all 20 data series:

This requires the following logic:

• Assuming we need lines for person n's relationships.
• This person's point will be (Xn, Yn) and has already been calculated earlier (at points on the graph around the circle).
• We only need 40 rows of data.
• Every odd row will have (Xn, Yn).
• For each even row:
• divide the line number by 2 to get the person's number (say m>);
• (Xn, Yn) if there is no relationship between n and m>;
• (Xm, Ym) if there are relations.

We need MOD and INDEX formulas to express this logic in Excel.

Once all line coordinates have been calculated, add them to our scatter plot as new series using the tool from the additional menu: “WORKING WITH DIAGRAMS” - “CONSTRUCTOR” - “Select data” in the “Select data source” window, use the “Add” button to adding all 43 rows.

We will implement the creation of such a ray diagram of connections in 3 stages:

1. Preparation of initial data.
2. Data processing.
3. Visualization.