An alternative way of showing the synchronous machine equivalent circuit is to use the Norton equivalent of the excitation voltage E_f and the synchronous reactance X_S, as shown in Figure 105-05.

Transforming to the Norton equivalent, we get:

Doing some algebraic transformations it is possible to demonstrate that we can obtain the following relation:

Where we define m as:

Where the variables are defined as:

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6.   Phasor Diagram of a Synchronous Generator

Since the voltages of a synchronous generator are AC voltages, they are usually expressed as phasors, which have magnitude and angle. Therefore, the relationships between them can be expressed by a graph two-dimensional. When the voltages on one phase (E_A, V_g, j X_S I_a and R_a I_a ) and the current I_a of this phase are plotted, resulting in a graph called phasor diagram showing the relationships between these quantities.

In Figure 105-06 we see the model of the equivalent electrical circuit per phase of the synchronous generator. Note that the internal resistance of the field loop and the variable external resistance have been combined on a single resistor R_F. Let's use this template to make the phasor diagram shown in Figure 105-07, when we use a resistive load and therefore have a factor of unit power.

From the circuit shown in Figure 105-06 we see that the difference between the total voltage E_A and the phase terminal voltage, V_g, is given by the resistive and inductive voltage drops. All voltages and currents are referred to V_g, whose angle is arbitrarily assumed to be 0°. So, we can write the equation that defines the terminal voltage V_g, and is given by eq. 105-06, or:

It should be noted that the circuit shown in Figure 105-06 presents a voltage source of direct current V_F, which feeds the rotor field circuit, which is modeled by inductance and series resistance of the field coil. In series with resistor R_F there is an adjustable resistor (not shown in the circuit) that controls the flow of the field current. To represent the three-phase system there are three circuits similar to the one shown, one for each phase. The voltages and currents generated by them are identical in magnitude, but they are out of phase with each other of 120°.

These three phases can be connected in delta or star configuration. When the per-phase equivalent circuit is used, it must be taken into account keep in mind an important fact: the three phases have the same voltages and currents only when the loads connected to them are balanced (or balanced). if generator loads are not balanced, so more sophisticated analysis techniques will be needed. These techniques are beyond the scope of this website.

This phasor diagram can be compared with the phasor diagrams of generators that work with leading and lagging power factors. In this case, we must note that:

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7.   Power and Torque in a Synchronous Generator

A synchronous generator converts mechanical power into three-phase electrical power. The power source mechanical, the prime mover, can be a diesel engine, a steam turbine, a hydraulic turbine or any similar device. Whatever the source, it must have the property basic assumption that its speed is almost constant regardless of the power demanded. Otherwise, the frequency of the resulting power system would vary.

Not all the mechanical power that enters a synchronous generator becomes electrical power at the output of the machine. The difference between the input power and the output represents the machine losses. The input mechanical power is the power on the generator shaft given by the product between the torque applied by the driving machine and the rotational speed of the machine, that is:

On the other hand, the power internally converted by the synchronous generator from the mechanical form to the electrical form is given by the product of the induced torque and the machine rotational speed, or it is:

An alternative way of writing this equation is as eq. 105-09.

In this equation the angle α is the angle between the induced voltage E_A and the armature current I_a. The difference between the input power of the generator and the power converted into it represents the mechanical, core and supplementary losses of the machine.

In Figure 105-10 we present a schematic showing the power flow in a synchronous machine.

The effective electrical power that appears at the output of the synchronous machine can be expressed in line quantities by eq. 105-10.

And in phase quantities by eq. 105-11, where we made V_g = V_F to emphasize the phase greatness.

We emphasize that, in this case, the angle θ is the angle between the phase or line voltage and the armature current, I_a, as shown in Figure 105-11 (below).

It is also possible to express the reactive power in line magnitudes, according to eq. 105-12.

And in phase magnitudes, according to eq. 105-13, where we made V_g = V_F to emphasize the phase greatness.

If the armature resistance R_A is ignored, considering X_S >> R_A, then we can deduce a very useful equation to give an approximate value of the power of generator output. To deduce this equation, let's examine the phasor diagram in Figure 105-11. This figure shows a simplified phasor diagram of a generator with the stator resistance R_A ignored.

Note that the vertical segment b-c can be expressed by eq. 105-14.

We can substitute in eq. 105-11 the value of I_a cos θ by the algebraic transformation of the above relation. Thus, we will get:

This equation stipulates an approximation because we have assumed that the generator resistances are equal to zero, that is, there are no electrical losses in the machine. So, we can state that P_out = P_conv.

It is also possible to calculate the δ angle by algebraically working eq. 105-15 and obtaining the eq. 105-16 below.

We should point out that the eq. 105-14 shows that the power produced by a synchronous generator depends directly on the δ angle. The angle δ is known as internal angle or torque angle of the machine. For the eq. 105-14 it is evident that the maximum power that the generator can supply is when δ = 90°, because sin δ = 1.

The maximum power indicated by this equation is called the static stability limit from the generator. Normally, the real generators never arrive nor close to that limit. Real machines have typical torque angles at full load from 20 to 30 degrees.

Examining the equations eq. 105-11, eq. 105-13 and eq. 105-15 carefully and, assuming that V_F is constant, the effective output power will be directly proportional to I_a cos θ and E_A sin δ and the reactive power output will be directly proportional to I_a sin θ.

These observations are useful when plotting phasor diagrams of synchronous generators with variable payload.

Taking into account eq.105-08 and eq. 105-15 we can easily write the equation that defines the induced torque on the generator rotor through eq. 105-17.

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8.   Determining the Parameters of a Generator

In the same way as we studied in the chapter referring to transformers, in synchronous machines we also use short-circuit and no-load tests to determine the parameters of synchronous machines.

The equivalent circuit that was deduced earlier for a synchronous generator contained three quantities that must be determined to fully describe the behavior of a real synchronous generator. Are they:

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    8.1   No-Load Test of a Synchronous Generator

This is the first step to be performed to determine the parameters of the synchronous generator. Therefore, we must ensure that we do not there is any kind of load connected to the generator terminals. That is, we have the generator in the condition of "empty". Next, we run the generator at its nominal speed. Since we have no load, then we know that I_a = 0 and in this case we have E_A = V_F . Knowing this information, it is possible to construct a plot of E_A versus I_F. The curve constructed on the graph is called the empty characteristic. In some literature also known as open circuit characteristic.

Using the characteristic curve we can find the generated internal voltage E_A for any current of field I_F given.

In Figure 105-12 we see the empty characteristic typical of a synchronous generator. Note that, at the beginning, the curve is almost perfectly straight until some saturation is observed with high field currents. The unsaturated iron of the synchronous machine has a reluctance that is several thousand times less than the reluctance of the air gap. Thus, in the beginning, almost all the magnetomotive force is in the air gap and the resulting flux increment is linear. When the iron finally saturates, the iron's reluctance increases dramatically and the flow increases a lot more slowly with increasing magnetomotive force. The linear portion of a empty characteristic is called the air gap line of the characteristic.

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    8.2   Short-Circuit Test of a Generator

After carrying out the empty test and determining the graph E_A versus I_F, we are able to carry out the short circuit test and determine the various parameters of the generator synchronous. Like first step, we must short-circuit the output of the generator with a set of ammeters and adjust the field current I_F for the value zero. Thus, the armature current I_a (or the line current I_L) is measured while the field current is gradually increased. Making a table of measurements and placing these values ​​on a graph, we will get the so-called short circuit characteristic and we can see this graph in the Figure 105-13.

Notice that it's basically a straight line. To understand this, we can look at Figure 105-06 and considering that the generator terminals are short-circuited (that is, V_g = V_F = 0 ), so we can write the value of I_a by eq. 105-18. In fact, the resulting magnetic induction field in the machine is very small, causing machine is not saturated, so short circuit characteristic is linear.

Of course we can write the value of the module of I_a by eq. 105-19.

On the other hand, when the machine is short-circuited, we have V_F = 0, and this allows us to write the machine's internal impedance, represented by Z_S and given by eq. 105-20.

However, in general, we know that we can consider Z_S >> R_A and so, considering R_A ≈ 0, we can rewrite the eq. 105-20 like:

In this equation we are representing the voltage at the output of the no load generator as V_F0. So, to determine the approximate value of the machine's synchronous reactance, X_S, with a given field current, we can follow the guide below:

If you are interested in determining the armature resistance, R_A, you can use the following process:

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    8.3   Short Circuit Ratio

Another parameter used to describe a synchronous generator is the so-called short circuit ratio. It can be set like:

The short-circuit ratio allows characterizing the quality of the synchronous machine. For the same power and rated currents, a machine with lower short-circuit ratio has less volume and weight and, consequently, lower cost.