What are Synchronous Machines?

The power that we consume at home – everyday – is delivered to us through long transmission lines from power plants. The role of power plants are to capture nature’s energy and convert it to first rotational energy, next to electrical energy by synchronous machines.

Here is a diagram of a synchronous machine:

Diagram: Showing key parts of a synchronous machine

Figure 1: Basic concept of a synchronous machine. Main parts are: the prime mover, slip rings, rotor and stator. The stator is a hollow steel cylinder that holds the 3-phsae windings around. The rotor is a DC electromagnet that is inserted in the Stator. The image shows a salient pole rotor that is essentially having multiple North and South poles. The colours symbolise the 3 separate phases. Each phase is one wire, just wrapped in 2 separate coils for more efficiency. X means the current goes in, the dot means current is coming out. Check this post if you are interested in how this works.

Energy hides in many places: coal, sun rays, wind, water etc. Taking water as an example:

  • Flowing water rotates a turbine (kinetic energy is transformed to rotational energy)
  • The turbine is attached to a synchronous generator that converts this rotational energy to electricity.

We can do the opposite at home: hook up a synchronous motor to the mains power and convert electricity into rotational energy!

Synchronous Machines are Reciprocal

Synchronous Machines consist of 2 main parts: a rotor and a stator. The stator consists of 3-phase coils while the rotor is a DC electromagnet.

A synchronous machine can operate both as a Generator & Motor.

    • Turn the prime mover ( attached to the rotor).
    • The rotor’s rotating magnetic field will cut the stator’s windings (wire), thus, generating an EMF based on Faraday’s Law of Electromagnetism.


    • The opposite of a generator.
    • Feed the machine with 3-phase electricity.
    • The alternating current in the rotor’s field windings will create a rotating magnetic field.
    • The rotor’s electromagnet’s magnetic field will lock onto the stator’s field and starts rotating. (North pole attracts South pole)

The Synchronous Reactance Model

Let’s draw the Synchronous Reactance Model of the Synchronous Generator. What is this? Basically, this is a circuit that represents the elements (losses, voltage drops, load) of the machine. So, this is how it looks like:

Synchronous Reactance Circuit Model for the Synchronous Generator

Figure 2: The Synchronous Reactance circuit model, where E is the EMF generated, jx and Ra are the resistances and ZL is the impedance of the load.


  • E: EMF generated
  • jx: Stator winding reactance
  • Ra: Stator armature resistance
  • Z(L): Load impedance

So, E is the actual EMF that is being generated when the rotor’s DC magnet cuts the stator windings. Ra and jx are representing losses.

The stator is made of copper coils, therefore, the coils have a natural resistance (Ra). But since it is a coil, it will also have a reactance (jX) too (see what it is below). In general, a coil has an impedance, which consist of 2 parts: resistance (real part) and reactance (imaginary part).

Impedance (Z) = Resistance (R) + Reactance (jX)

So, there’s going to be a voltage drop across jx, that represents the coil’s reactance and Ra, that represents the coil’s resistance. Finally, the voltage across Z(L) will be the voltage what we get at the generator’s output terminals. Note, Z(L) is represented as it has an impedance. Most motors or gear that has a coil in them will have an impedance.

What is ‘jx’, the reactance in synchronous machines?
(Background Theory)

It is similar to resistance. We know what that is: the structure of the wire resisting the flow of electrons. Well, we call it resistance when the voltage and current are in phase. Like across a resistor.

Reactance‘ is when the voltage and current are not in phase. The current can either be ahead of the voltage (leading) or behind (lagging).

There are 3 components:

  • resistors (V&I in phase),
  • capacitors (Current leading voltage by 90°)
  • inductors (Current lagging voltage by 90°)
Voltage and Current Relationshop in Resistive, Capacitive and inductive Load

Figure 3: Voltage & Current phase relationship in various types of loads.

Resistors have resistance but capacitors and inductors have reactance. Synchronous machines have coils to generate a magnetic field. These are essentially inductors. jx in the above circuit represents the reactive part of the coil in the generator.

X is the value, j only indicates that it is an imaginary component. I will go into this a bit further down.

Why Does Current Lag in an Inductor?

So, current and voltage are not in phase across an inductor. In fact, current lags by 90°! Let’s see what is happening:

Diagram: of a coil showing the flow of current and the related magnetic field generated as well as the back EMF.

Figure 4: The image shows the magnetic field generated around a coil when current is fed into it. Initially when the magnetic field is growing, there will be a back EMF generated that will oppose the current flow. This is the reason behind the 90° current lag.

  • Current enters the coil and generates a magnetic field. If you grab the wire at any point, if your thumb point in the direction of the current, your fingers will point in the direction of the magnetic field (yellow arrows).
  • The field lines always point from North to South outside the coil.
  • The magnetic field is not instantaneous, so it grows.
  • This growing of the magnetic field means the coil is exposed to a changing magnetic field (while it grows, which in real time is super quick). What does this suggest?
  • The Law of Electromagnetic Induction tells us, if a wire is in a changing magnetic field, there will be an EMF generated. This is the Back EMF and will oppose the current flow.

So, the current is held back for a little while. Of course once the magnetic field settles (stops growing), the back EMF disappears.

Note, when power is cut from the coil (no electromotive force that makes current move), the reducing magnetic field will yet again induce a back EMF now in the other direction. If we still have a closed circuit, this force will push the current for a little while longer.

Analysing the Inductor’s Voltage Equation

The above is also apparent from the inductor’s voltage equation:

Voltage Equation and Voltage Current phase rlationship of an inductor.

Figure 5: Voltage equation of an inductor. The graph shows the phase relationship of current and voltage. Current lags by 90°.

From the equation:

V has maximum value if di/dt has the max value. di/dt simply means current change. So, di/dt is a value and it represents the amount the current changes over a given time period. The Y axis represents this value.

The X axis represents time (it is phase in the image but they are related). This di/dt change is measured with a tangent line.

So, let’s place a tangent line onto the current curve at 180°. We would get a line that is in parallel with the X axis. If we take a very small step on the X axis, the corresponding Y value wouldn’t change. So, there is no change in current at 180°. di/dt =0.

The steeper the tangent line is, the large the change is of the Y value. This happens at 90° and 270°.

So, at 90° di/dt = maximum. Therefore, from the equation, the voltage has the maximum value. Drawing this relationship, we see the current lagging by 90°.

What does ‘j’ mean in ‘jx’?
(Background Theory)

j is the ‘representative’ of an imaginary number. Back in the days, people used to think in 1 dimension: positive and negative numbers. Because of this, of course they failed to answer the question what x*x = -1 is. What is the value of x? What is the number that we multiply by itself we get -1?

The reason this was unsolvable is because a positive number multiplied by itself gives a positive number. So is a negative number multiplied by itself! The result is a positive number!

Let’s think in 2 dimensions for a change.

Diagram of the Complex Plane showing the real and imaginary numbers.

Figure 6: The origin of imaginary numbers. The once unsolvable question of what the value is for 1*x*x = -1 is answered in the two dimensional complex plane where x is describes an anticlockwise rotation of 90°.

The above diagram shows a complex plane, which is 2-dimensional. So what do we do to the number 1 twice to turn it to -1? We rotate it! X is a 90° rotation. If X is a 90 degree rotation anti-clockwise, we rotate the number 1 twice and we get -1. Problem solved!

  • j indicates a 90° rotation anticlockwise and
  • -j a rotation clockwise.

So, in an inductor (coil) the current is 90° late compared to the voltage. Thus we denote it by j and call it impedance.

Phasor Diagram of Synchronous Machines

Synchronous Generator

In a synchronous generator we have the following components:

  • E: EMF generated in stator windings by the rotor
  • Ra: Armature (Coil) Resistance
  • jX(s): Armature Reactance (stator/winding reactance)
  • Ia: Armature Current
  • V: Terminal Voltage output

The phasor diagram shows visually the phase representation between voltages. We take V, the terminal output voltage as reference. As we’re talking about a machine that has impedance, Ia lags behind V. The cosine of the angle between V and Ia is called the power factor.

Phasor Diagram of the Synchronous Generator

Figure 7: Phasor Diagram of a Synchronous Generator. It is a visual representation of the phase relationship between voltages.

How to Draw the Phasor Diagram

  • We draw V first. This is the terminal voltage and we draw everything else in reference to this.
  • The current leads, so we draw it with an angle between. This is the power angle but more on this later.
  • The good thing about vectors is that we can move them around. IaRa will be the part of the voltage (since V=IR) that is in phase with the current. So, this will be in parallel to the current.
  • The out of phase voltage component is 90° off from the current. It is JIaX(s). Current*reactance. It is Ohm’s Law, simple. Draw this from the IaRa vector.
  • From this we can draw E, the EMF. E is the vector sum of V, the terminal votlage, and the voltage drop due to losses (JIaXs + IaRa).

The EMF is a voltage too, a force that wants to move the electrons. V is what remains from E after all the losses. The alpha angle is called the load angle which is the angle between E and V.

Ignoring the Ra component (it is very small)

The winding resistance is very small compared to the reactance in the coil. So to simplify our diagram, we assume Ra = 0 so IaRa = 0.

Simplified Phasor Diagram of the Synchronous Generator, where we assume the armature reisitance is o Ohm.

Figure 8: Simplified phasor Diagram of a Synchronous Generator where the RaIa voltage is ignored. This is because in generator windings the reactance greatly dominates over resistance.

It is clear for the graph that the EMF generated by the prime mover will be equal to the sum of the terminal voltage (voltage after the losses) and the losses in the coil.

Equation: EMF is equal to the sum of the terminal voltage and the voltage dropped due to losses.

Power Output of Synchronous Machines

We have 3 phases in a 3-phase synchronous generator. Generally speaking, if we only have resistance in a circuit, the power is V =IR. But in case of an impedance, we multiply IR by the power factor to get the power. The power factor is the angle between the current and the voltage.

Since we have 3 phases, the power will be:

Equation: Synchronous Machine Power 3-phase power equation


  • Vp: the phase voltage.
  • Ia: the armature current
  • cosΦ: the power factor

Finding the power equation in terms of E, Xs and load angle α.

Let’s draw an extended phase diagram:

Analysing the angles of the Phasor Diagram of a Synchronous Machine with a help of a b vector. Aim is to find a power and torque equation.

Figure 9: Phasor diagram of a synchronous machine showing a single phase only. E represents the EMF generated by the prime mover, Ia the armature current that is drawn by EMF, V the final output terminal voltage from the generator.

  • We add a b vector to act as a guide. It is perpendicular to V.
  • From trigonometry we know that in case of a right angled triangle, the sin of an angle is the opposite side/hypotenuse. So let’s find the length of our guide vector:
Derivation: finding expression for the b vector.
  • So we got a value for b. Now let’s write down the equation for sinα and substitute b in:
Derivation: finding an expression for the load angle.
  • We substitute equation 2 to equation 1 to get the Power equation in terms of phase voltage, EMF, load angle and the reactive component.
  • The above calculation was in terms of a single coil. For a 3-phase generator we multiply by 3. For a star configuration, where the phase current = line current, and VL = sqrt(Vp):
Formula: for the 3-phase power delivered by a synchronous machine in terms of the EMF, Reactance and load angle.

Note, the above equation was derived by assuming the armature resistance (resistance of the coil) = 0Ω.

From the above equation it is clear that the Power is maximised if sin α =1. This the case when α=90°.

What is α? α is the load angle and it increases with increasing load. If α>90° the generator will lose synchronism.

It is the safest to operate the generator at 15-60°.

Graph: Power vs Load Angle

Figure 10: Power vs Load Angle curve. Power is maximised if the load angle is 90°. It is increased by increasing the load’s current draw.

The following relationships are important to note. P is the active, Q is the reactive power:

Active and Reactive Power relationship with load and power angle.

Torque of Synchronous Machines

Derivation: Formula of Torque applied by a 3-phase synchronous machine in terms of load angle, reactance and EMF.


  • Vp: the phase voltage.
  • Ep: EMF generated in a single phase
  • ω: angular speed
  • α: the load angle
  • Xs: reactance of the coil
  • Ns: Synchronous speed

Both the torque and the power flow is proportional to the load angle.

Take a look at the phase diagram once again. Do you see that by increasing α, I will have a steeper angel and so does jIaXs as it is perpendicular to Ia. In this case, E will be larger. So, in other words, the output voltage will be the same, but we need a larger E to make it happen in the case of an increased load.

Phasor Diagram of Synchronous Machine.

Figure 11: Phasor diagram of a synchronous machine showing a single phase only. E represents the EMF generated by the prime mover, Ia the armature current that is drawn by EMF, V the final output terminal voltage from the generator.

Phasor Diagram of Synchronous Motors

The Phasor Diagram of the Synchronous Motor is very similar to the generator’s. In the generator we had lagging power factor, so the current was lagging behind the voltage.

In the case of the motor we can have both lagging and leading power factor. This depends on the excitation.

Excitation means how much current is fed in the stator’s DC magnet’s coil. Remember, this field generates the magnetic flux that induces the EMF in the stator coils. The diagram demonstrates a leading power factor.

Circuit model of Synchronous Motor and its related phasor diagram.

Figure 12: Circuit model for a Synchronous Motor and the related phasor diagram. Note, the armature resistance has been ignored as the reactance dominates in the stator windings. We have a leading power facto.

How can current lead voltage in synchronous motors?

It is a little bit controversial as we did mention earlier that in coils the current lags behind the voltage by 90°. What’s happing then?

  • Current is fed from an external supply to the stator windings and a rotating magnetic field will be generated.
  • The stator has a DC electromagnet. This magnet’s poles will lock onto the rotating field.
  • Now the DC magnet’s field is rotating too which will cut the stator’s windings.
  • Based on the Law of Electormagnetic Induction, a current will be generated that reinforces the current flow in the stator.

This reinforcement makes the current lead the voltage. Note, this is only the case when the field current, If is overexcited.

Diagram: A rotor inside the stator of a synchronous motor. The diagram shows the currents & fields generated.

Figure 13: A 2-pole synchronous motor showing the magnetic fluxes for both stator and rotor. As the rotor rotates, it’s magnetic field will cut the stator windings generating an EMF that will induce a current. This current reinforces the stator current flow, which originally lags, and make it lead instead. This is the overexcited case, when the Field Current, If is large.

V-Curve & Inverted V-Curve of the Synchronous Motor

The V-Curve and the Inverted V-Curve reflect how the motor operates in under or over excited conditions. This means how much current is applied to the rotor windings. This is called the field current.

We see the conditions: no, half and full load. When we apply a load, the machine will draw more armature current Ia. According to the unity power factor compounding curve when the load is increased we have to increase the field current to maintain the same power factor.

Graph: V and Inverted V-curve of Synchronous Motor

Figure 14: V and inverted V-Curve of Synchronous Motor. Ia represents the armature current (current in stator winding) and If represents the field current, which is the current in the stator’s DC magnet coil.

Reactive Power and its Effects

Due to the nature of capacitance and inductance, there are cases where the current and phase are not in phase with each other. Power generated from systems like such have both a real power and a reactive power component.

The problem with reactive power is that it bounces back and forth in the system from generator to load not doing any work. Current is a part of power. Extra, unnecessary current in the grid generates P = I^2*R losses.

Graph: Power in a Resistive, Capacitive and Inductive load generated by an in-phase, leading and lagging current respectiely.

Figure 15: Comparing power generated in a resistive, fully capacitive and fully inductive load.

The power factor reflects how much reactive power an inductive system generates. A power factor of 0.8 is usually the norm. There is a penalty for those companies who produce large reactive powers.

Capacitors can counter-balance inductive systems. Although for large inductive machines we’d need enormous capacitors that are not feasible.

One way to come across this is to operate an unloaded, over-excited synchronous motor as it has a capacitive effect.