In order to describe how the transformer works, it is necessary to review briefly some simple elements of electricity.
1. Electromagnetic Fields.
       A current of electricity flowing through a wire produces not only heat, but also a magnetic field about the wire. This may be proved by placing a compass in the vicinity of the current-carrying conductor Fig. (1).  
Fig. (1) Magnetic field around a conductor.

       The magnetic field around a single wire carrying a current may be rather weak. By winding the wire into a ring, the magnetic lines of force (sometimes called magnetic flux), are concentrated in the small space inside the coil and the magnetic effect is much increased. (Fig. 2) The grouping of the lines of force is known as a magnetic field.
(Fig. 2) Magnetic field about a wire loop.

       A coil of wire is nothing but a succession of these rings stacked one after the other (Fig. 3-A). Each adds its quota to the magnetic field. Most of the magnetic lines of force pass straight through the coil. Each line makes a complete circuit, returning by a path outside the coil. A coil carrying a current is in fact a magnet. Where the lines come out is referred to (for identification) as the north (N) pole, where they enter as the south (S) pole (Fig. 3-B).
Fig. (3)  Magnetic field about a coil.

       The strength of the magnetic field inside a coil depends on the strength of the current flowing and the number of turns. Its strength is, therefore, expressed in ampere-turns, that is, amperes multiplied by the numbers of turns. Thus, a single turn carrying a very large current may produce the same effect as a great many turns carrying a small current.
       A coil with an air core, however, produces a comparatively weak field. The strength is enormously increased by putting in a core of iron. (Fig. 4.) This is generally referred to as an electromagnet.
Fig. (4) Coil with iron core.

2. Electromagnetic Induction..
       One relation between electricity and magnetism has already been demonstrated, that of producing magnetism with the aid of electric current. There is another important relation and that is the production of electricity with the aid of magnetism.
       When a conductor is moved through a magnetic field an electrical pressureor voltageis produced in the conductor. If the strength of the magnetic field is increased (say, by replacing a simple magnet with a more powerful electromagnet), it will be found that a greater electrical pressure is now induced in the conductor cutting the lines of force. Similarly, the greater the length of the conductor, the more voltage is produced because more lines of force are cut. Also, if the speed at which the conductor is moved through the magnetic field is increased, it is found that the voltage induced in the conductor is also increased. The magnitude of the electrical pressure induced in a conductor while it is moving through a magnetic field, therefore, is determined by the rate of cutting of the lines of force of the magnetic field.
       If a conductor which is part of a closed circuit is moved through a magnetic field, an electric current will flow in the conductor. The current results from what is called the induced electromotive force (EMF). But this happens only if the directions (1) of the current, (2) of the magnetic field, and (3) of the motion are at right angles to each other.
       If a loop of wire (as illustrated in Fig. 5) is rotated at uniform speed in a magnetic field, a voltage is induced in the conductor that makes up the loop. If the voltage for different positions of the loop is measured and the angular position of the loop, (in Fig. 1-7) the waveform of the induced voltage can be obtained. This waveform is called a sine wave. Note that the instantaneous values of the voltage changes continuously from zero to maximum, from maximum back to zero, from zero to a maximum in the opposite direction, and then back to zero, thus completing one cycle. The maximums, in either direction of the voltage wave are called the amplitude of the voltage, and the number of cycles per second the frequency of the wave. The part of the curve from zero-to-maximum-back-to-zero is called an alternation. Such a waveform is an example of the form of voltage and current in the alternating current circuit.
Fig. (5) Sine-wave voltage induced in coil as it rotates in magnetic field.

       In a 60-cycle system, each cycle or two alternations takes place in 1/60 second.
       An electrical pressure can be induced in a conductor by moving it through a magnetic field as described above. A voltage can also be induced electromagnetically by moving a magnetic field across the conductor. (Fig. 6.).
Fig. (6)    A basic alternator

       It makes no difference whether the conductor is moved across the magnetic field or if the magnetic field is moved across the conductor. A stationary conductor which has a magnetic field sweeping across it is cutting a magnetic field just the same as though the conductor was moving across the magnetic field.

2.1 Inductance.
       A conductor carrying alternating current has a magnetic field around it which alternates its characteristics in accordance with the alternations of the current flowing through the conductor. Following the sine wave characteristic of the alternating current, the magnetic field produced is zero at the start, builds up to a maximum in one direction at the first-quarter cycle, as shown in Fig. 7A, B, C, and D  respectively, and reduces to zero at the second-quarter cycle, as shown in Fig. 7D, C, B, and A respectively, builds up again to a maximum in the third-quarter cycle but in a direction opposite to the previous maximum, and reduces to zero again to complete the cycle.

2.2 Self Inductance 
       The expansion and contraction of the magnetic field constitutes a moving magnetic field which cuts the conductor carrying the current producing it. Such action will produce a voltage in the conductor distinct from, and tending to oppose that causing the original current to flow. This second voltage also produces a current, which in turn affects the original current, and which in turn affects the magnetic field around the conductor, thus affecting the whole set up; this finally stabilizes at some point.
Fig. (7) Expansion and contraction of a magnetic field about a conductor.

       If the two voltages in the conductor are now combined (as actually they cannot exist separately), the resulting voltage will be different in magnitude and the voltage wave will be displaced somewhat when referred to the wave representing the current flow, or, put another way, it would appear that the current wave now lags the voltage wave (Fig.  8). This action prevents the current and voltage (which produce power) from acting together for each point of the cycle, with the net effect being a reduced production of power. The net power produced compared to the 100 percent produced if both current and voltage acted together fully is known as power factor. Further, the action can be looked at as something which prevents the free flow of electricity in the circuit, somewhat like the resistance produced by the friction of the water flow in a pipe. This magnetic reaction in a conductor is called self-inductance since it is caused by the magnetic field about itself.
Fig. (8)   Effect of inductance on voltage and current in a conductor (not drawn to scale)

2.3 Mutual Inductance
       This same reaction may be caused by the magnetic fields of adjacent conductors, in which case it is called mutual inductance, since both conductors affect each other (Fig. 9).
Fig. (9)  The effect of the magnetic field about a conductor on an adjacent conductor.

2.4 Inductive Reactance
       The obstructive effect of these phenomena of induction within a conductor and between conductors carrying alternating current is called inductive reactance and, like resistance in a circuit, is measured in ohms.     
2.5 Capacitance-Capacitive Reactance.
       When two conductors are near each other, there is also another reaction between them, but it is not due to the magnetic field. As an alternating current flows through a conductor, it will cause that conductor to have a scarcity of electrons during the first half cycle and an excess during the second half cycle. In attempting to restore a balance, it will therefore tend to attract and repel electrons from the adjacent conductor. This to-and-fro motion of electrons sets up an alternating current in the second conductor. If both conductors are carrying alternating currents, they will react upon each other.
       The amount of this reaction, called capacitance will depend on the area of the conductors exposed to each other, the distance between them, the kind of insulating material between them, and the voltage (and current) in the conductors. Like the inductance in a circuit, the current set up in a conductor because of this capacitive effect will tend to cause the resultant current to be displaced from the voltage wave (Fig. 1-12). The obstructive effect of this capacitance phenomenon between conductors is called capacitive reactance, and is also measured in ohms. 
Fig. (10)  Effect of capacitance on voltage and current in a conductor (not drawn to scale).

2.6 Impedance 
       The total obstruction to the flow of current in an ac circuit may, therefore, be caused by resistance, inductance, and capacitance or more generally, resistance and reactance. This total obstruction, which impedes the flow of electricity, is called impedance (usually denoted by the letter Z). In ac circuits, then, Ohm's Law becomes
       where I is the current, E the voltage, and Z the impedance.
       Resistance and reactance are not added directly to arrive at the impedance of the circuit. Rather, their relationship may be compared to a boat traveling in a body of water affected by both current and wind (Fig. 11); it is pushed by the current (resistance) of the water, and by the wind (reactance). If the wind in one direction is at right angles to the boat, it may be referred to as inductive reactance; if in the other direction, it may be referred to as capacitive reactance; the net effect of the two is the overall reactance. The relative strengths of these three components will determine the direction the boat would take; similarly, the relative position of the current wave with respect to the voltage wave will be determined by the relative values of the three components of impedance: resistance, inductance, and capacitance of the circuit. It is obvious that if the inductance and capacitance values are equal, then the net reactance is equal to zero and the only quantity left is resistance. When this condition occurs, the circuit is said to be in resonance.
Fig. (11) Water analogy of resistance and reactance relationship to

3.  Generation of Voltage in a Transformer Coil.
       It has been demonstrated how an electrical pressure or voltage may be generated in a conductor adjacent to another one carrying an alternating current. This process is called induction, and, in the conductors it acted to obstruct the normal flow of current. This same effect of induction, however, can prove useful in another way.  
       Instead of two wires adjacent to each other, assume there are two coils of wire adjacent to each other and an alternating current of electricity flows in one of them. As was shown previously, the voltage induced in the second coil will depend on the length of the conductor, the relative speed between conductor and the magnetic field, and the strength of the magnetic field. The entire magnetic field set up by the first coil can be assumed to cut the turns of the second coil if both coils are wound on an iron core (Fig. 12). The relative speed between the conductor and magnetic field is fixed, being dependent on the frequency of the alternating current flowing through the first coil. The voltage in the second coil will therefore depend on the length of the conductor, or on the number of turns. If the magnetic field and the rate of     cutting are the same for both coils, then each turn of each coil will have the same voltage produced in it. Therefore, to obtain the desired voltage in the second coil, the volts per turn are determined from the first coil. If a voltage of 1000 volts is applied to a coil of 1000 turns, then 1000 volts will be generated in the entire coil, each turn generating 1 volt. Now, if only a voltage of 100 volts is desired in the second coil, only 100 turns will be required as each turn generates 1 volt.
Fig. (12) Diagram of transformer having two windings insulated from each other and wound on a common iron core.
       It must be remembered that the voltage in the second coil is an induced voltage and will therefore be displaced from the voltage in the first coil by a half cycle (Fig. 8). The currents will also be displaced by a half cycle as shown in Fig. 12.

Fig. (130  Current in secondary of transformer displaced from current in primary of transformer by one-half cycle.

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