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A Semester of First Year Physics with Peter Eyland
Lecture 12 (Complex e.m.f.s)
In this lecture the following are introduced:
• Generating alternating current
• Adding two alternating potential differences
• Complex e.m.f.s
A major reason for using alternating current in power networks is that its time variation allows the use of transformers. These are devices that convert electrical power at low potential and high current to high potential (typically 320 kV) and low current to make longdistance electrical power transmission practicable. At the end of the transmission transformers also change the high potential down to low potential suitable for the consumer (typically 240 Volts).
Generating Alternating Current
The input mechanical work of rotating a coil in a magnetic field produces output electrical work. A plane coil of N turns is shown rotating in a magnetic induction, B.
The magnetic flux through the crosssectional area of a single closed conducting loop has the unit Tesla.m^{2}, and is (from Lecture 10):
The area presented to the magnetic induction (and so the flux) will depend on cosθ, where θ is the angle between the magnetic induction and the area vector. (The area vector is perpendicular to the plane of the area). 
The magnetic flux through a coil of N turns is
.
A coil rotating with constant speed in a uniform magnetic induction will present an area that depends on the angle θ, but now the angle is increasing linearly with time. The angle θ = ωt where ω depends on the speed of rotation and so the area (and the flux) will change as cosωt. An initial phase angle can be added which is determined by the starting conditions, i.e. the flux will change as cos (ωt + α ).
Faraday's law can now be written:
This e.m.f. alternates sinusoidally in potential difference, with This is shown graphically on the right. 
The alternating e.m.f. induces an alternating current in the external circuit.
Example
A coil of diameter 60 mm, having 150 turns, rotates at 500 revolutions per minute in a uniform magnetic field
of 0.7 T. Find the peak e.m.f.
A more practical alternating current generator is shown in cross section below.

The central shaft of the rotor is rotated by steam or hydroelectric turbines etc. 
Adding two Alternating Potential Differences
The A.C. output of a single generator may not be enough in some circumstances. A higher output e.m.f. can be obtained by connecting two A.C. generators in series. If both the A.C. generators output a sinusoidal potential, then each are specified by their amplitude; initial phase; and frequency. Also, if they have the same angular frequency then the output potential is sinusoidal; with a resultant amplitude and a resultant initial phase.
The emfs expressed mathematically: 
Graphically expressed: 
Adding gives: 
where the components of the amplitude E_{R} are and 
From the components by dividing: 
and by squaring: 
In other words, two sinusoidal emfs of the same frequency add together to form another sinusoidal emf with the same frequency, but different initial phase and amplitude. The resultant initial phase and amplitude can be found from the initial phases and amplitudes of the component emfs.
The results (above) are mathematically the same as if adding two vectors (shown below).
The tangent relation is seen from the vertical and horizontal vector components.
The amplitude relation is seen from the cosine rule applied to the vector triangle.
However the A.C. quantities have phase angles (which say how far through the cycle they start) and not real angles in space. Because of this, the trigonometric diagrams that are used to add A.C. quantities together are called phasor diagrams. Using phasor diagrams becomes complicated when the number of elements and branches increase, so the use of complex numbers will simplify things :) because they contain both amplitude and phase information.
Complex e.m.f.s
A complex A.C. e.m.f. is pictured as a phasor rotating in the complex plane. 
, is the complex e.m.f. and the physical e.m.f. ( e ), output from an A.C. generator is represented by the imaginary part of the complex e.m.f.. 
Example
Two e.m.f.s are connected in series with each other.
They have the same frequency (60 Hz) and the same maximum potential difference (150V),
but the second one is 60^{0} ahead in phase. Find
(a) the resultant e.m.f., and
(b) the phase angle between the resultant and each of the two e.m.f.s, and
(c) The values of each of the two e.m.f.s when the resultant is zero.
In what follows, e^{jθ}
is written as exp[jθ], so that the exponent can be more easily seen on the screen.
The first step below is to put the initial phase of the second e.m.f into Cartesian form ( cosθ + j·sinθ ).
After addition, the Cartesian result is transformed into Polar form ( E·exp[jωt + α] )
to get the initial phase of the result.
The resultant is 30^{0} ahead of the first e.m.f. and 30^{0} behind the second.
When e_{R} = 0:
Any value of n will do, because each represents the same part of the cycle.
Summarising:
The mechanical work of rotating a coil in a magnetic field produces AC.
Two sinusoidal emfs of the same frequency add together to form another sinusoidal emf
with the same frequency, but different initial phase and amplitude.
The resultant initial phase and amplitude can be found from the initial phases and amplitudes of the component emfs, i.e.
where: and
A complex e.m.f. contains both amplitude and phase terms. The physical e.m.f. output from an A.C. generator is represented by the imaginary part of the complex e.m.f..
Acknowledgement: These notes are based in part on "Alternating Current Circuit Theory" by G.J.Russell and K.Mann NSWUP 1969.
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