Period, T is the time for one complete cycle of the alternating current (a.c.)
Frequency, f is the number of oscillations per unit time; f = 1/T
Peak value, I0/V0 is the highest point on the graph
Instantaneous current/voltage, I/V is the current/voltage at a particular instant
I = I0sinωt
V = V0sinωt
where ω = 2πf
The root-mean-squared (r.m.s) value, Irms/Vrms is the value of steady current/voltage that produces the same power in a resistor as the alternating current/voltage
Irms = I0/√2
Vrms = V0/√2
Mean Power in an a.c. Supply
For a sinusoidal alternating current, the peak power is twice the average power
P = IV and using Irms and Vrms
P = I0/√2 × V0/√2 = ½IV
For example:
An alternating voltage is represented by the equation: V = 220sin(120πt)
For this alternating voltage, determine
peak voltage
Simply using the equation, the peak voltage, V = 220 V
the r.m.s voltage
Vrms = V0/√2 = 220/√2
Vrms = 156 v
the frequency
The quantity is sin() is equal to ωt:
∴ ω = 120π
Also, ω = 2πf, so:
f = 120π/2π
∴ f = 60 Hz
Transformer
A Transformer is a device used to increase or decrease the current or voltage of an alternating current
An Ideal Transformer has no power loss in the transformer
Input power = Output power
The p.d VP across the primary coil causes an alternating current IP to flow, producing a magnetic field in the soft iron core
The secondary coil is thus in a changing magnetic field, and an alternating current IS is induced in it, producing an alternating e.m.f. VS across the secondary coil
Step-up Transformer: The primary coil has fewer turns than the secondary coil, hence the output voltage is greater (current decreases by the same factor)
Step-down Transformer: The primary coil has greater turns than the secondary coil, hence the output voltage is lower (current increases by the same factor)
Transformer Relationships:
IPVP = ISVS
NS/NP = VS/VP = IP/IS
or simply use ratios
Phase Difference in VP/VS and IP/IS/Φ
The alternating current in the primary coil is not in phase with the alternating e.m.f. induced in the secondary coil:
Current in the primary coil gives rise to the magnetic field
The magnetic field in the core is in phase with the current in the primary coil
The magnetic flux cuts the secondary coil inducing the e.m.f. in the secondary coil
The e.m.f. induced is proportional to the rate of change of field, so not in phase
VP and VS have a phase difference of 90° with IS, IP and Φ
Eddy Currents
If a metallic conductor moves in a magnetic field, an e.m.f. is induced which will make free electrons in the metal move, causing electric current — eddy currents
The eddy currents will oppose change in flux linkage of the conductor by Lenz’s law and energy of motion will be dissipated as heat
Energy Loss in a Practical Transformer
Some power is lost due to resistance in the coils of transformers causing them to heat up
Some power is lost as the magnetic flux flows back and forth. To minimize this, a soft magnetic material is used where magnetic flux direction can change easily
Losses also occur in the core due to eddy currents: induced currents flows through the iron core and dissipate energy due to its resistance. Currents can be reduced y making the core out of thin laminated sheets; flux can easily flow but eddy currents cannot
Transmission of Electrical Energy
Electricity transmission lines have resistance, therefore, energy will be lost through heating in the wires
Electricity transmitted at high voltage a.c. supply:
High Voltage: For same power, current is smaller, so less heating and voltage loss in cables/wires
A.c. Supply: This can change the output voltage efficiently using transformers
Half-wave Rectification
For one half of the time, the voltage is 0, this means that the power available from a half-wave rectified supply is reduced
Full-wave Rectification
The four diodes are known as a bridge diode
When current is flowing for the first half of period
When current is flowing for the second half of period
Smoothing
In order to produce steady d.c. from ‘bumpy’ d.c. that results from rectification, a smoothing capacitor is required
The capacitor charges and maintains the voltage as a.c. voltage rises, (first half of the wave)
As the wave slopes downward, the capacitor begins to discharge in order to maintain the voltage
A small capacitor discharges more rapidly than a large capacitor and gives rise to a greater ripple in the output
If the load resistor is small, the capacitor will also discharge rapidly
CR is the time constant of a capacitor resistor: It is the time taken for a charge to fall 1/e times the original value
The value should be much greater than the time period of a.c. supply so the capacitor does not have sufficient time to discharge significantly
In general, the greater the value R × C, the smoother the rectified a.c.