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#### Alternating Currents

###### Sinusoidal Current • 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:
1. An alternating voltage is represented by the equation: V = 220sin(120πt)
For this alternating voltage, determine

1. peak voltage
• Simply using the equation, the peak voltage, V = 220 V
2. the r.m.s voltage
• VrmsV0/√2220/√2
• Vrms = 156 v
3. the frequency
• The quantity is sin() is equal to ωt:
• ∴ ω = 120π
• Also, ω = 2πf, so:
• f = 120π/
• ∴ 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.
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