Chapter 10 Alternating Currents

Question 1

An alternating current (a.c) is defined as

Answer 1

• A current which periodically varies from positive to negative and changes its magnitude continuously with time
• This means the direction of an alternating current varies every half cycle
• The variation of current, or p.d., with time can be described as a sine curve ie. sinusoidal
• Therefore, the electrons in a wire carrying a.c. move back and forth with simple harmonic motion

Question 2

As with SHM, the relationship between time period T and frequency f of an alternating current is given by:

Answer 2

Question 3

Peak current (I0), or peak voltage (V0), is defined as:

Answer 3

The maximum value of the alternating current or voltage

Question 4

Peak current, or voltage, can be determined from the amplitude of the graph

Answer 4

Question 5

Mains electricity is supplied as alternating current

Answer 5

• Power stations produce alternating current
• This is the type of current supplied when devices are plugged into sockets

Question 6

The equation representing alternating current which gives the value of the current I at any time t is:

Answer 6

• I = I0 sin(⍵t)
• Where:
  • I = current (A)
  • I0 = peak current (A)
  • ⍵ = angular frequency of the supply (rad s-1)
  • t = time (s)
  • Note: this a sine function since the alternative current graph is sinusoidal

Question 7

A similar equation can be used for representing alternating voltage:

Answer 7

V = V0 sin(⍵t)

Where:
V = voltage (V)
V0 = peak voltage (V)
Recall the relation the equation for angular frequency ⍵:

Question 8

Root-Mean-Square Current

Answer 8

• Root-mean-square (r.m.s) values of current, or voltage, are a useful way of comparing a.c current, or voltage, to its equivalent direct current, or voltage
• The r.m.s values represent the d.c current, or voltage, values that will produce the same heating effect, or power dissipation, as the alternating current, or voltage

Question 9

• The r.m.s value of an alternating current is defined as:

Answer 9

The value of a constant current that produces the same power in a resistor as the alternating current

• The r.m.s current I_r.m.s is defined by the equation:
• So, r.m.s current is equal to 0.707 × I₀, which is about 70% of the peak current I₀
• The r.m.s value of an alternating voltage is defined as:

The value of a constant voltage that produces the same power in a resistor as the alternating voltage

Question 10

• The r.m.s voltage V_r.m.s is defined by the equation:

Answer 10

• Where:
  
  • I₀ = peak current (A)
  • V₀ = peak voltage (V)

Question 11

• The r.m.s value is therefore defined as:

Answer 11

The steady direct current, or voltage, that delivers the same average power in a resistor as the alternating current, or voltage

Question 12

• A resistive load is any electrical component with resistance eg. a lamp

Answer 12

V_r.m.s and peak voltage. The r.m.s voltage is about 70% of the peak voltage

Question 13

Mean Power

Answer 13

• In mains electricity, current and voltage are varying all the time
• This also means the power varies constantly, recall the equations for power:
• Where:
  
  • I = direct current (A)
  • V = direct voltage (A)
  • R = resistance (Ω)

Question 14

The r.m.s values means equations used for direct current and voltage can now be applied to

Answer 14

• alternating current and voltage
• These are also used to determine an average current or voltage for alternating supplies
• Recall the equation for peak current:

l₀= √2 I_r.m.s

• Therefore, the peak (maximum) power is related to the mean (average) power by:

P_mean= I_r.m.sR

P_mean= P/2

Question 15

The mean power in a resistive load

Answer 15

is half the maximum power for a sinusoidal alternating current or voltage

Question 16

• Rectification is defined as:

Answer 16

The process of converting alternating current and voltage into direct current and voltage

Question 17

Rectification is used in electronic equipment which requires a

Answer 17

direct current

• For example, mains voltage must be rectified from the alternating voltage produced at power stations

Question 18

There are two types of rectification

Answer 18

• Half-wave rectification
• Full-wave rectification

Question 19

For half-wave rectification

Answer 19

• The graph of the output voltage V_out against time is a sine curve with the positive cycles and a flat line (V_out = 0) on the negative cycle
• This is because the diode only conducts in the positive direction

Question 20

For full-wave rectification:

Answer 20

• The graph of the output voltage V_out against time is a sine curve where the positive cycles and the negative cycles are both curved ‘bumps’

Question 21

The difference between the graphs of full-wave and half-wave rectification

Answer 21

Question 22

Half-wave rectification consists of a

Answer 22

• single diode
  
  • An alternating input voltage is connected to a circuit with a load resistor and diode in series
• The diode will only conduct during the positive cycles of the input alternating voltage,
  
  • Hence there is only current in the load resistor during these positive cycles
• The output voltage V_out across the resistor will fluctuate against against time in the same way as the input alternating voltage except there are no negative cycles
• This type of rectification means half of the time the voltage is zero
• So, the power available from a half-wave rectified supply is reduced

Question 23

Full-wave rectification requires a

Answer 23

• bridge rectifier circuit
  
  • This consists of four diodes connected across an input alternating voltage supply
• The output voltage V_out is taken across a load resistor
• During the positive cycles of the input voltage, one terminal if the voltage supply is positive and the other negative
  
  • Two diodes opposite each other that are in forward bias will conduct
  • The other two in reverse bias will not conduct
  • A current will flow in the load resistor with the positive terminal at the top of the resistor
• During the negative cycles of the input voltage, the positive and negative terminals of the input alternating voltage supply will swap
  
  • The two diodes that were forward bias will now be in reverse bias and not conduct
  • The other two in reverse bias will now be in forward bias and will conduct
  • The current in the load resistor will still flow in the same direction as before

Question 24

In full-wave both the positive and negative cycles s, the current in the load resistor is the…

Answer 24

• same
• Each diode pair is the same as in half-wave rectification
  
  • Since there are two pairs, this equates to full-wave rectification overall
• The main advantage of full-wave rectification compared to half-wave rectification is that there is more power available
  
  • Therefore, a greater power is supplied on every half cycle

Question 25

When A is positive and B is negative, diodes 2 and 3 will conduct and 1 and 4 will not. When A is negative and B is positive, diodes 1 and 4 will conduct and diodes 2 and 3 will not. The current in the load resistor R will flow downwards

Answer 25

Question 26

Smoothing

Answer 26

• In rectification, to produce a steady direct current or voltage from an alternating current or voltage, a smoothing capacitor is necessary
• Smoothing is defined as:

The reduction in the variation of the output voltage or current

Question 27

Smoothing works in the following ways:

Answer 27

• A single capacitor with capacitance C is connected in parallel with a load resistor of resistance R
• The capacitor charges up from the input voltage and maintains the voltage at a high level
• As it discharges gradually through the resistor when the rectified voltage drops but the voltage then rises again and the capacitor charges up again
• The resulting graph of a smoothed output voltage V_out and output current against time is a ‘ripple’ shape

Question 28

A smoothing capacitor connected in parallel with the load resistor. The capacitor charges as the output voltage increases and discharges as it decreases

Answer 28

Question 29

A smooth, rectified current graph creates a ‘rippling’ shape against time

Answer 29

Question 30

The amount of smoothing is controlled by the

Answer 30

• capacitance C of the capacitor and the resistance R of the load resistor
  
  • The less the rippling effect, the smoother the rectified current and voltage output
• The slower the capacitor discharges, the more the smoothing that occurs ie. smaller ripples
• This can be achieved by using:
  
  • A capacitor with greater capacitance C
  • A resistance with larger resistor R
• Recall that the product RC is the time constant τ of a resistor
• This means that the time constant of the capacitor must be greater than the time interval between the adjacent peaks of the output signal