Alternating Current & Voltage (Cambridge International A-Level Physics)

Properties of Alternating Current & Voltage

• An alternating current (a.c) is defined as: 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

• As with SHM, the time period of an alternating current is defined as:

$T=\frac{1}{f}$

• Where:

  • T = time period (s)
  • f = frequency (Hz)

• Peak current (I₀), or peak voltage (V₀), is defined as: The maximum value of the alternating current or voltage

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

Time period and peak current on a current-time graph

Graph of alternating current against time with a time period of 20 ms and peak current of 2 A

• Mains electricity is supplied as alternating current

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

Worked Example The variation with time t of the output voltage V of an alternating voltage supply is shown in the graph below.

Use the graph to calculate the frequency of the supply.

Answer:

Step 1: Write down the period-frequency relation

$f=\frac{1}{T}$

Step 2: Calculate the time period from the graph

• The time period is the time taken for one complete cycle
• From the graph, this is equal to 0.2 ms
• Therefore, the time period is

T = 0.2 ms = 0.2 × 10^-3 s

Step 3: Substitute into frequency equation

$f=\frac{1}{0.2\times10^{-3}}=5000\mathrm{Hz}=5\mathrm{kHz}$

Exam Hint Remember to double check the units on the alternating current and voltage graphs. These are often shown in the range of milli-seconds (ms) instead of seconds (s) on the x axis.

Using Sinusoidal Representations

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

$I = I_{0} \sin ( \\omega t )$

• Where:

  • I = current (A)
  • I₀ = peak current (A)
  • ω = angular frequency of the supply (rad s^-1)
  • t = time (s)

• Note: this a sine function since the alternative current graph is sinusoidal

• A similar equation can be used for representing alternating voltage:

$V = V_{0} \sin ( \\omega t )$

• Where:

  • V = voltage (V)
  • V₀ = peak voltage (V)

• Recall the relation between the equation for angular frequency ω:

$ω=\frac{2π}{T}=2πf$

• Where:

  • f = frequency of the supply (Hz)

Worked Example An alternating current I varies with time t as shown in the graph below.

Using the graph and the equation for alternating current, calculate the value of the current at a time 0.48 s.

Answer:

Step 1: Write out the equation for alternating current

$I = I_{0} \sin ( \\omega t )$

Step 2: Write out the equation for angular frequency

$ω=\frac{2π}{T}$

Step 3: Measure the time period T and peak current I ₀ from the graph **

• The time period is the time taken for one full cycle, T = 0.10 s
• Peak current (amplitude),I₀ = 17 A

Step 4: Substitute values into alternating current equation at time t**

• Using the time given in the question, t = 0.48 s

$I = I_{0} \sin ( \\omega t ) = I_{0} \sin ( \frac{2 \pi t}{T} )$

$I = 17 \sin ( \frac{2 \pi ( 0.48 )}{0.1} ) = - 16.168 = - 16 \mathrm{A} ( 2 \mathrm{s} . \mathrm{f} )$

Exam Hint These equations are written as $x = x_{0} \sin ( \\omega t )$, the x is replaced with I or V, depending on the question.

Remember to check that your calculator is in radians mode when using any of these equations. This is because the angular frequency ω is measured in rad s^-1.

Current can also be positive or negative, depending on its direction. This is why the answer in the worked example is negative. At t = 0.48 s the current will be in the negative direction. The mark schemes in the exams are normally not too picky about this, but the size of the current (16 A in the worked example) is what will get the mark.