# 7.10.1 Alternating Current & Voltage

### Sinusoidal & Root-Mean-Square Current & Voltage

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

A current which periodically varies between a positive to a negative value 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 relationship between time period T and frequency  f for a.c is:

• 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 a current-time or voltage-time graph
• The peak-to-peak current or voltage is the distance between a positive and consecutive negative peak. This means:

peak voltage V0 = peak-to-peak voltage ÷ 2

Graph of alternating current against time showing the time period, peak current and peak-to-peak current

#### 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 and the peak voltage.

#### Root-Mean-Square Current & Voltage

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

The square root of the mean of the squares of all the values of the current in one cycle

• Or:

The equivalent direct current that produces the same power

• The rms current Irms is defined by the equation:

• Where:
• I0 = peak current (A)
• The rms value of an alternating voltage is defined as:

The square root of the mean of the squares of all the values of the voltage in one cycle

• Or:

The equivalent dc voltage that produces the same power

• The rms voltage Vrms is defined by the equation:

• Where:
• V0 = peak voltage (V)

• Rms current is equal to 0.707 × I0, which is about 70% of the peak current I0
• This is also the case for rms voltage
• The rms value is therefore defined as:

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

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

Vrms and peak voltage. The rms voltage is about 70% of the peak voltage

• The average power of a supply is the product of the rms current and voltage:

Average power = Irms × Vrms

#### Worked Example

An electric oven is connected to a 230 V root mean square (rms) mains supply using a cable of negligible resistance.

Calculate the peak-to-peak voltage of the mains supply.

Step 1: Write down the Vrms equation

Step 2: Rearrange for the peak voltage, V0

V0 = √2 × Vrms

Step 3: Substitute in the values

V0 = √2 × 230

Step 4: Calculate the peak-to-peak voltage

• The peak-to-peak voltage is the peak voltage (V0) × 2

Peak-to-peak voltage = (√2 × 230) × 2 = 650.538 = 651 V (3 s.f)

#### Exam Tip

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

### Applications of Alternating Current & Voltage

• Mains electricity is supplied as alternating current by the National Grid
• This means that power stations produce alternating current
• This is the type of current supplied when devices are plugged into sockets
• In the UK, the mains electricity supplied to households is 230 V at 50 Hz. This is its rms value
• However, this varies depending on the customer (larger buildings and factories will require more)
• The mains voltage varies throughout the day depending on the demand and supply of electricity
• It is only lamps, heaters, cookers and devices with large electric motors (such as vacuum cleaners) that use a.c from the mains
• Most other devices such as televisions, computers and games consoles work with d.c
• This means they are built with a step-down transformer that converts the 230 V a.c into (for example) 12 V d.c
• The peak and peak-to-peak values for the current and voltage for mains electricity are used to calculate the rms value and vice versa

### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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