Define voltage gain as a ratio, calculate it, and express it in decibels using the gain equation

What this dot point is asking

SEAB wants you to define the voltage gain of an amplifier as the ratio of output voltage to input voltage, to calculate it, and to express it in decibels. The central insight is that gain measures how many times bigger the output is than the input, and that because gains range over a huge span, they are often quoted on the logarithmic decibel scale.

The answer

What gain means

An amplifier produces an output signal that is a larger copy of its input signal. The voltage gain tells you how many times larger the output is than the input:

Gain is a pure number with no unit, because it is the ratio of two voltages. A gain of $200$ means the output voltage is $200$ times the input voltage.

Calculating gain

To find the gain, divide the output voltage by the input voltage, making sure both are in the same unit first. A small input of $20\ \text{mV}$ and an output of $4\ \text{V}$ give a gain of $4 / 0.020 = 200$. The same formula rearranges to find the output from the input and gain, $V_{out} = A_v V_{in}$, or the input from the output and gain.

Gain in decibels

Gains can be enormous, so they are often expressed on a logarithmic scale in decibels ($\text{dB}$). For a voltage gain:

The factor is $20$ for voltage (and current) ratios. Some useful values: a gain of $10$ is $20\ \text{dB}$, a gain of $100$ is $40\ \text{dB}$, and a gain of $1000$ is $60\ \text{dB}$. Each extra factor of ten in the ratio adds $20\ \text{dB}$.

Amplification and attenuation

If the output is larger than the input, the gain is greater than one and the decibel value is positive: this is amplification. If the output is smaller than the input, the gain is less than one and the decibel value is negative: this is attenuation. A gain of exactly one is $0\ \text{dB}$, meaning no change in size.

Examples in context

Example 1. A hearing aid. A hearing aid amplifies a faint sound into one loud enough to hear, with its amplification often quoted in decibels because the ear itself responds logarithmically to loudness. A gain of $40\ \text{dB}$ means the signal voltage is multiplied by $100$, turning a whisper into a clearly audible sound.

Example 2. An audio mixing desk. Each channel of a mixing desk has a gain control calibrated in decibels. Setting a channel to $+6\ \text{dB}$ roughly doubles its voltage, while $-6\ \text{dB}$ roughly halves it. The decibel scale lets engineers add and compare gains across many stages without juggling huge ratios.

Try this

• Cue. An amplifier has $V_{in} = 50\ \text{mV}$ and $V_{out} = 5.0\ \text{V}$. Find the gain. Convert and divide: $A_v = 5.0 / 0.050 = 100$.

• Cue. Express a voltage gain of $10$ in decibels. $A_{dB} = 20\log_{10}(10) = 20 \times 1 = 20\ \text{dB}$.

• Cue. State what a gain of $0\ \text{dB}$ means. The output equals the input ($A_v = 1$), so there is no amplification or attenuation; the signal size is unchanged.