How to Calculate Gain in dB and Linear: Formulas and Worked Examples

To calculate gain, you divide the output signal magnitude by the input signal magnitude. For voltage gain in decibels (dB), the formula is G_dB = 20 × log₁₀(V_out / V_in). For power gain in decibels, the formula is G_dB = 10 × log₁₀(P_out / P_in). Linear gain is simply the raw ratio (Output / Input) without the logarithmic scaling.

Whether you are biasing a 2N3904 transistor, setting the feedback resistors on an OPA2134 op-amp, or measuring the output of a Mini-Circuits RF amplifier, mixing up the 10log and 20log multipliers or failing to track your milli-units will brick your design. Below is the complete mathematical framework, rearranged for reverse-engineering, followed by bench-tested worked examples.

The Core Gain Formulas and Symbol Definitions

Gain is fundamentally a dimensionless ratio when expressed in linear terms, but it takes on the decibel (dB) unit when expressed logarithmically. The distinction between voltage/current gain and power gain is critical: power is proportional to the square of voltage (P = V²/R). Because log(x²) = 2 × log(x), the voltage multiplier is 20, while the power multiplier is 10.

Table 1: Gain Formula Symbols and Definitions
Symbol	Definition	Standard Unit	Notes / Assumptions
A_v	Linear Voltage Gain	V/V (Dimensionless)	Can be negative for inverting topologies.
A_p	Linear Power Gain	W/W (Dimensionless)	Always a positive value ≥ 0.
G_dB	Gain in Decibels	dB	Negative dB indicates attenuation (loss).
V_out	Output Voltage	Volts (V)	Must be same domain (RMS, Peak, or DC) as V_in.
V_in	Input Voltage	Volts (V)	Must be same domain as V_out.
P_out	Output Power	Watts (W) or mW	Assumes matched load unless specified.
P_in	Input Power	Watts (W) or mW	Often expressed in dBm in RF systems.

The Primary Equations

Linear Voltage Gain: A_v = V_out / V_in
Linear Power Gain: A_p = P_out / P_in
Voltage Gain (dB): G_dB = 20 × log₁₀(V_out / V_in)
Power Gain (dB): G_dB = 10 × log₁₀(P_out / P_in)

Rearranged Forms for Reverse Engineering

On the bench, you rarely just calculate forward. Usually, you have a target output and a known input, and you need to find the required gain, or you have a datasheet gain spec and need to predict the output voltage. Here are the algebraic rearrangements solving for every variable.

Solving for Voltage Variables

Find V_out (from dB): V_out = V_in × 10^{(G_dB / 20)}
Find V_in (from dB): V_in = V_out / 10^{(G_dB / 20)}
Find V_out (from Linear): V_out = V_in × A_v
Find V_in (from Linear): V_in = V_out / A_v

Solving for Power Variables

Find P_out (from dB): P_out = P_in × 10^{(G_dB / 10)}
Find P_in (from dB): P_in = P_out / 10^{(G_dB / 10)}
Find P_out (from Linear): P_out = P_in × A_p
Find P_in (from Linear): P_in = P_out / A_p

Worked Examples with Strict Unit Tracking

The most common point of failure in gain calculations is unit mismatch. If V_in is in millivolts and V_out is in volts, dividing them directly yields a ratio that is off by a factor of 1,000, resulting in a 60 dB error. Always normalize to base SI units (Volts, Watts) before executing the logarithm.

Example 1: Audio Preamplifier Voltage Gain

Scenario: You are testing a non-inverting op-amp stage built with an NE5532. You inject a 15 mV_RMS sine wave at the input. Your oscilloscope reads 1.2 V_RMS at the output. Calculate the linear voltage gain (A_v) and the gain in decibels (G_dB).

Step 1: Normalize units to Volts.

V_in = 15 mV = 0.015 V
V_out = 1.2 V (already normalized)

Step 2: Calculate Linear Gain (A_v).

A_v = V_out / V_in
A_v = 1.2 V / 0.015 V
A_v = 80 V/V

Step 3: Calculate Gain in Decibels (G_dB).

G_dB = 20 × log₁₀(80)
G_dB = 20 × 1.90309
G_dB = 38.06 dB

Example 2: RF Low-Noise Amplifier (LNA) Power Gain

Scenario: You are cascading a Mini-Circuits MAR-6 RF amplifier. The datasheet specifies a typical gain of 18 dB. Your signal generator feeds an input power of -30 dBm into the 50-ohm system. Calculate the output power in dBm, and then convert it to milliwatts (mW). For a deeper look at RF power math, refer to the Mini-Circuits dBm application note.

Step 1: Calculate Output Power in dBm.

In the decibel domain, multiplication becomes addition. You simply add the gain (in dB) to the input power (in dBm).

P_out(dBm) = P_in(dBm) + G_dB
P_out(dBm) = -30 dBm + 18 dB
P_out(dBm) = -12 dBm

Step 2: Convert dBm to milliwatts.

The formula for dBm is: P_(mW) = 10^{(P_dBm / 10)}

P_(mW) = 10^{(-12 / 10)}
P_(mW) = 10^-1.2
P_(mW) = 0.063 mW (or 63 μW)

Assumptions, Unit Traps, and Realistic Magnitudes

Formulas assume ideal conditions. Real-world circuits introduce parasitic capacitance, clipping, and impedance mismatches that warp your theoretical numbers.

When the Formulas Apply (and When They Don't)

Linear Region Assumption: The formulas assume the active device is operating in its linear region. If your op-amp is powered by ±5V rails and your calculation predicts a 12V output, the op-amp will clip at ~3.5V (due to headroom limits). Your measured gain will artificially collapse.
Impedance Matching: The equivalence between voltage dB and power dB (where a 20dB voltage gain equals a 20dB power gain) only holds true if the input and output impedances are identical. In RF systems (50Ω in/out), this is standard. In audio (high-Z input, low-Z output), voltage gain and power gain in dB will be vastly different numbers.

Unit Mistakes That Break the Math

Warning: The RMS vs. Peak-to-Peak Trap
If your oscilloscope reads V_pp (peak-to-peak) but your multimeter reads V_RMS, your gain calculation will be wrong by a factor of 2.828 (which is 2√2). This introduces a 9 dB error. Always ensure both V_in and V_out are measured in the exact same domain (both RMS, both Peak, or both DC) before dividing.

Realistic Answer Magnitudes

If your calculation yields a number outside these typical ranges, double-check your decimal placement:

Op-Amp Open-Loop Gain: 100 dB to 130 dB (A_v = 100,000 to 3,000,000 V/V).
Audio Preamp (Closed-Loop): 20 dB to 40 dB (A_v = 10 to 100 V/V).
RF Low-Noise Amplifier (LNA): 15 dB to 25 dB.
Passive Attenuator (Loss): -3 dB to -40 dB (Negative gain).

For foundational theory on how these logarithmic scales map to human hearing and circuit behavior, the All About Circuits textbook chapter on Decibels provides excellent baseline context, while Texas Instruments' Op Amps for Everyone details how open-loop gain degrades at higher frequencies.

Frequently Asked Questions

How to calculate gain from a Bode plot?

A Bode plot displays gain (in dB) on the Y-axis and frequency (logarithmic Hz) on the X-axis. To calculate the linear gain at a specific frequency, locate the frequency on the X-axis, trace up to the magnitude curve, and read the dB value on the Y-axis. Convert that dB value back to linear using the formula A_v = 10^{(G_dB / 20)}. For example, if the plot shows 40 dB at 1 kHz, the linear voltage gain at 1 kHz is exactly 100 V/V.

How to calculate gain in dB when voltage is negative?

The logarithm of a negative number is undefined in real math. When dealing with inverting amplifiers (where V_out is 180 degrees out of phase with V_in, resulting in a negative linear gain), you must take the absolute value of the ratio before applying the log. The formula becomes G_dB = 20 × log₁₀(|V_out / V_in|). The inversion is a phase shift, not an amplitude reduction; a gain of -10 V/V is still +20 dB of magnitude gain.

How to calculate power gain if input and output impedances are different?

If Z_in and Z_out are not equal (common in audio and sensor interfaces), you cannot simply use the voltage ratio to find power gain. You must calculate the actual power at both nodes using P = V² / R. First, find P_in = (V_in)² / Z_in. Next, find P_out = (V_out)² / Z_out. Finally, apply the power gain formula: G_dB = 10 × log₁₀(P_out / P_in). This will yield a drastically different number than the voltage gain in dB.

How to calculate total gain of cascaded amplifier stages?

When amplifiers are wired in series (cascaded), the math is highly dependent on your format. If you are using linear gain, you multiply the individual stage gains together (A_total = A₁ × A₂ × A₃). If you are using decibels, you simply add the gains of each stage (G_total = G₁ + G₂ + G₃). This additive property is the primary reason RF and audio engineers prefer the dB scale for system-level block diagrams.