The Core Voltage Gain Equation and Symbol Definitions

Voltage gain ($A_v$) quantifies the ratio of an amplifier's output voltage to its input voltage. In linear terms, it is a dimensionless ratio. In logarithmic terms, it is expressed in decibels (dB), which compresses massive dynamic ranges into manageable numbers for RF and audio engineering.

Linear Gain:
A_v = V_out / V_in

Logarithmic Gain (Decibels):
A_v(dB) = 20 × log_10(V_out / V_in)

Table 1: Symbol Definitions and Typical Ranges
Symbol Definition Standard Unit Typical Magnitude
A_v Linear voltage gain Dimensionless (V/V) 0.1 to 100,000
A_v(dB) Logarithmic voltage gain Decibels (dB) -20 dB to +120 dB
V_out Output voltage Volts (V), mV, or μV Depends on load and supply rails
V_in Input voltage Volts (V), mV, or μV Typically mV for small-signal

Assumptions and Application Limits

The voltage gain equation is not a universal law; it relies on specific circuit conditions to yield valid data:

  • Linear Operation: The amplifier must not be clipping. V_out must remain strictly within the supply rails (e.g., ±15V for a standard op-amp). If your math predicts a 20V output from a 12V single-supply rail, the equation has failed because the physical circuit is saturating.
  • Impedance Context: The basic equation assumes we are measuring open-circuit voltage or that load effects are already factored into V_out. In low-frequency audio, this is usually fine. In 50Ω RF systems, load mismatch will skew your expected voltage.
  • Frequency Range: The formula applies strictly within the amplifier's mid-band bandwidth. At the -3dB cutoff frequencies, the actual gain drops by 30% due to parasitic capacitance.
  • AC vs DC Measurement: For AC signals, V_out and V_in must be measured in the exact same domain (both RMS, both peak, or both peak-to-peak). When measuring on an oscilloscope, ensure your scope is set to AC coupling if you are analyzing small-signal gain riding on a DC bias.

Rearranged Forms of the Voltage Gain Equation

When debugging a circuit on the bench or specifying a component from a datasheet, you rarely solve for A_v directly. Here are the algebraic rearrangements you will actually use:

  • Solve for Output Voltage (V_out):
    V_out = A_v × V_in
    V_out = V_in × 10^(A_v(dB) / 20)
  • Solve for Input Voltage (V_in):
    V_in = V_out / A_v
    V_in = V_out / 10^(A_v(dB) / 20)
  • Solve for Linear Gain (A_v) from dB:
    A_v = 10^(A_v(dB) / 20)

Worked Examples with Unit Tracking

Problem 1: Converting Linear Gain to Decibels for an Audio Preamp

Scenario: You are testing a microphone preamp built with an NE5532 op-amp. Using a signal generator, you inject a 2 mV_RMS test tone. You measure the output with a Keysight 34461A digital multimeter and read 1.5 V_RMS. Find the linear gain and the gain in dB.

  1. Align units:
    V_in = 2 mV = 0.002 V
    V_out = 1.5 V
  2. Calculate linear gain (A_v):
    A_v = 1.5 V / 0.002 V = 750 V/V
  3. Calculate logarithmic gain (A_v(dB)):
    A_v(dB) = 20 × log_10(750)
    A_v(dB) = 20 × 2.875 = 57.5 dB

Problem 2: Finding Output Voltage from a Datasheet dB Specification

Scenario: An RF amplifier module (e.g., Mini-Circuits ZX60-33LN+) specifies a typical gain of 14 dB at 1 GHz. If your input signal from a software-defined radio is -20 dBm (which translates to 22.4 mV_RMS into a 50Ω load), what is the output voltage?

  1. Convert dB gain to linear gain (A_v):
    A_v = 10^(14 / 20) = 10^0.7 ≈ 5.01 V/V
  2. Apply the linear gain to the input voltage:
    V_out = A_v × V_in
    V_out = 5.01 V/V × 22.4 mV
    V_out = 112.2 mV_RMS

Bench Note: In RF, we often just add dB values: -20 dBm + 14 dB = -6 dBm. Converting -6 dBm into a 50Ω system yields 112.2 mV_RMS, confirming the voltage-domain math. For a deeper dive into RF power math, consult the Microwaves10 Decibel Encyclopedia.

Realistic Magnitudes and Unit Mistakes That Break the Math

What a Realistic Answer Looks Like

If your calculator spits out a number, you need a sanity check. Here is what realistic voltage gain looks like across standard topologies, as detailed in the TI Analog Engineer's Pocket Reference:

Table 2: Expected Gain by Amplifier Topology
Amplifier Topology Typical Linear Gain (A_v) Typical dB Gain Phase Shift
Common Collector (Emitter Follower) ≈ 1 V/V ≈ 0 dB 0°
Common Emitter (BJT) -50 to -300 V/V 34 to 50 dB 180°
Inverting Op-Amp (Closed Loop) -1 to -100 V/V 0 to 40 dB 180°
Non-Inverting Op-Amp +1 to +1000 V/V 0 to 60 dB 0°

The Sanity Check: If you are designing a single-stage audio power amplifier and your math yields 120 dB (which is 1,000,000 V/V), you have a math error. That would require kilovolt supply rails to avoid instant clipping. Open-loop op-amps can hit 100 dB, but closed-loop practical circuits rarely exceed 60 dB per stage.

Unit Mistakes That Break the Equation

  1. Mixing Peak-to-Peak with RMS: If V_in is measured in V_pp on your Rigol oscilloscope and V_out is measured in V_RMS on your multimeter, the ratio is garbage. Convert both to RMS (V_pp / 2√2 for sine waves) or both to peak before dividing.
  2. Using the Power Multiplier (10) for Voltage: The decibel equation for power is 10 × log_10(P_out / P_in). Because P = V^2 / R, the square brings a 2 to the front, making it 20 × log_10(V_out / V_in). Using 10 instead of 20 will result in a gain value exactly half of what it should be in dB. See Analog Devices MT-044 Tutorial for the formal derivation.
  3. Ignoring Impedance in RF: In low-frequency audio, voltage gain is mostly independent of load. In 50Ω RF systems, a voltage gain of 2 V/V (6 dB) into a matched load results in a power gain of 6 dB only if the input and output impedances are identical. If they differ, voltage gain and power gain diverge.

Frequently Asked Questions

How do you calculate the voltage gain equation for a common emitter amplifier?

For a basic BJT common emitter amplifier with an unbypassed emitter resistor, the approximate linear voltage gain is A_v ≈ -R_C / R_E, where R_C is the collector resistor and R_E is the emitter resistor. If the emitter resistor is fully bypassed by a capacitor, the gain becomes A_v ≈ -R_C / r'_e, where r'_e is the internal AC emitter resistance (typically 25 mV / I_E). The negative sign denotes a 180° phase inversion between the base input and collector output.

Why does the voltage gain equation use 20 log instead of 10 log?

The decibel was originally defined for power ratios: 10 × log_10(P_out / P_in). Since power is proportional to the square of voltage (P = V^2 / R), substituting voltage into the power equation yields 10 × log_10((V_out / V_in)^2). Using the logarithmic power rule (log(x^y) = y × log(x)), the exponent 2 moves to the front, resulting in 20 × log_10(V_out / V_in). This ensures that a +3 dB increase represents a doubling of power, whether you calculate it using watts or volts (assuming constant impedance).

Can the voltage gain equation yield a negative number?

Yes, in two distinct ways. First, linear gain (A_v) can be negative, which does not mean the signal is attenuated; it indicates a 180° phase shift (inversion) between input and output, common in inverting op-amp topologies and common-emitter BJT stages. Second, logarithmic gain (A_v(dB)) is negative when V_out < V_in (attenuation). For example, a passive voltage divider that halves the signal has an A_v of 0.5, which translates to 20 × log_10(0.5) = -6.02 dB.

What happens to the voltage gain equation at high frequencies?

The standard mid-band equation assumes ideal components. At high frequencies, parasitic capacitances (like the Miller effect in transistors or stray PCB trace capacitance) create low-pass filter poles. The gain magnitude rolls off at -20 dB/decade per pole. To model this accurately, the scalar A_v is replaced by a complex transfer function A_v(jω), where ω = 2πf. At the cutoff frequency (f_c), the magnitude of the gain drops to 0.707 of its mid-band value, representing the -3 dB point.