A differential amplifier is a circuit that amplifies the difference between two input signals while rejecting any signal common to both. The Bipolar Junction Transistor (BJT) differential amplifier is the fundamental building block of modern operational amplifiers (Op-Amps). One of the most critical performance metrics for this circuit is the Common Mode Rejection Ratio (CMRR).
CMRR is a measure of how effectively the amplifier rejects "common-mode" signals—noise or interference present on both inputs simultaneously. In this post, we discuss the challenges of improving CMRR and the professional solutions used in integrated circuits.
Basic BJT Differential Amplifier Circuit
The diagram below illustrates a dual-input, balanced-output BJT differential amplifier.
In this configuration, $Q_1$ and $Q_2$ form the differential pair. The emitter resistance $R_E$ provides the biasing current. For a deep dive into this construction, see our tutorial on Basic BJT Differential Amplifier Analysis.
Understanding CMRR
CMRR is defined as the ratio of differential-mode gain ($A_d$) to common-mode gain ($A_c$):
$$CMRR = \left| \frac{A_d}{A_c} \right|$$It is typically expressed in decibels (dB):
$$CMRR_{dB} = 20 \log_{10} \left| \frac{A_d}{A_c} \right|$$A high CMRR is essential because it allows the amplifier to extract small signals from noisy environments. While a basic amplifier might achieve 40 dB, precision applications typically require 60 dB to 100 dB.
The Conflict: CMRR vs. Biasing
To understand how to improve CMRR, we look at the gain equations for a balanced output:
$$A_d = \frac{-h_{fe}R_C}{h_{ie} + R_s}$$$$A_c = \frac{-h_{fe}R_C}{R_s + h_{ie} + 2R_E(1 + h_{fe})}$$From these equations, we can see that $A_d$ is independent of $R_E$, but $A_c$ is inversely proportional to $R_E$. Therefore, to increase CMRR, we must increase $R_E$ to minimize the common-mode gain.
The Problem: Increasing $R_E$ directly affects the DC biasing (Q-point):
$$I_{CQ} \approx \frac{V_{EE} - V_{BE}}{2R_E}$$If $R_E$ is made very large to improve CMRR, the collector current $I_{CQ}$ drops significantly, potentially pushing the transistors into cutoff. If we increase $V_{EE}$ to compensate, we eventually hit practical power supply limits and increase power dissipation.
Professional Solutions
To achieve a "virtually infinite" $R_E$ without disrupting the DC bias, we replace the passive resistor with active circuitry:
- Constant Current Bias: Replaces $R_E$ with a transistor-based constant current source. This provides high AC impedance (improving CMRR) while maintaining a stable DC current.
- Current Mirror Circuit: Provides extremely stable biasing current that is resistant to temperature and supply voltage fluctuations.
- Active Loads: Replacing collector resistors ($R_C$) with current mirrors to further increase differential gain.
Summary
Improving the CMRR of a BJT differential amplifier requires a high emitter resistance to provide negative feedback against common-mode signals. Since physical resistors are limited by biasing requirements, active current sources and mirrors are used in integrated circuits to provide the high impedance necessary for superior noise rejection without sacrificing Q-point stability.
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