# 3.7: Case Study- Distributed Biasing of Differential Amplifiers

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In this case study a broadband distributed balun-like section is presented as an alternative to inductor-biasing of a pseudo-differential amplifier (PDA). The distributed biasing circuit discriminates between differential-and common-mode signals, resulting in rejection of common-mode signals. A PDA is shown in Figure $$\PageIndex{1}$$, where the inductors present high RF impedances to the transistors while providing low-impedance paths for bias currents. However, inductive biasing of pseudo-differential circuits presents the same environment to common- and differential-mode signals so that the CMRR is $$1$$.

Differential amplifiers have large differential gain, $$A_{d}$$. At the same time it is desirable to minimize the common-mode gain, $$A_{c}$$, as the resulting high CMRR provides immunity to substrate-induced noise. Considering that each transistor has transconductance, $$g_{m}$$, and that even- and odd-mode impedances, $$Z_{\text{EVEN}}$$ and $$Z_{\text{ODD}}$$, are presented to the drains of the transistors, then the gains are approximately

$\label{eq:1}A_{d}=g_{m}Z_{\text{ODD}}\quad\text{and}\quad A_{c}=g_{m}Z_{\text{EVEN}}$

and so

$\label{eq:2}\text{CMRR}=A_{d}/A_{c}=Z_{\text{ODD}}/Z_{\text{EVEN}}$

The desired amplifier characteristics are thus obtained by synthesizing the even- and odd-mode impedances.

First consider the inductively biased circuit in Figure $$\PageIndex{1}$$. Modal analysis of the inductor biasing circuit results in the circuit model shown in Figure $$\PageIndex{2}$$, from which the total even-mode impedance is

$\label{eq:3}Z_{\text{EVEN}}(s)=(s_{L}+R_{DD}/2)//R_{L}$

where $$//$$ indicates a parallel connection, and the total odd-mode impedance is

$\label{eq:4}Z_{\text{ODD}}(s)=sL//R_{L}$

Since $$R_{DD}$$ is usually negligible and $$L$$ is a bias or choke inductor so that $$sL$$ is very large, $$Z_{\text{ODD}}\approx R_{D}\approx Z_{\text{EVEN}}$$ and so the CMRR is $$1$$. However, a coupled-line network can provide different model impedances.

Now consider the Marchand balun-like structure in Figure $$\PageIndex{3}$$ that replaces the drain bias circuit in Figure $$\PageIndex{1}$$. The Marchand Balun structure Figure $$\PageIndex{1}$$: A PDA amplifier with bias inductors, $$L$$, at the drains, parasitic supply resistance, $$R_{DD}$$, and single-ended load impedance, $$R_{L}$$. $$L$$ is also known as a choke inductor chosen so that its impedance is very large at the maximum operating frequency, i.e. $$|sL| ≫ R_{L}$$. Figure $$\PageIndex{2}$$: Modal subcircuits of the inductor-based biasing circuit of Figure $$\PageIndex{1}$$, including single-ended load resistance $$R_{L}$$. Figure $$\PageIndex{3}$$: Marchand balun-like biasing circuit with single-ended load resistance $$R_{L}$$. External DC bias is applied at Ports $$b$$ using decoupling capacitors to ensure RF ground.

presents different impedances for common- and differential-mode signals. The synthesis of this biasing circuit is described in [8, 9] and follows a procedure similar to that for filter design. So high CMRR performance is the result of presenting different even- and odd-mode impedances to the active devices. The final results of the design are shown in Figure 3.8.1, first for inductive-biasing of the PDA and then for the coupled-line balun-like biasing circuit.

3.7: Case Study- Distributed Biasing of Differential Amplifiers is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.