Cascaded Amplifiers

The available power gain of amplifiers or amplifier stages can be increased by using cascaded amplifiers or amplifier stages. In this section, we will discuss the behavior of cascade connections of amplifiers. Fig. 57 shows a cascade connection of two two-ports. The output of the first two-port is connected to the input port of the second.

Fig. 57 A cascade connection of two two-ports.

If we want to design a system that consists of two or more cascaded subsystems, we need error distribution methods for specification of the performance of each subsystem. In this section, we will give expressions for the evaluation of the total error due to the limitations of noise, speed and power. These expressions form the basis for error distribution in designing cascaded systems.

The transmission parameters of a two-port that consists of the cascade connection of two two-ports, as shown in Fig. 57 are:

\[\begin{split}\left( \begin{array} [c]{c} V_{i}\\ I_{i} \end{array} \right) =\left( \begin{array} [c]{cc} A_{1}A_{2}+B_{1}C_{2} & A_{1}B_{2}+B_{1}D_{2}\\ A_{2}C_{1}+C_{2}D_{1} & B_{2}C_{1}+D_{1}D_{2} \end{array} \right) \left( \begin{array} [c]{c} V_{o}\\ I_{o} \end{array} \right) ,\end{split}\]

where \(A_{1},B_{1},C_{1},D_{1}\) and \(A_{2},B_{2},C_{2},D_{2}\)\ are the transmission parameters of the first and the second two-port of the cascade connection, respectively.

Port isolation

In section Modeling of the power supply isolation, we have seen that the non-ideal port isolation of an amplifier is usually described with a few parameters. Under well-specified conditions for source, load and power supply, such simplified descriptions can be meaningful. Fig. 58 shows two cascaded differential voltage amplifiers that have been connected to one power supply. Under appropriate drive conditions, the port isolation properties of the amplifier chain can be estimated from the port isolation properties of the individual amplifiers. We will give some expressions in the following paragraphs.

Fig. 58 Cascaded differential voltage amplifiers.

Rejection Factor

If \(n\) amplifiers form a cascaded amplifier chain, the rejection factor of the chain \(F_{tot}\) can be obtained from the rejection factors \(F_{i}\) of the amplifiers that constitute the chain:

\[F_{tot}=F_{1}F_{2}...F_{n}.\]

Please consider the remarks about the completeness of such a description set down in section Modeling of the power supply isolation.

Common-Mode Rejection Ratio

If \(n\) amplifiers form a cascaded amplifier chain, the common-mode rejection ratio of the chain \(CMRR_{tot}\) can be obtained from the common-mode rejection ratios \(CMRR_{i}\) and the rejection factors \(F_{i}\) of the amplifiers that constitute the chain:

\[\frac{1}{CMRR_{tot}}=\frac{1}{CMRR_{1}}+\frac{1}{F_{1}CMRR_{2}}+...+\frac{1}{F_{1}F_{2}F_{n-1}CMRR_{n}}.\]

The index number \(i\) refers to the amplifier’s position in the chain; \(i=1\) refers to the first amplifier and \(i=n\) to the last amplifier of the chain.

Please consider the remarks about the completeness of such a description as set down in section Modeling of the power supply isolation.

If the rejection factor of all amplifiers is much larger than unity, and their common-mode rejection ratios are in the same order of magnitude, then the total common-mode rejection ratio approximates that of the first amplifier in the chain.

Power Supply Rejection Ratio

If \(n\) amplifiers in a cascaded amplifier chain are connected to the same power supply source, the power supply rejection ratio of the chain \(PSRR_{tot}\) can be obtained from the power supply rejection ratios \(PSRR_{i}\) and the gains \(A_{i}\) of the individual amplifiers. The index number \(i\) refers to amplifier’s position in the chain; \(i=1\) refers to the first amplifier and \(i=n\) to the last amplifier of the chain.

We will give the expression for voltage amplifiers (\(A_{vi}\) is the voltage gain of the \(i-th\) voltage amplifier):

\[\frac{1}{PSRR_{tot}}=\frac{1}{PSRR_{1}}+\frac{1}{A_{v1}PSRR_{2}}+...+\frac{1}{\left( A_{v1}...A_{v\left( n-1\right) }\right) PSRR_{n}}.\]

Please consider the remarks about the completeness of such a description as they have been made in section Modeling of the power supply isolation.

If the gain of all amplifiers is much larger than unity, and their power supply rejection ratios are in the same order of magnitude, then the total power supply rejection ratio equals that of the first amplifier in the chain.

Noise behavior

The noise figure of a system that consists of cascaded amplifiers that have a finite nonzero available power gain, can be calculated from the noise figures and the available powers gains of the individual amplifiers. This was shown by Friis (see [19]).

For \(n\) cascaded subsystems from which the input of the first system is connected to the signal source and the input of each following subsystem is connected to the output of the previous subsystem, the total noise factor \(NF_{tot}\) can be expressed as:

(20)\[NF_{tot}=NF_{1}+\frac{NF_{2}-1}{A_{p1}}+...+\frac{NF_{n}-1}{A_{p1} ...A_{p(n-1)}}, \label{ex-friis}\]

where \(NF_{i}\)\ [-] is the noise factor of subsystem \(i\), calculated with respect to the output impedance of its driving subsystem, and \(A_{pi}\)\ [-] the available power gain of subsystem \(i.\)

Given a unilateral voltage amplifier with voltage gain \(A_{v}\), an input resistance \(R_{i}\) and an output resistance \(R_{o}\), and driven from a source impedance \(R_{s}\), the available power gain can be written as

\[A_{p}=\left( \frac{R_{i}}{R_{s}+R_{i}}\right) ^{2}A_{v}^{2}\frac{R_{s} }{R_{o}}.\]

Expression (20) is only useful for situations in which the cascaded amplifiers have a nonzero, finite output resistance. This is often the case in so-called characteristic impedance systems. If this is not the case, the noise of cascaded amplifiers can be evaluated with the aid of the techniques described in section Modeling of the noise behavior.

Small-signal dynamic behavior

If \(n\) amplifiers form a cascaded amplifier chain, the small-signal transfer function \(H_{tot}(j\omega)\ \)can be obtained as the product of the transfer functions \(H_{i}(j\omega)\) of all amplifiers in the chain:

(21)\[H_{tot}(j\omega)=H_{1}(j\omega)H_{2}(j\omega)...H_{n}(j\omega). \label{ex-cascaded-transfer}\]

The pole-zero patterns of all constituting amplifiers thus have to be added.

Static nonlinear behavior

The small-signal gain of a chain of \(n\)\ cascaded amplifiers that all operate at their quiescent operating point is the product of the small-signal gains of the individual amplifiers. This simply follows from expression (21). If the amplifiers show nonlinear behavior, the gain at an excursion from the quiescent point differs from that at the quiescent operating point. This can be expressed with the aid of the differential gain error. Let \(A_{i}\) be the gain at the quiescent operating point of an amplifier located at position \(i\) in the amplifier chain, and let \(\epsilon_{i}(y_{i})\) be the differential gain error of that amplifier at an output signal excursion \(y_{i}\) from the quiescent operating point. Then, if the amplifier’s nonlinearity is small, the small-signal gain \(A_{i}(y_{i})\) at output excursion \(y_{i}\) can be approximated by

\[A_{i}(y_{i})=A_{i}\left( 1+\epsilon_{i}(y_{i})\right) .\]

If the signal excursion from the quiescent point at the output of the last amplifier in the chain equals \(y\), the excursion at the output of the \(i-th\) amplifier in the chain can be approximated by

\[y_{i}=\frac{y}{A_{i+1}A_{i+2}...A_{n}}.\]

We then obtain the total differential gain \(\epsilon_{tot}\) of the amplifier chain as

\[\epsilon_{tot}=\epsilon_{1}\left( \frac{y}{A_{2}A_{3}...A_{n}}\right) +\epsilon_{2}\left( \frac{y}{A_{3}A_{4}...A_{n}}\right) +...\]