1

The following is a figure from 8.47 in Design of Analog CMOS Integrated Circuit, page 301:

enter image description here

The book tends to suggests that feedback network

enter image description here

However, I have a different opinion.

enter image description here

enter image description here

kile
  • 798
  • 2
  • 11

3 Answers3

2

The two-port model analysis of the network is helpful here. In this analysis, you model the forward amplifier block and the feedback network as equivalent two-port networks, and work to characterize the properties at the terminals on either side of the network.

The key question there is what are the terminals of the forward amplifier network and what are the terminals of the feedback network, and in what configuration are they connected.

To clarify this, I add to the original schematic the name of the Rs-Rf junction a label for future reference, which I am calling point \$P\$. I also add a blue box to identify one possible choice of the feedback network. You could also choose to include RD2 as part of the feedback network. In the end, which choice you make won't affect the final results for the circuit analysis.

enter image description here

The important thing in this analysis to identify the terminals of the forward amplifier and the feedback network. The feedback network has terminals at \$P\$ and at ground on the left (amplifier input) side, and at the \$V_{out}\$ terminal and ground on the right (amplifier output) side. The forward amplifier has terminals at \$V_{in}\$ and \$P\$ on the left, and at \$V_{out}\$ and ground on the right. They are the same output terminals as the feedback network on the right side, because they are connected in parallel (shunt).

This connection is referred to as a 'Series-Shunt' connection, where on the input side the amplifier terminals and feedback terminals are connected in series, and on the output side of the amplifier, they are connected in parallel (shunt).

The appropriate two-port network model for the feedback network is the \$H\$ model, shown below (Fig 8.50 from the same Razavi reference from OP). In that figure \$V_1\$ corresponds to the voltage at \$P\$ in the figure relative to ground. The usual approximation is to neglect the forward dependent source \$H_{21}\$. Calculating how the feedback network loads the input side amounts to calculating the resistor \$H_{11}\$ in the diagram, which represents the impedance seen looking into the feedback network from the amplifier input side (left side of the diagram) with the dependent source \$H_{12}\$ turned off. The way to turn off \$H_{12}\$ is to short \$V_2\$ in this diagram, which is \$V_{out}\$ in the original circuit.

From Razavi, Fig 8.50, p. 303

So, to get the \$H_{11}\$ resistance, we want to know what the resistance is between the \$P\$ terminal and ground when we set \$V_{out}\$ to zero. That is why the right side of resistor Rf is grounded in Fig 8.54b of your original post when analyzing the impedance of the feedback network from the amplifier input side.

Looking at the feedback network from the amplifier-output side (the right side of the circuit), we see from the two-port model in the figure that the impedance \$H_{22}\$ is obtained by zeroing the dependent current source \$H_{21}\$, which amounts to severing the connection between feedback terminal and the amplifier terminal on the left (input) side of the circuit, since they are connected in series on that side. The reason you sever the connection is that you want to turn off the current \$I_1\$ in the two-port model in the figure above. That is what leads to the diagram on the right in figure 8.54 shown in the original post, reproduced here for convenience. It is when you sever the connection to the forward amplifier at \$P\$ that you are left with the \$H_{22}\$ impedance when looking at the feedback network from the right side.

From Razavi Fig. 8.54

From these considerations, you can see that the critical question is not so much the assignment of resistors to either the feedback network or the amplifier, but rather what load the feedback network presents on either side of the two-port equivalent model. It is important not to try to associate components in the two-port model with individual components in your circuit, but to realize that they are just model parameters that can be used to characterize the response at those two terminals.

So my suggestion is not to focus on what resistors constitute the feedback network, but rather on what are the terminals, and how are they connected to the amplifier terminals. This gives the information that is really required, which is how the feedback network loads each side of the circuit.

rpm2718
  • 2,812
  • 1
  • 6
  • 14
  • What is $V_1$ and $H_{11}$ in circuit https://i.stack.imgur.com/8Pdgf.png? Is $H_{12}$ voltage source? – kile Mar 29 '24 at 21:12
  • Is $H_{21}$ equal to $M_2$ here? Is $V_1$ equal to $V_{in}$? – kile Mar 29 '24 at 21:18
  • The network in the first fig is the two-port representation of the feedback network only. V1 in that figure is the voltage between the Rs/Rf junction and ground, and $H_{11}$ is the equivalent resistance seen looking into the Rs/Rf junction and ground. $H_{21}$ is ignored, approximated as zero. I will expand the answer soon to clarify better. – rpm2718 Mar 30 '24 at 00:08
  • Substantial edit done to try to explain things more clearly. – rpm2718 Mar 30 '24 at 00:48
  • Can you mark "ground on the right (amplifier output) side" in the graph? – kile Mar 30 '24 at 09:13
  • Is $H_{11}$ the impedance at the gate of $M_1$ or at reference point P? Where should I add the test voltage? Could you draw a new diagram on where you add the test the voltage to get $H_{11}$? – kile Mar 30 '24 at 09:16
  • I believe those questions are answered clearly in my post. The bottom two terminals in the two-port model correspond to ground in the original circuit. $H_{11}$ is determined by looking at the three bottom resistors in isolation (disconnected from M1 and M2), and shorting the $V_{out}$ terminal to ground, and measuring the resistance at $P$. – rpm2718 Mar 30 '24 at 13:29
  • Is this https://i.imgur.com/hBP3pkm.png what you mean for $H_{11}$? – kile Mar 30 '24 at 22:33
  • Yes, that would give $H_{11}$ at the measurement port you show. Notice that it takes RD2 out of it, leaving the parallel combination of Rs and Rf, just like the book shows. – rpm2718 Mar 31 '24 at 02:40
  • What is $I_1$? Is it the current flowing through $M_1$? – kile Mar 31 '24 at 08:05
  • No, focus on what the terminals are of that two-port model. The model represents the feedback network only, and the terminals on the left are ground and $P$, and on the right $V_{out}$ and $P$. So $I_1$ in that model would be the current going into point $P$ if you disconnected those lower three resistors from M1 and M2, grounded $V_{out}$, and put a voltage $V_1$ at $P$. This gives $H_{11}$, the resistance that the feedback network presents as a load to the left side of the circuit. It ends up being the parallel combo of Rs and Rf. – rpm2718 Mar 31 '24 at 12:02
  • There are three currents flowing through P, which one is $I_1$? https://i.imgur.com/FWIxnTR.png – kile Mar 31 '24 at 13:54
  • Remember that the two-port model I posted represents the feedback network only. You should imagine it disconnected from M1 and M2. So it is $I_t$ in the graphic you posted here https://i.imgur.com/hBP3pkm.png. – rpm2718 Mar 31 '24 at 19:33
  • I think it would help you to study or review two-port models. Very useful. – rpm2718 Mar 31 '24 at 19:49
  • Take a look at my circuit. https://i.imgur.com/SUcDcb7.png – kile Mar 31 '24 at 21:59
1

The book includes Rs because, although its presence is required for the open loop circuit to work anyway, the amount of feedback will be related to Rs/(Rs+Rf). Not knowing Rs would make it difficult to determine Rf.

Cristobol Polychronopolis
  • 10,599
  • 10
  • 18
1

In Razavis book this circuit was chosen as an example for the special case where it is not clear how the feedback network should/can be identified.

The two alternatives depend on the question if Rd2 is considered to be part of the feedback or not. (On page 307 the 2nd option is discussed).

More than that - note that (a) it makes really no sense not to ground Rf and (b) that the shown circuits only serve the purpose to find an expression for the gain without feedback (open-loop gain Ao) - in contrast to the quantity we call "loop gain" which is defined as Ao*beta (beta=feedback factor).

Another (simple) example: In a BJT voltage follower (common collector) - is the resistor RE part of the amplifier or part of the feedback circuit - or both?

LvW
  • 26,610
  • 3
  • 23
  • 56