Merge pull request #31 from simi1505/simi

Read through Xmas update Sec. 8.7 / 11 / 12

content/_sec_basic_ota.qmd

@@ -312,7 +312,7 @@ Being a single-stage amplifier stability is usually not an issue, as only the ou
 
 {{< include ./figures/_fig_ota_se_r2r.qmd >}}
 
-Another popular version is shown in @fig-ota-se-twostage. Here, we have a two-stage amplifier able to provide higher voltage gain. The first stage (the diffpair loaded by a current mirror) is followed by a single-stage common-source amplifier with current-source load. Being a two-stage amplifier with two high-ohmic nodes, stability is a concern, so usually we need some form of compensation (in @fig-ota-se-twostage a very simplistic Miller-compensation using $C_\mathrm{M}$ is shown; often, we would want to implement a more advanced scheme). An advantage of this two-stage amplifier (compared with the simple 5T-OTA) is the dc-balanced load on top of the differential pair, as each side sees a voltage drop from $\VDD$ of one $\VGS$ ($\VGS[3]$ on the left side, $\VGS[5]$ on the right side).
+Another popular version is shown in @fig-ota-se-twostage. Here, we have a two-stage amplifier able to provide higher voltage gain. The first stage (the diffpair loaded by a current mirror) is followed by a single-stage common-source amplifier with current-source load. Being a two-stage amplifier with two high-ohmic nodes, stability is a concern, so usually we need some form of compensation. @fig-ota-se-twostage shows a very simplistic Miller-compensation using $C_\mathrm{M}$. Often, we would want to implement a more advanced scheme. Some examples can be found in [@Baker_compensating_OPAMP]. An interesting technique is the indirect compensation with cascoded input differential pairs (see @fig-improved-ota) for higher power supply rejection. An advantage of this two-stage amplifier (compared with the simple 5T-OTA) is the dc-balanced load on top of the differential pair, as each side sees a voltage drop from $\VDD$ of one $\VGS$ ($\VGS[3]$ on the left side, $\VGS[5]$ on the right side).
 
 {{< include ./figures/_fig_ota_se_twostage.qmd >}}
 
@@ -326,4 +326,4 @@ $$
 V_\mathrm{out} \approx V_\mathrm{ref} \left( 1 + \frac{R_1}{R_2} \right)
 $$
 
-if the gain of the OTA is sufficiently high. The quiescent current through $R_{1,2}$ established a minimum load current for the LDO, which is often a good thing for stability. More information on LDOs can be found in [@Razavi_2019_ldo].
+if the gain of the OTA is sufficiently high. The quiescent current through $R_{1,2}$ establishes a minimum load current for the LDO, which is often good for stability. More information on LDOs can be found in [@Razavi_2019_ldo].

content/_sec_biasing.qmd

@@ -94,13 +94,19 @@ $$
 V_\mathrm{ref} = \VBE + \frac{R_2}{R_1} \frac{k T}{q} \ln m.  
 $$
 
-By proper selection of $R_1$, $R_2$ and $n$ we can satisfy @eq-bandgap-voltage-with-temp to result in @eq-bandgap-voltage.
+By proper selection of $R_1$, $R_2$ and $m$ we can satisfy @eq-bandgap-voltage-with-temp to result in @eq-bandgap-voltage.
 
 ::: {.callout-note title="Improved Bandgap Reference"}
 For an improved implementation of @fig-bandgap-simple, the current mirrors should be cascoded, and a startup circuit should be included to guarantee proper operation after enabling it. Further, @eq-vbe-vs-temp and @eq-delta-vbe-vs-temp build on the relationship $I_\mathrm{C} = f(\VBE)$, while we control $I_\mathrm{E}$ in this circuit. If $\beta$ is large then $I_\mathrm{C} \approx I_\mathrm{E}$, but this is not the case for the used PNPs.
 :::
 
-The circuit of @fig-bandgap-simple has been implemented in Xschem and is shown in @fig-bandgap-schematic. The current sources have been improved by using cascodes. No base current compensation is implemented, as it is assumed that the $\beta$ of the PNP are similar although they are operated at different emitter current densities. Note the addition of a startup branch with $M_\mathrm{startup}$ which is inactive during normal operation but will inject a startup current if no proper bias point has yet been found. There is no circuitry added for enabling/disabling the circuit, which would also be needed for a practical implementation. As usual, the MOSFET sizing has been done in this [notebook](sizing/sizing_bandgap_simple.ipynb).
+The circuit of @fig-bandgap-simple has been implemented in Xschem and is shown in @fig-bandgap-schematic. The current sources have been improved by using cascodes. We are using the low-voltage current mirror type already introduced in @sec-improved-ota. The bias voltages for the cascodes are generated via the voltage drops of $R_3$ and $R_4$, respectively. 
+
+No base current compensation for the BJTs is implemented, as it is assumed that the $\beta$ of the PNP are similar although they are operated at different emitter current densities.
+
+Note the addition of a startup branch with $M_\mathrm{startup}$ which is inactive during normal operation but will inject a startup current if no proper bias point has yet been found.
+
+There is no circuitry added for enabling/disabling the circuit, which would also be needed for a practical implementation. As usual, the MOSFET sizing has been done in this [notebook](sizing/sizing_bandgap_simple.ipynb).
 
 The resistors $R_{1-4}$ have been implemented out of unit elements of ca. $5\,\text{k}\Omega$ for optimum matching. Building a bandgap for the first time on silicon likely will show a slightly deviating temperature coefficient, which is why we keep a few dummy resistors around in $R_2$ to compensate the TC in a redesign.
 

content/_sec_differential_ota.qmd

@@ -34,6 +34,8 @@ For differential operation, the differential pair of $M_{1,2}$ is loaded by $R_\
 
 As soon as we implement a two-stage amplifier we need to look into stability. We likely have more than two poles, and in the case of the amplifier in @fig-diff-ota-resload we have a low-frequency pole at the drains of $M_{1,3}$ and $M_{2,5}$ and a further low-frequency pole at the output. Any additional pole will add further phase shift making stability critical. We now need a method to stabilize this amplifier and thus we will look into *Miller compensation*.
 
+Of course, loop gain analysis of differential OTAs can and should be carried out. An exemplary testbench where the loop gain is simulated with Rosenstark's, Middlebrook's, and Tian's method can be found [here](xschem/ota-differential_tb-loopgain.sch). Note that there is no underlying circuit right now and it should only show how it could be done.
+
 ## Miller Compensation
 
 A popular way to stabilize a multi-pole feedback-system is to make one pole dominant, and try to shift the other poles to sufficiently high frequencies, that we have enough [phase margin](https://en.wikipedia.org/wiki/Phase_margin) in the closed-loop system. We may strive for $60^\circ$ as this will only cause a minor peaking in the frequency response [@Gray_Meyer_5th_ed].

references.bib

@@ -6,7 +6,10 @@
 
 %% Saved with string encoding Unicode (UTF-8) 
 
-
+@online{Baker_compensating_OPAMP,
+author={R. J. Baker},
+title={Bad Circuit Design 3 - Compensating an Op-Amp},
+url={https://cmosedu.com/cmos1/bad_design/bad_design3/bad_design_3.htm}}
 
 @article{Razavi_2021_bandgap,
 	author = {Razavi, Behzad},

xschem/ota-differential.sym

@@ -0,0 +1,65 @@
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+}
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