A general method for TF It’s systematic Uses Mason’s formula Will use the two stage op amp to illustrate Will modify to obtain exact TF for lead compensated op amp

First step Identify all interesting node At each node (other than input), denote current going into the node and voltage of the node Find total admittance from each node to ground Find inter-node admittance

vs vb cc VDD VDD VDD M9 M12 M11 vo Iref M1 M2 Vin- vin+ CL M3 M4 R M5

Second step Find current into each node Find voltage at each node gm effects Inter-node admittance Find voltage at each node Current into the node / admittance to ground Arrange into a directed graph from input to output

vs vb cc VDD VDD VDD M9 M12 M11 vo Iref M1 M2 Vin- vin+ CL M3 M4 R M5

Mason’s Gain Formula A forward path: a path from input to output Use Mk: total gain along the k-th path A loop is a closed path in which you can start at any point, follow the arrows, and come back to the starting point Use Li: total gains along the i-th loop Loop i and loop j are non-touching if they do not share any nodes or branches

The determinant of the total graph Δ: Δk: The determinant of the residue graph excluding the k-th forward path Mason’s Gain formula:

vs vb cc Mirror node: G7, G8, D5 VDD VDD VDD M9 M12 M11 vo Iref M1 M2 vin M1 M2 vin+= voQ CL vb M3 M4 cc R M5 M6 M10 Mirror node: G7, G8, D5 M7 M8 vin2

Think of each cascade pair as a composite transistor with the same gm, but very small go. Then we can ignore the cascade transistors for small signal analysis. R is in series with ro of M1M3 pair. R is Veff/Id, so is much much smaller. Cgd1 no longer connects to M node. Cgd2 no longer connects to Vo1 node. So set both to 0.

 

vs vb cc VDD VDD VDD M9 M12 M11 vo Iref M1 M2 Vin- vin+ CL M3 M4 R M5 Rc R M5 M6 M10 M7 M8 vin2

With Miller compensation, the inter-node admittance: With lead compensation, the inter-node admittance: To get new TF, only need to divide sCc by (1+sRcCc)