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Deriving Expression for Core Power Supply Voltage Vcore

 
IR Drop

 
Derivations

The following terminology is used for power, voltage and current at nominal and worst-case conditions.

Pnom Pmin Chip power consumption
Vdd Vddmin External voltage
Inom Imin Total current supplied to chip core
Ipad Ipadmin Average current through each core supply pad
  Vcore Minimum voltage at edge of core
  Vmin Minimum voltage at centre of core

The values of Pnom, Vdd, Vddmin and Vmin are part of the chip spec. The other values are related by the following expressions, where Npad is the number of core Vdd pads or core Vss pads.

Pmin =  Pnom ×   Vddmin2
Vdd2
Inom =  Pnom
Vdd
Imin =  Pmin  =  Pnom×Vddmin  = Inom ×  Vddmin
Vddmin Vdd2 Vdd
Ipad =  Inom  =  Pnom
Npad Vdd×Npad
Ipadmin =  Imin  =  Inom × Vddmin  =  Pnom×Vddmin
Npad Npad Vdd Npad×Vdd2

Referring to the figure on the right, the voltage at the edge of the core Vcore is related to the external voltage Vddmin, the current flow Ipadmin through the supply pad and the resistances of the package leadframe, the bond wires and the core power supply pad.

Vcore =  Vddmin−2×Ipadmin×(Rpkg+Rbond+Rpad)
=  Vddmin−2×Ipad× Vddmin ×(Rpkg+Rbond+Rpad)
  Vdd  
= 
Vddmin(1−2× Ipad ×(Rpkg+Rbond+Rpad))
  Vdd  

IVcore related to Vddmin and package, bond and pad resistances

Picture showing the relationship between Vcore, the external voltage and the package, bond and core supply pad resistances.

We can express Ipadmin as a function of the IR drop and then equate this to the value on the left where it is a function of the core power. This way we have an expression that links the core power to the IR drop.

Ipadmin = 
(Vcore−Vmin)  as a function of IR drop
Rsup×Npad
 = 
Pnom×Vddmin  as a function of core power
Npad×Vdd2

where Rsup=Rvdd=Rvss, taking the supply power to be symmetrical.

Rearranging gives

Pnom =  (Vcore−Vmin)×Vdd2  ×  1
Vddmin Rsup