# Spiral 2 7. Capacitance, Delay and Sizing. Mark Redekopp

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1 2-7.1 Spiral 2 7 Capacitance, Delay and Sizing Mark Redekopp

2 2-7.2 Learning Outcomes I understand the sources of capacitance in CMOS circuits I understand how delay scales with resistance, capacitance and voltage I can determine appropriate width of PMOS and NMOS transistors based on the configuration of the transistors and given current conduction parameters I understand how fan in and fan out affect the delay of a circuit I understand how to use sizing to drive larger fan out loads I understand the sources of static and dynamic power consumption and how they are affect by changes in various parameters

3 WHAT IS CAPACITANCE? 2-7.3

4 2-7.4 Capacitance Capacitors are formed by separating two substances with an Capacitors charge Capacitance measures how much charge is needed to achieve a certain voltage (electric potential) C = Capacitance measured in Farads (F) + - Res Connected to a source, charge will be stored on the conductive plates creating a positive voltage between the conductive plates Res. Conductive Material Insulator Material E y C Capacitor Schematic Symbol To change the voltage at the capacitor we must change the voltage (if we turn off the voltage source charge will drain off the capacitor)

5 2-7.5 Charging/Discharging Capacitors Charging a capacitor gets more as more charge is added Eventually, no more charge can be added and the capacitor acts like an open circuit Voltage of Capacitor As more charge (voltage) builds up that charge repels like charge and makes it harder to add more (like pushing a spring) Time

6 2-7.6 Capacitor I V Relationship Fact Also recall, or Thus, substituting Current is linearly related (slope = C) to the in voltage (not the absolute voltage) No voltage (constant voltage) means no

7 2-7.7 Measures of Capacitance C = is the permittivity of the insulator substance (intrinsic material property) defined as 1 for a vacuum Silicon dioxide (separates gate from silicon) = 3.9 Pure silicon = is the of the conductive materials is the (or thickness of the capacitor)

8 2-7.8 First order RC circuit step response RC CIRCUIT ANALYSIS

9 2-7.9 Resistance / Capacitance Analogy Charge = Water Resistance = Limit of water flow Voltage Source = Water Pressure Capacitance = Total Water Needed Switching Time = Time to fill or drain the capacitor ( bucket ) of charge Thus, increase the voltage or decrease the resistance/capacitance.

10 Voltage, Resistance & Capacitance Let's analyze a simple circuit Known as an RC circuit (resistor & capacitor in series) Assume t < 0, For t > 0, Vs = (voltage source turns on) Current through R must be same as current "through" C Now let's solve for dvc/dt We can solve this differential equation 0 For Vc(0)=0 we have

11 Voltage, Resistance & Capacitance Let's analyze a simple circuit Known as an RC circuit (resistor & capacitor in series) Assume t < 0, Vs = Vdd then Vc=Vdd For t > 0, Vs = 0 (voltage source = GND) Current through R must be same as current "through" C Now let's solve for dvc/dt 0 We can solve this differential equation 0 For Vc(0)=Vdd we have

12 Time Constant Notice the charging (discharging) time is determined by product of R*C We refer to this as the, τ τ= RC As the product of RC increases we get switching times We can show that the time it takes to charge/discharge a capacitor to a fraction of Vdd is given in the table below Voltage Range Time 0 to 50% (t p = prop. delay) *RC 0 to 63% (τ) RC 10% to 90% (t r =rise time/delay) *RC

13 DELAY

14 MOSFET Parasitic Capacitance In order to examine the delay of a MOSFET, we have to determine nature and amount of capacitance associated with MOS transistor Parasitic: Unintentional, capacitors It's not that we want caps, we're due to the structure of the MOSFET. The oxide layer separating gate and substrate is a more obvious capacitor However the around the source and drain also form capacitors Gate Source Drain 14

15 Transistor R and C values Observation: Output of one transistor usually drives input of another Sources of resistance connecting Sources of capacitance of the next transistor Other small capacitances 3V Source Charge must be conducted through this path and build up at the gate input to raise the voltage to the necessary value Metal Wire Gate Gate Source Drain Source Drain

16 Resistor and Capacitor Delay Outputs connect to other inputs To change the output voltage (really the input of the next gate), we must conduct enough charge to raise or lower its present voltage Resistance limits the amount of charge that can be transferred per unit time Capacitance determines how much charge must be present to attain a certain voltage Time it takes to attain a certain voltage is proportional to R*C Wire from an output of another gate + Wire + Desired Transition: 0V -> Vdd + Switching Time ~ Resistance*Capacitance

17 Where RC Circuits Occur Consider a CMOS gate driving others ( ) The output connects to the gate inputs of the loads (fanout) and thus can be modeled as a capacitive load (C L )

18 Where RC Circuits Occur Depending on the inverter input the PMOS or NMOS will be in mode and can be modeled as resistors Thus we have an RC circuit either charging or discharging the output (C L )

19 Defining Delays We model The next gate(s) and other parasitic capacitances as a lumped capacitance The PDN or PUN transistors as resistors (since they are operating in linear mode when the input is near VDD or GND) t PLH and t PHL refer to the delay of a circuit when the output changes from low to high ( ) [t PLH ] or high to low ( ) [t PHL ] We say the delay of the gate is then: t PD (t PLH + t PHL ) / 2 Important: What reduces delay? Vin Vout Vin Vout

20 CMOS SIZING

21 Sizing PMOS is Slower than NMOS! Recall the equations for current through a transistor in linear mode 2 Ohm's law says I = V/R so in the equation above Problem (K N K P ) PMOS are at conducting than NMOS This will leave to imbalances in (i.e. ) To balance the delay when pulling up vs. pulling down we can play with Solution: Make by about a factor of 2 or 2.5 IN VDD OUT CL

22 Sizing Inverter n p p n p p n n n p W W W W W K W K R R L KW R Assume Find the ratio and W p and W n that balances the delay of output during falling and rising transitions 2 n p K K

23 Sizing Simple CMOS Gates Goal: Make any gate have the same worst case resistance as an inverter The ratio of the {W/L} PUN / {W/L} PDN should be about two (or higher) to make up for slow PMOS Important Notes: For parallel transistors consider only the case if 1 is on (Remember if R R then Reff=R/2 which is a better case so we assume only one is on) Series transistors in series add lengths/ resistance All paths in a PxN should have same resistance NAND NOR

24 Sizing Complex CMOS Gates PUN: make each path W/L = 2) A B C D D A PDN: make each path W/L = 1 B C 24

25 Compound Gate Example Y = D (A + B + C) PUN: make each path W/L = 2) A B PDN: make each path W/L = 1 A C B D D C Y

26 Compound Gate Example PUN: make each path W/L = 2) Y = AB+CD PDN: make each path W/L = 1

27 FANIN & FANOUT

28 Fanout Fanout refers the number of gates an output connects to As the fanout increases C L and means the delay This inverter has a fanout (# of loads) = 1 This inverter has a fanout (# of loads) = 3

29 Increasing Drive Strength So far we ve always modeled an idea inverter for our transistor sizes Ideal inverter => NMOS: 1/1 and PMOS 2/1 or 3/1 We can counteract the increase in CL by through increasing transistor NMOS and PMOS widths increase by: INV (W n /L n = 1) INV (W n /L n = 2) INV (W n /L n = 4)

30 Fan in Fan in refers to the number of inputs to a gate Each input adds intrinsic, parasitic t phl =0.69*R n ( ) To discharge C2 requires going 2L (i.e. 2R N ) To discharge C3 requires 3L (i.e. 3R N ) This means delay grows quadratically with fan in but linearly with fanout t pd ~ Important: Rarely want FI > Fanin = 2 Fanin = 5

31 Mitigating Delay May increase widths Bottom transistor has width Order transistors to allow the latest arriving input control the transistor to the output If Z is the latest arriving input we want to put it to the output

32 INTERCONNECT DELAY

33 Ideal vs. Realistic Wire Ideal wire should have R=0 and little capacitance In real life it has some small R and C R = ρl/a ρ = resistivity of material (some intrinsic property) L = length or wire A = cross section area of the material As technology scales (we build smaller devices) and means resistance goes ρ = Intrinsic property of material L A

34 Modeling Interconnect Delay Interconnect delay is starting to (already is) switching delay Important design considerations Long wire traces slow a signal down, thus global signals on a chip require special attention A real wire can be modeled as Ideal wire Lumped Model (overestimates delay) Distributed Model (better estimate)

35 Dealing With Interconnect Clock, reset, and other signals must be routed carefully and a whole tree of buffers inserted to decrease the delay

36 DYNAMIC POWER

37 Power Power consumption decomposed into: Static: Power constantly being dissipated (grows with # of transistors) Dynamic: Power consumed for switching a bit (1 to 0) P DYN = I DYN *V DD ½C TOT V DD2 f Recall, I = C dv/dt V DD is the logic 1 voltage, f = clock frequency Dynamic power favors parallel processing vs. higher clock rates V DD value is tied to f, so a reduction/increase in f leads to similar change in Vdd Implies power is proportional to f 3 (a cubic savings in power if we can reduce f) Take a core and replicate it 4x => 4x performance and 4x power Take a core and increase clock rate 4x => 4x performance and 64x power Static power Leakage occurs no matter what the frequency is

38 Temperature Temperature is related to power consumption Locations on the chip that burn more power will usually run hotter Locations where bits toggle (register file, etc.) often will become quite hot especially if toggling continues for a long period of time Too much heat can destroy a chip Can use sensors to dynamically sense temperature Techniques for controlling temperature External measures: Remove and spread the heat Heat sinks, fans, even liquid cooled machines Architectural measures Throttle performance (run at slower frequencies / lower voltages) Global clock gating (pause..turn off the clock) None results can be catastrophic Fun video to be played now!

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