# E40M Capacitors. M. Horowitz, J. Plummer, R. Howe

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1 E40M Capacitors 1

3 Why Are Capacitors Useful/Important? How do we design circuits that respond to certain frequencies? What determines how fast CMOS circuits can work? Why do we often put a large bypass capacitor between Vdd and Gnd? 3

4 Capacitors What is a capacitor? It is a new type of two terminal device It is linear Double V, you will double I We will see it doesn t dissipate energy Stores energy Rather than relating i and V Relates Q, the charge stored on each plate, to Voltage Q = CV Q in Coulombs, V in Volts, and C in Farads Like all devices, it is always charge neutral Stores +Q on one lead, -Q on the other lead 4

5 iv for a Capacitor We generally don t work in Q, we like i and V But current is charge flow, or dq/dt So if Q = CV, and i=dq/dt i= C dv/dt This is a linear equation but between i and dv/dt. If you double i for all time, dv/dt will also double and hence V will double. C = εa d where ε is the dielectric constant 5

6 Capacitors Only Affect Time Response not Final Values Capacitors relate i to dv/dt This means if the circuit settles down and isn t changing with time, a capacitor has no effect (looks like an open = = 6

7 So What Do Capacitors Do? They affect how fast a voltage can change Current sets dv/dt, and not V Fast changes require lots of current For very small Δt capacitors look like voltage sources They can supply very large currents And not change their voltage But for large Δt Capacitors look like open circuits (they don t do anything) 7

8 Capacitor Energy The Power that flows into a charging capacitor is P = iv = C dv V dt And the energy stored in the capacitor is E = Pdt E = Pdt V 0 = CVdV = 1 2 CV2 This energy is stored and can be released at a later time. No energy is lost. 8

9 Capacitor Types There are many different types of capacitors Electrolytic, tantalum, ceramic, mica,... They come in different sizes Larger capacitance Generally larger size Higher voltage compliance Larger size Electrolytic have largest cap/volume But they have limited voltage They are polarized One terminal must be + vs. other 9

10 Gate of MOS Transistor Is a capacitor between Gate and Source To change the gate voltage You need a current pulse (to cause dv/dt) If the current is zero (floating) dv/dt = 0, and the voltage remains what it was! 10

11 All Real Wires Have Capacitance It will take some charge to change the voltage of a wire Think back to our definition of voltage Potential energy for charge To make a wire higher potential energy Some charge has to flow into the wire, to make the energy higher for the next charge that flows into it This capacitance is what sets the speed of your computer And determines how much power it takes! x ox C S R L S H W SiO 2 Si C I L 11

12 Capacitor Info, If You Know Physics E&M Models the fact that energy is stored in electric fields Between any two wires that are close to each other A capacitor is formed by two terminals that are not connected But are close to each other The closer they are, the larger the capacitor To create a voltage between the terminals Plus charge is collected on the positive terminal Negative charge is collected on the negative terminal This creates an electric field (Gauss s law) Which is what creates the voltage across the terminals There is energy stored in this electric field 12

13 Capacitors in Parallel and Series i T i 1 i 2 i 3 C 1 C 2 C 3 13

14 Capacitors in Parallel and Series i T C 1 C 2 C 3 14

15 CAPACITOR RESISTOR CIRCUITS 15

16 Capacitors and Logic Gate Speeds When the input changes from low to high The pmos turns off, and the nmos turns on The output goes from high to low But in this model The output changes as soon as the input changes 16

17 Gates Are NOT Zero Delay It would be great if logic gates had zero delay But they don t Fortunately, it is easy to figure out the delay of a gate It is just caused by the transistor resistance Which we know about already And the transistor and wire capacitance 17

18 Improved Model Just add a capacitor to the output node Its value is equal to the capacitance of the gates driven Plus the capacitance of the wire itself 18

19 RC Circuit Equation When the input to the inverter is low, the output will be at V dd Right after the input rises, here is the circuit Want to find the capacitor voltage verses time 5V Just write the nodal equations: We just have one node voltage, V out V out i RES = V out /R 2 i CAP = CdV out /dt From KCL, the sum of the currents must be zero, so dv out dt = V out R 2 C 19

20 RC Circuit Equations Solving, dv out V out This is an exponential decay V 5 The x axis is in time constants t dt = so that ln V 0 R 2 C out The y axis has been normalized to 1 Slope always intersects 0 one tau later (τ = RC) ( ) ln( 5V) = t R 2 C V out = 5V ( e t/r C )

21 What Happens When Input Falls? Now the voltage across the capacitor starts at 0V i = (V dd V out )/R 1 dv out /dt = i/c dv out dt ( = V dd V out ) R 1 C Not quite the right form Need to fix it by changing variables Define V new = V dd V out dv out /dt = = - dv new /dt, since V dd is fixed 5V V out V 5 dv new V new t dt = so that ln V 0 R 1 C new ( ) ln( 5) = t R 1 C V out = 5V ( 1 e t/r C ) 1 21

22 RC Circuits in the Time Domain 5V V out = 5V ( 1 e t/r C ) 1 V out = 5V ( e t/r C ) 2 In capacitor circuits, voltages change slowly, while currents can be instantaneous. 22

23 Simple RC Circuit Demo EveryCircuit Demo CMOS Inverter 23

24 Interesting Aside Exponentials never reach their final value So if this logic gate is driving another gate, when does the next gate think its input is 0 or 1? This is one of the reasons why logic levels are defined as a range of values. 1 X 0 Vdd Gnd 24

25 Learning Objectives Understand what a capacitor is i=c dv/dt It is a device that tries to keep voltage constant Will supply current (in either direction) to resist voltage changes Understand how voltages and current change in R C circuits Voltage waveforms are continuous Takes time for their value to change Exponentially decay to final value (the DC value of circuit) Currents can charge abruptly In the next few lectures we ll return to representing signals in the frequency domain and learn about a new way to solve capacitor circuits. 25

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