#import "@preview/mannot:0.3.1" #import "@preview/cetz:0.4.2" #import "@preview/zap:0.5.0" #import "../lib/common_rewrite.typ" : * #import "../lib/truthtable.typ" : * #import "../lib/fetModel.typ" : * #show math.integral: it => math.limits(math.integral) #show math.sum: it => math.limits(math.sum) #set page( paper: "a4", margin: ( bottom: 10mm, top: 5mm, left: 5mm, right: 5mm ), flipped:true, footer: context [ #grid( align: center, columns: (1fr, 1fr, 1fr), [#align(left, datetime.today().display("[day].[month].[year]"))], [#align(center, counter(page).display("- 1 -"))], [Thanks to Daniel for the circuit Symbols], [#align(right, image("../images/cc0.png", height: 5mm,))] ) ], ) #let pTypeFill = rgb("#dd5959").lighten(10%); #let nTypeFill = rgb("#5997dd").lighten(10%); #place(top+center, scope: "parent", float: true, heading( [Digitaltechnik] )) #let SeperatorLine = line(length: 100%, stroke: (paint: black, thickness: 0.3mm)) #let MathAlignLeft(e) = { align(left, block(e)) } #let colorBoolscheLogic = color.hsl(105.13deg, 92.13%, 75.1%) #let colorOptimierung = color.hsl(202.05deg, 92.13%, 75.1%) #let colorRealsierung = color.hsl(280deg, 92.13%, 75.1%) #let colorState = color.hsl(356.92deg, 92.13%, 75.1%) //#let colorIntegral = color.hsl(34.87deg, 92.13%, 75.1%) #let LNot(x) = math.op($overline(#x)$) #columns(4, gutter: 2mm)[ #bgBlock(fill: colorBoolscheLogic)[ #subHeading(fill: colorBoolscheLogic)[Allgemein] *Moorsches Gesetz:* 2x der Anzahl der Transistoren pro Fläche (in 2 Jahren) Flächenskalierung eines Transistors: $1/sqrt(2)$ *Kombinatorisch:* Kein Gedächtnis *(Synchrone) sequenentielle:* Mit Gedächtnis *Fan-In:* Anzahl der Inputs eines Gatters *Fan-Out:* Anzahl der Output Verbindungen eines Gatters ] #bgBlock(fill: colorBoolscheLogic)[ #subHeading(fill: colorBoolscheLogic)[Boolsche Algebra] *Dualität* $LNot(0) = 1$, $LNot(1) = 0$ *Äquivalenz* $LNot((LNot(A)))=A$\ $A dot A = A$, $A + 0 = A$ \ *Konstanz* $A dot 1 = A$ $A + 1 = 1$ *Komplementärgesetz* \ $A dot LNot(A) = 0$, $A + LNot(A) = 1$ *Kommutativgesetz* \ $A dot B = B dot A$, $A + B = B + A$ *Assoziativgesetz*\ $A dot (B dot C) = (A dot B) dot C$\ $A + (B + C) = (A + B) + C$ *Distributivgesetz*\ $A dot (B + C) = A dot B + A dot C$ \ $A + (B dot C) = (A + B) dot (A + C)$ *De Morgan*\ $LNot((A + B)) = LNot(A) dot LNot(B)$\ $LNot((A dot B)) = LNot(A) + LNot(B)$ *Absorptionsgesetz*\ $A + (A dot B) = A$\ $A dot (A + B) = A$ *Resolutionsgesetz (allgemein)*\ $X dot A + LNot(X) + B = X dot A + LNot(X) dot B + bold(A dot B)$ *Resolutionsgesetz (speziell)*\ $X dot A + LNot(X) dot A = A$\ $(X + A) dot (LNot(X) + A) = A$ ] #bgBlock(fill: colorBoolscheLogic)[ #subHeading(fill: colorBoolscheLogic)[Boolsche Funktionen] $f: {0,1}^n -> {0,1}$ Variablenmenge: ${x_0, x_1, ..., x_n}$\ Literalmenge: ${x_0, ..., x_n, LNot(x_0), ... LNot(x_n)}$ \ Einsmenge: $F = {underline(v) in {0,1}^n | f(underline(v)) = 1}$ Nullmenge: $overline(F) = {underline(v) in {0,1}^n | f(underline(v)) = 0}$ Don't-Care-Set: ${underline(v) in {0,1}^n | f(underline(v)) = *}$ Funktionsbündel: $underline(y) = underline(f)(underline(x))$ \ $underline(f): {0,1}^n -> {0,1}^m$ *Kofaktoren* aka Bit $n$ fixen\ $x_i : f_x_i = f(x_1, ..., 1, ..., x_n)$\ $overline(x)_i : f_overline(x)_i = f(x_1, ..., 0, ..., x_n)$ *Substitutionsregel* $x_i dot f = x_i dot f_x_i$ $overline(x)_i dot f = overline(x)_i dot f_overline(x)_i$ $x_i + f = x_i + f_overline(x)_i$ $overline(x)_i + f = overline(x)_i + f_x_i$ *Boolsche Expansion*\ $f(underline(x)) = x_i dot f_x_i + overline(x)_i dot f_overline(x)_i$ $f(underline(x)) = (x_i + f_overline(x)_i) dot (overline(x)_i + f_x_i)$ $overline(f(underline(x))) = overline(x)_i dot overline(f_overline(x)_i) + x_i dot overline(f_x_i)$ $overline(f(underline(x))) = (overline(x)_i + overline(f_x_i)) dot (x_i + overline(f_overline(x)_i)) $ *Eigentschaften:* tautologisch: $f(underline(x)) = 1, forall underline(x) in {0,1}^n$\ kontradiktorisch: $f(underline(x)) = 0, forall underline(x) in {0,1}^n$\ unabhängig von $x_i <=> f_x_i = f_overline(x)_i$\ abhängig von $x_i <=> f_x_i != f_overline(x)_i$\ ] #bgBlock(fill: colorOptimierung)[ #subHeading(fill: colorOptimierung)[Hauptsatz der Schaltalgebra] Jede $f(x_0, ...,x_n)$ kann als... - *Minterme $m$:* $ = LNot(x)_0 dot x_1 dot ...$\ VerODERungen von VerUNDungen\ $f(underline(x)) = m_0 + m_1 + ... + m_n$ - *Maxterme $M$:* $ = LNot(x)_0 + x_1 ü ...$\ VerUNDungen von VerODERungen\ $f(underline(x)) = m_0 dot m_1 dot ... dot m_n$ ... dargestellt werden *DNF:* Disjunktive Normalform, *Minterme* - Term $tilde.equiv$ $1$-Zeile - $LNot(x)_0 dot x_1 + x_0 dot x_1 +...$\ - $1 tilde.equiv x_0$, $0 tilde.equiv overline(x_0)$ *KNF:* Konjunktive Normalform, *Maxterme* - Term $tilde.equiv$ $0$-Zeile - $(LNot(x)_0 + LNot(x)_1) dot (x_0 + x_1) dot...$\ - $1 tilde.equiv overline(x_0)$, $0 tilde.equiv x_0$ Kanonische: In jedem Term müssen alle enthalten sein. *KDNF:* Kanonische DNF\ *KKNF:* Kanonische KNF *DMF:* Disjunktive #underline("Minimal")-Form: \ $ --> LNot(x_0)x_1 + LNot(x_1)$\ *KMF:* Konjunktive #underline("Minimal")-Form: \ $ --> (LNot(x_0) + x_1) dot LNot(x_1)$ $f(underline(x)) -->$ *KKNF* / *KDNF* mit Boolsche Expansion ] // Dotierung #bgBlock(fill: colorRealsierung)[ #table( columns: (auto, 1fr), [N-Type], [ - Dotierung: Phosphor (V) - Negative Ladgunsträger ($e^-$) - mehr Elektron als Si ], [P-Type], [ - Dotierung: Bor (III) - Postive Landsträger (Löcher) - mehr Löcher als Si ] ) #zap.circuit({ import cetz.draw : * import zap : * diode("A", (0,1.7), (3,1.7), fill: black, i: (content: $i_d$, anchor: "south")) rect((0,0),(1,1), fill: pTypeFill, stroke: none) rect((2,0),(3,1), fill: nTypeFill, stroke: none) rect((1,0), (1.5,1), fill: color.lighten(pTypeFill, 50%), stroke: none) rect((1.5,0), (2,1), fill: color.lighten(nTypeFill, 50%), stroke: none) line((2, 0), (2, 1), stroke: (dash: "dotted")) line((1, 0), (1, 1), stroke: (dash: "dotted")) line((1.5, 0), (1.5, 1), stroke: (dash: "densely-dotted")) cetz.decorations.brace((2,-0.1),(1,-0.1)) content((1.5, -0.6), "RLZ") content((2.5, 0.5), "N") content((0.5, 0.5), "P") content((1.25, 0.5), "-") content((1.75, 0.5), "+") }) #grid( columns: (1fr, 1fr), column-gutter: 6mm, align: center, [#align(center)[*NMOS*]], [#align(center)[*PMOS*]], grid.cell(inset: 2mm, align(center, zap.circuit({ import "../lib/circuit.typ" : * registerAllCustom(); fet("T", (0,0), type: "N", scale: 150%); }) ) ), grid.cell(inset: 2mm, align(center, zap.circuit({ import "../lib/circuit.typ" : * registerAllCustom(); fet("T", (0,0), type: "P", scale: 150%); }), ) ), scale( x: 75%, y: 75%, zap.circuit({ import cetz.draw : * import zap : * rect((1.5,0),(4-1.5, 0.1), fill: rgb("#535353"), stroke: none) rect((0,0),(4,-1), fill: pTypeFill, stroke: none) rect((0.5,-0),(1.5, -0.5), fill: nTypeFill, stroke: none) rect((4 - 1.5,-0),(4-0.5, -0.5), fill: nTypeFill, stroke: none) rect((1.5,-0),(2.5, -0.5), fill: none, stroke: (paint: black, dash: "dotted", thickness: 0.06)) line((3, 0.3), (3, 0)) line((1, 0.3), (1, 0)) line((2, 0.3), (2, 0.1)) cetz.decorations.brace((2.5,-0.6),(1.5,-0.6)) content((2, -1.3), "Channel") content((3, -0.25), $"n"^+$) content((1, -0.25), $"n"^+$) content((0.5, -0.75), "p") content((3, 0.5), "S") content((1, 0.5), "D") content((2, 0.5), "G") }) ), scale( x: 75%, y: 75%, zap.circuit({ import cetz.draw : * import zap : * rect((1.5,0),(4-1.5, 0.1), fill: rgb("#535353"), stroke: none) rect((0,0),(4,-1), fill: nTypeFill, stroke: none) rect((0.5,-0),(1.5, -0.5), fill: pTypeFill, stroke: none) rect((4 - 1.5,-0),(4-0.5, -0.5), fill: pTypeFill, stroke: none) rect((1.5,-0),(2.5, -0.5), fill: none, stroke: (paint: black, dash: "dotted", thickness: 0.06)) line((3, 0.3), (3, 0)) line((1, 0.3), (1, 0)) line((2, 0.3), (2, 0.1)) cetz.decorations.brace((2.5,-0.6),(1.5,-0.6)) content((2, -1.3), "Channel") content((3, -0.25), $"p"^+$) content((1, -0.25), $"p"^+$) content((0.5, -0.75), "n") content((3, 0.5), "S") content((1, 0.5), "D") content((2, 0.5), "G") }) ), ) *Drain Strom:* NMOS: $I_"Dn" = cases( gap: #0.6em, 0 & 0 < U_"GS" < U_t, beta_n (U_"GS" - U_t - U_"DS" / 2) U_"DS" quad & cases(delim: #none, U_"GS" >= U_t, 0 < U_"DS" < U_"GS" - U_t), beta_n/2 (U_"GS" - U_"th")^2 & cases(delim: #none, U_"GS" >= U_t, U_"DS" > U_"GS" - U_t) )$ PMOS: $I_"Dp" = cases( gap: #0.6em, 0 & 0 > U_"GS" > U_t, beta_p (U_"GS" - U_t - U_"DS" / 2) U_"DS" quad & cases(delim: #none, U_"GS" <= U_t, 0 > U_"DS" > U_"GS" - U_t), beta_p/2 (U_"GS" - U_"th")^2 & cases(delim: #none, U_"GS" <= U_t, U_"DS" < U_"GS" - U_t) ) $ ] // Quine McCluskey #bgBlock(fill: colorOptimierung)[ #subHeading(fill: colorOptimierung)[Quine McCluskey] ] // NMOS/PMOS #bgBlock(fill: colorRealsierung)[ #subHeading(fill: colorRealsierung)[CMOS] $hat(=)$ Complemntary MOS #table( columns: (1fr, 1fr), zap.circuit({ import zap : * import cetz.draw : content import "../lib/circuit.typ" : * set-style(wire: (stroke: (thickness: 0.025))) registerAllCustom(); fet("N0", (0,0), type: "N", angle: 90deg); fet("P0", (0,1), type: "P", angle: 90deg); wire("N0.G", (rel: (-0.1, 0)), (horizontal: (), vertical: "P0.G"), "P0.G") node("outNode", (0,0.5)) node("inNode", (-0.6,0.5)) wire((-1, 0.5), "inNode") wire((0.2, 0.5), "outNode") node("N2", (0,-0.5)) node("N2", (0,1.5)) wire((-1, -0.5), (0.5, -0.5)) wire((-1, 1.5), (0.5, 1.5)) content((-1, 0.5), scale($"X"$, 60%), anchor: "east") content((0.45, 0.5), scale($overline("X")$, 60%), anchor: "east") content((-0.9, 1.5), scale($"U"_"DD"$, 60%), anchor: "east") content((-0.9, -0.5), scale($"GND"$, 60%), anchor: "east") }), [ *Inverter* $overline(X)$ ], zap.circuit({ import zap : * import cetz.draw : content import "../lib/circuit.typ" : * set-style(wire: (stroke: (thickness: 0.025))) registerAllCustom(); fet("P0", (0.5,0.25), type: "P", angle: 90deg); fet("P1", (0.5,1.25), type: "P", angle: 90deg); fet("N0", (0,-1), type: "N", angle: 90deg); fet("N1", (1,-1), type: "N", angle: 90deg); content((-0.7, 1.75), scale($"V"_"DD"$, 60%), anchor: "east") content((-0.7, -1.5), scale($"GND"$, 60%), anchor: "east") content("N0.G", scale($"B"$, 60%), anchor: "east") content("P0.G", scale($"B"$, 60%), anchor: "east") content("N1.G", scale($"A"$, 60%), anchor: "east") content("P1.G", scale($"A"$, 60%), anchor: "east") wire((-0.75, -1.5), (1.5, -1.5)) wire((-0.75, 1.75), (1.5, 1.75)) wire("N0.S", "N1.S") node("N2", "P0.D") wire("N2", (horizontal: (), vertical: "N0.S")) node("N3", "N0.D") node("N4", "N1.D") node("N5", "P1.S") node("N6", (horizontal: (), vertical: "N0.S")) wire("N2", (horizontal: (rel: (0.5, 0)), vertical: "N2")) content((horizontal: (rel: (0.65, 0)), vertical: "N2"), scale($"Y"$, 60%)) }), [ *NOR* $overline(A +B) = Y$ ], zap.circuit({ import zap : * import cetz.draw : content import "../lib/circuit.typ" : * set-style(wire: (stroke: (thickness: 0.025))) registerAllCustom(); content((-0.7, 0.5), scale($"V"_"DD"$, 60%), anchor: "east") content((-0.7, -2.75), scale($"GND"$, 60%), anchor: "east") fet("P0", (0, 0), type: "P", angle: 90deg); fet("P1", (1, 0), type: "P", angle: 90deg); fet("N0", (0.5,-1.25), type: "N", angle: 90deg); fet("N1", (0.5,-2.25), type: "N", angle: 90deg); wire((-0.75, 0.5), (1.5, 0.5)) wire((-0.75, -2.75), (1.5, -2.75)) wire("P0.D", "P1.D") node("N2", (horizontal: "N1.D", vertical: "P0.D")) node("N3", "N0.S") wire("N2", "N3") wire("N3", (rel: (0.5, 0))) content((horizontal: (rel: (0.65, 0)), vertical: "N3"), scale($"Z"$, 60%)) node("4", "P0.S") node("4", "P1.S") node("4", "N1.D") content("N0.G", scale($"B"$, 60%), anchor: "east") content("P0.G", scale($"B"$, 60%), anchor: "east") content("N1.G", scale($"A"$, 60%), anchor: "east") content("P1.G", scale($"A"$, 60%), anchor: "east") }), [ *NAND* $overline(A dot B) = Z$ ], ) ] // CMOS #bgBlock(fill: colorRealsierung)[ #subHeading(fill: colorRealsierung)[CMOS Verzögerung] *Inverter*\ $t_("p"/"nLH") ~ (C_"L" t_"ox" L_"p/n")/(W_"p/n" mu_"p/n" epsilon(V_"DD" - abs(V_"Tpn"))) $ #grid( columns: (1fr, 1fr), [ *Steigend mit* - Last $C_L$ - Oxyddicke $T_"ox"$ - Kandlalänge $L_"p/n"$ - Schwellspannung $V_"Tp/n"$ ], [ *Sinkend mit* - Kanalweite - Landsträger Veweglichkeit $mu_"p/n"$ ], ) $t_p ~ C_L/(beta(V_"DD" - abs(V_"T")))$ $t_p ~ C_L/(W(V_"DD" - abs(V_"T")))$ ] #bgBlock(fill: colorState)[ #subHeading(fill: colorState)[Latches, Flipflops und Register] ] #bgBlock(fill: colorState)[ #subHeading(fill: colorState)[Timing] *Register Bedinungen* #cetz.canvas(length: 0.5mm, { import cetz.draw: * let cycle_time = 38 let cycle_start = cycle_time*0.8 let cycle_end = cycle_time*4 let signal_hight = 10 let switch_offset = cycle_time/13 let signal_storke = (paint: rgb("#2e2e2e"), thickness: 0.3mm) let t_c2q = 0.6 let t_setup = 0.6 let t_hold = 0.4 // clk1 line((1*cycle_time + switch_offset/2, signal_hight + 1), (1*cycle_time + switch_offset/2, -40), stroke: (paint: rgb("#0004ff"), thickness: 0.4mm, dash: "densely-dashed")) // q change line((cycle_time*(t_c2q + 1) + switch_offset/2, -15 + signal_hight + 1), (cycle_time*(t_c2q + 1) + switch_offset/2, -40), stroke: (paint: rgb("#0004ff"), thickness: 0.4mm, dash: "densely-dashed")) // d change line((cycle_time*(t_setup + 2) + switch_offset/2, -30 + signal_hight + 1), (cycle_time*(t_setup + 2) + switch_offset/2, -40), stroke: (paint: rgb("#0004ff"), thickness: 0.4mm, dash: "densely-dashed")) // clk line((cycle_time*3 + switch_offset/2, signal_hight + 1), (cycle_time*3 + switch_offset/2, -40), stroke: (paint: rgb("#0004ff"), thickness: 0.4mm, dash: "densely-dashed")) // hold time line((cycle_time*(3+t_hold) + switch_offset/2, -30 + signal_hight + 1), (cycle_time*(3+t_hold) + switch_offset/2, -40), stroke: (paint: rgb("#0004ff"), thickness: 0.4mm, dash: "densely-dashed")) content(( cycle_start -7, 5), "clk") line((cycle_start,0), (cycle_time,0), (cycle_time + switch_offset,signal_hight), (cycle_time*2, signal_hight), (cycle_time*2 + switch_offset, 0), (cycle_time*3, 0), (cycle_time*3 + switch_offset, 10), (cycle_end, signal_hight), stroke: signal_storke) translate((0, -15)) content((cycle_start -7, 5), "Q") line( (cycle_start,0), (cycle_time*(t_c2q + 1), 0), (cycle_time*(t_c2q + 1) + switch_offset, signal_hight), (cycle_time*(t_c2q + 3),signal_hight), (cycle_time*(t_c2q + 3) + switch_offset, 0), (cycle_end + switch_offset, 0), stroke: signal_storke ) line( (cycle_start,signal_hight), (cycle_time*(t_c2q + 1), signal_hight), (cycle_time*(t_c2q + 1) + switch_offset, 0), (cycle_time*(t_c2q + 3),0), (cycle_time*(t_c2q + 3) + switch_offset, signal_hight), (cycle_end + switch_offset, signal_hight), stroke: signal_storke ) translate((0, -15)) content((cycle_start -7, 5), "D") line( (cycle_start,0), (cycle_time*(t_setup + 2), 0), (cycle_time*(t_setup + 2) + switch_offset, signal_hight), (cycle_end + switch_offset, signal_hight), stroke: signal_storke ) line( (cycle_start,signal_hight), (cycle_time*(t_setup + 2), signal_hight), (cycle_time*(t_setup + 2) + switch_offset, 0), (cycle_end + switch_offset, 0), stroke: signal_storke ) }) ] #bgBlock(fill: colorState)[ #subHeading(fill: colorState)[Pipeline/Parallele Verarbeitungseinheiten] ] #bgBlock(fill: colorState)[ #subHeading(fill: colorState)[Zustandsautomaten] ] #colbreak() #bgBlock(fill: colorRealsierung)[ #subHeading(fill: colorRealsierung)[Verlustleistung/Verzögerung] $t_p ~ C_L / (V_"DD" - V_"Tn")$ $P_"stat" ~ e^(-V_T)$ $P_"dyn"~ V_"DD"^2$ *Dynamisch:* Bei Schlaten \ 1. Kapazitiv Verlustleistung $I_C$ \ 2. Querstrom Verlustleistung $I_Q$ \ #zap.circuit({ import zap : * import cetz.draw : content import "../lib/circuit.typ" : * set-style(wire: (stroke: (thickness: 0.025))) registerAllCustom(); fet("N0", (0,0), type: "N", angle: 90deg); fet("P0", (0,1), type: "P", angle: 90deg); wire("N0.G", (rel: (-0.1, 0)), (horizontal: (), vertical: "P0.G"), "P0.G") node("outNode", (0,0.5)) node("inNode", (-0.6,0.5)) wire((-1, 0.5), "inNode") wire((0.5, 0.5), "outNode") wire((0, -0.5), (0, -1)) node("N2", (0,-1)) node("N2", (0,1.5)) wire((-1, -1), (0.5, -1)) wire((-1, 1.5), (0.5, 1.5)) content((-1, 0.5), scale($"X"$, 60%), anchor: "east") content((0.8, 0.5), scale($overline("X")$, 60%), anchor: "east") content((-0.9, 1.5), scale($"U"_"DD"$, 60%), anchor: "east") content((-0.9, -1), scale($"GND"$, 60%), anchor: "east") }), - Quer/Kurzschluss Strom $i_q$ \ $P_"short" = a_01 f beta_n tau (V_"DD" - 2 V_"Tn")^3$ \ $tau$: Kurzschluss/Schaltzeit - Lade Strome des $C_L$ $i_c$ $P_"cap" = alpha_01 f C_L V_"DD"^2$ *Statisch:* Konstant - Leckstom (weil Diode) - Gatestrom ] #bgBlock(fill: colorRealsierung)[ #subHeading(fill: colorRealsierung)[Verlustleistung] $alpha = "#Schaltvorgänge"/"#Takte (#Clk Flanken)"$ $P_"cap" = alpha dot f_"clk" dot C dot U_"DD"$ ] #SIPrefixesTable ] #place(bottom, truth-table( outputs: ( ("NAND", (1, 1, 1, 0)), ("NOR", (1, 0, 0, 0)), ("XNOR", (1, 0, 0, 1)), ("XOR", (0, 1, 1, 0)), ("AND", (0, 0, 0, 1)), ("OR", (0, 1, 1, 1)), ), inputs: ("A", "B") ), float: true )