📚 A-Level CAIE Engineering: Quick Reference Handbook of Formulae and Theorems | A-Level CAIE 工程：公式定理速查手册
Engineering at A-Level requires a solid command of essential formulae, theorems, and principles that span across mechanics, materials, thermodynamics, and electrical systems. This quick reference handbook consolidates the most vital equations and concepts you need to master for the CAIE Engineering syllabus, presented in a clear, exam-focused format for rapid revision and deep understanding.
A-Level 阶段的工程学科要求学生熟练掌握涵盖力学、材料学、热力学和电气系统的核心公式、定理与原理。本速查手册汇集了 CAIE 工程教学大纲中最关键的方程式与概念，并以清晰、面向考试的形式呈现，便于快速复习和深度理解。
1. Fundamental Mechanics and Newton’s Laws | 基础力学与牛顿定律
Newton’s three laws of motion form the bedrock of classical mechanics. The First Law states that an object remains at rest or in uniform motion unless acted upon by a resultant force. The Second Law quantifies this relationship as F = ma, where F is the resultant force in newtons (N), m is mass in kilograms (kg), and a is acceleration in metres per second squared (m/s²). The Third Law states that for every action there is an equal and opposite reaction, meaning forces always occur in pairs acting on different bodies.
牛顿三大运动定律构成了经典力学的基石。第一定律指出，除非受到合外力的作用，否则物体将保持静止或匀速直线运动状态。第二定律将这一定量关系表述为 F = ma，其中 F 为合外力（单位牛顿 N），m 为质量（单位千克 kg），a 为加速度（单位米每二次方秒 m/s²）。第三定律指出，每一个作用力都有一个大小相等、方向相反的反作用力，这意味着力总是成对出现并作用在不同的物体上。
The principle of conservation of momentum is derived directly from Newton’s laws and is essential for analysing collisions and explosions. For a system with no external forces, the total momentum before an event equals the total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, where u represents initial velocities and v represents final velocities. In perfectly elastic collisions, kinetic energy is also conserved, whereas in perfectly inelastic collisions, the bodies stick together and move with a common velocity.
动量守恒原理直接源自牛顿定律，是分析碰撞和爆炸问题的关键。对于一个没有外力作用的系统，事件发生前的总动量等于事件发生后的总动量：m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂，其中 u 表示初速度，v 表示末速度。在完全弹性碰撞中，动能也守恒；而在完全非弹性碰撞中，物体粘在一起并以共同速度运动。
Friction forces oppose motion and are calculated using the coefficient of friction μ. The maximum static friction is given by Fₛ ≤ μₛR, where R is the normal reaction force. The kinetic friction experienced during sliding is Fₖ = μₖR. The coefficient of friction is a dimensionless quantity that depends on the materials in contact and the surface roughness.
摩擦力阻碍运动，可使用摩擦系数 μ 进行计算。最大静摩擦力由公式 Fₛ ≤ μₛR 给出，其中 R 为法向反力。滑动过程中经历的动摩擦力为 Fₖ = μₖR。摩擦系数是一个无量纲量，取决于接触材料及其表面粗糙度。
2. Stress, Strain, and Material Properties | 应力、应变与材料性质
Stress is defined as force per unit area, expressed as σ = F / A, where σ is the stress in pascals (Pa), F is the applied force, and A is the cross-sectional area. Strain is the measure of deformation representing the displacement between particles in the body relative to a reference length, given by ε = ΔL / L₀, where ΔL is the change in length and L₀ is the original length. Strain is dimensionless.
应力定义为单位面积上的力，表达式为 σ = F / A，其中 σ 为应力（单位帕斯卡 Pa），F 为施加的力，A 为横截面积。应变是描述变形的量度，表示物体内粒子相对于参考长度的位移，由公式 ε = ΔL / L₀ 给出，其中 ΔL 为长度变化量，L₀ 为原始长度。应变为无量纲量。
Young’s modulus E describes the stiffness of a material and is the ratio of stress to strain within the elastic limit, according to Hooke’s Law: E = σ / ε = (F L₀) / (A ΔL). A material with a higher Young’s modulus is stiffer and deforms less under a given load. The elastic limit is the point beyond which permanent plastic deformation occurs, and the yield point marks the onset of significant plastic strain.
杨氏模量 E 描述了材料的刚度，根据胡克定律，它是弹性限度内应力与应变之比：E = σ / ε = (F L₀) / (A ΔL)。杨氏模量越高的材料刚度越大，在给定载荷下变形越小。弹性限度是材料发生永久塑性变形的临界点，屈服点则标志了显著塑性应变的开始。
The ultimate tensile strength (UTS) is the maximum stress a material can withstand while being stretched before necking or fracturing. Ductile materials such as mild steel exhibit a large plastic region and necking before fracture, whereas brittle materials such as cast iron fail suddenly with little plastic deformation. The area under a stress-strain curve represents the energy absorbed per unit volume, known as toughness.
极限抗拉强度（UTS）是材料在拉伸过程中发生颈缩或断裂前所能承受的最大应力。韧性材料（如低碳钢）在断裂前表现出大范围的塑性区和颈缩现象，而脆性材料（如铸铁）则在几乎不发生塑性变形的情况下突然失效。应力-应变曲线下的面积代表单位体积所吸收的能量，称为韧性。
The factor of safety is a crucial design parameter calculated as: Factor of Safety = Ultimate Stress / Working Stress. This ratio ensures that structures can accommodate unexpected loads, material imperfections, and in-service degradation. Typical factors of safety range from 1.5 to 5, depending on the application and the consequences of failure.
安全系数是一个关键的设计参数，计算公式为：安全系数 = 极限应力 / 工作应力。该比值确保结构能够应对意外载荷、材料缺陷和使用过程中的退化。典型的安全系数范围为 1.5 到 5，具体取决于应用场景和失效后果的严重程度。
3. Kinematics and Projectile Motion | 运动学与抛体运动
The equations of uniformly accelerated motion, often called the SUVAT equations, are fundamental tools for analysing linear motion. The five variables are: s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). The four key equations are: v = u + at, s = ut + ½at², s = ½(u + v)t, and v² = u² + 2as. These equations apply only when acceleration is constant.
匀加速运动方程常被称为 SUVAT 方程，是分析直线运动的基本工具。五个变量分别是：s（位移）、u（初速度）、v（末速度）、a（加速度）和 t（时间）。四个关键方程为：v = u + at，s = ut + ½at²，s = ½(u + v)t，以及 v² = u² + 2as。这些方程仅在加速度恒定时适用。
Projectile motion is analysed by resolving the motion into independent horizontal and vertical components. The horizontal component of velocity remains constant (assuming no air resistance) as u cos θ, while the vertical component follows accelerated motion under gravity: vᵧ = u sin θ – gt. The maximum height reached is H = (u² sin² θ) / (2g), the range on horizontal ground is R = (u² sin 2θ) / g, and the time of flight is T = (2u sin θ) / g.
抛体运动通过将运动分解为相互独立的水平分量和竖直分量进行分析。速度的水平分量保持恒定（假设无空气阻力），为 u cos θ；竖直分量则在重力作用下做加速运动：vᵧ = u sin θ – gt。所达到的最大高度为 H = (u² sin² θ) / (2g)，水平地面上的射程为 R = (u² sin 2θ) / g，飞行时间为 T = (2u sin θ) / g。
Angular motion has close parallels with linear motion. Angular displacement θ is measured in radians, angular velocity ω in rad/s, and angular acceleration α in rad/s². The analogous equations are: ω = ω₀ + αt, θ = ω₀t + ½αt², θ = ½(ω₀ + ω)t, and ω² = ω₀² + 2αθ. The conversion between linear and angular quantities is given by s = rθ, v = rω, and a = rα, where r is the radius of rotation.
角运动与直线运动有着密切的对应关系。角位移 θ 以弧度为单位，角速度 ω 以 rad/s 为单位，角加速度 α 以 rad/s² 为单位。对应的方程为：ω = ω₀ + αt，θ = ω₀t + ½αt²，θ = ½(ω₀ + ω)t，以及 ω² = ω₀² + 2αθ。线量与角量之间的转换由以下公式给出：s = rθ，v = rω，以及 a = rα，其中 r 为旋转半径。
4. Energy, Work, and Power | 能量、功与功率
Work is done when a force moves its point of application through a distance. The work done W is calculated as W = F × d × cos θ, where θ is the angle between the force vector and the direction of displacement. When the force is parallel to the displacement, W = Fd. The SI unit of work and energy is the joule (J), where 1 J = 1 N·m = 1 kg·m²/s².
力使其作用点移动一段距离时即做功。所做功 W 的计算公式为 W = F × d × cos θ，其中 θ 为力矢量与位移方向之间的夹角。当力与位移平行时，W = Fd。功和能量的国际单位制单位是焦耳（J），1 J = 1 N·m = 1 kg·m²/s²。
Kinetic energy is the energy possessed by a body due to its motion, given by Eₖ = ½mv². Gravitational potential energy is the energy stored by virtue of a body’s position in a gravitational field: Eₚ = mgh, where h is the height above a reference level and g is the gravitational field strength (9.81 m/s² on Earth). The principle of conservation of mechanical energy states that in the absence of dissipative forces, the sum of kinetic and potential energy remains constant.
动能是物体因其运动而具有的能量，由公式 Eₖ = ½mv² 给出。重力势能是物体因其在重力场中的位置而储存的能量：Eₚ = mgh，其中 h 为相对于参考水平面的高度，g 为重力场强（地球为 9.81 m/s²）。机械能守恒原理指出，在没有耗散力的情况下，动能和势能的总和保持恒定。
Power is the rate of doing work, defined as P = W / t or P = Fv for motion at constant velocity. The SI unit of power is the watt (W), where 1 W = 1 J/s. Efficiency is the ratio of useful output power to total input power, often expressed as a percentage: Efficiency = (Pₒᵤₜ / Pᵢₙ) × 100%. In real systems, efficiency is always less than 100% due to energy losses from friction, heat, and sound.
功率是做功的速率，定义为 P = W / t，对于匀速运动也可表示为 P = Fv。功率的国际单位制单位是瓦特（W），1 W = 1 J/s。效率是有用输出功率与总输入功率之比，通常以百分比表示：效率 = (Pₒᵤₜ / Pᵢₙ) × 100%。在实际系统中，由于摩擦、发热和噪声造成的能量损失，效率始终低于 100%。
5. Statics, Moments, and Equilibrium | 静力学、力矩与平衡
A body is in static equilibrium when the net force and the net moment acting on it are both zero. For coplanar forces, this means ΣFₓ = 0, ΣFᵧ = 0, and ΣM = 0 about any point. The moment of a force about a pivot is the product of the force magnitude and the perpendicular distance from the line of action to the pivot: M = F × d. The SI unit of moment is the newton-metre (N·m).
当作用在物体上的合外力和合力矩均为零时，物体处于静力平衡状态。对于共面力，这意味着 ΣFₓ = 0、ΣFᵧ = 0，且对任意点均有 ΣM = 0。力对某支点的力矩等于力的大小乘以作用线到支点的垂直距离：M = F × d。力矩的国际单位制单位为牛顿·米（N·m）。
The principle of moments states that for a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments about any pivot point. This principle is used extensively in analysing levers, beams, and structures. A couple is a pair of equal and opposite parallel forces separated by a perpendicular distance, producing a pure turning effect with a moment equal to one force multiplied by the separation distance.
力矩原理指出，当物体处于转动平衡状态时，绕任意支点的顺时针力矩总和等于逆时针力矩总和。该原理广泛应用于杠杆、横梁和结构的分析中。力偶是一对大小相等、方向相反的平行力，它们之间存在垂直间距，能够产生纯转动效应，其力矩等于其中一个力乘以间距。
Free-body diagrams are essential tools for resolving forces and analysing equilibrium. Each force vector is represented by an arrow with its magnitude and direction clearly labelled. Beams subjected to loads develop shear forces and bending moments that vary along their length. A simply supported beam with a centrally applied point load W experiences a maximum bending moment at the centre equal to WL / 4, where L is the beam span.
自由体图是分解力和分析平衡的重要工具。每个力矢量均用箭头表示，并清晰标注其大小和方向。受载梁沿其长度方向产生的剪力和弯矩是变化的。一根简支梁在跨中承受集中载荷 W 时，其中央位置承受的最大弯矩为 WL / 4，其中 L 为梁的跨度。
6. Fluid Mechanics and Hydraulic Systems | 流体力学与液压系统
Pressure in a fluid is defined as force per unit area: P = F / A. The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m². In a static fluid, the pressure at a depth h below the surface is given by P = ρgh, where ρ is the fluid density and g is the gravitational field strength. This hydrostatic pressure acts equally in all directions at a given depth and increases linearly with depth.
流体中的压强定义为单位面积上的力：P = F / A。压强的国际单位制单位为帕斯卡（Pa），1 Pa = 1 N/m²。在静止流体中，表面以下深度 h 处的压强由公式 P = ρgh 给出，其中 ρ 为流体密度，g 为重力场强。在给定深度处，该液体静压力在所有方向上均相等，并随深度线性增加。
Pascal’s principle states that a pressure change applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of its container. This is the basis for hydraulic systems, where a small force applied to a small-area piston creates a large force on a large-area piston: F₁/A₁ = F₂/A₂. The mechanical advantage is the ratio of output force to input force: MA = A₂/A₁, which illustrates how hydraulics provide force amplification.
帕斯卡原理指出，施加在密闭流体上的压强变化会毫无衰减地传递到流体的每一部分以及容器壁上。这是液压系统的基础，施加在小面积活塞上的小力能够在大面积活塞上产生大力：F₁/A₁ = F₂/A₂。机械增益是输出力与输入力之比：MA = A₂/A₁，这说明了液压系统如何实现力放大。
Archimedes’ principle states that a body wholly or partially submerged in a fluid experiences an upthrust equal to the weight of the fluid displaced. This buoyant force F_b = ρ_fluid × V_displaced × g. An object floats when the upthrust equals its weight; if the upthrust is less than the weight, the object sinks. The principle is critical in ship design and buoyancy calculations.
阿基米德原理指出，完全或部分浸没在流体中的物体会受到一个等于所排开流体重量大小的上浮力。该浮力 F_b = ρ_fluid × V_displaced × g。当上浮力等于物体的重量时，物体漂浮；若上浮力小于重量，则物体下沉。该原理在船舶设计和浮力计算中至关重要。
For an ideal fluid in steady flow, the continuity equation states that the mass flow rate is constant: A₁v₁ = A₂v₂, assuming the fluid is incompressible. Bernoulli’s equation for inviscid, incompressible flow is P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, combining pressure energy, kinetic energy per unit volume, and potential energy per unit volume. This equation explains phenomena such as the Venturi effect and aerodynamic lift.
对于稳定流动的理想流体，连续性方程指出质量流率恒定：A₁v₁ = A₂v₂，假定流体不可压缩。适用于无黏性不可压缩流动的伯努利方程为 P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂，该方程综合了压能、单位体积动能和单位体积势能。该方程解释了文丘里效应和空气动力升力等现象。
7. Thermodynamics and Heat Transfer | 热力学与热传递
The First Law of Thermodynamics is a statement of energy conservation: ΔU = Q – W, where ΔU is the change in internal energy of the system, Q is the heat added to the system, and W is the work done by the system. For a thermodynamic cycle, the net work output equals the net heat input. The Second Law states that heat cannot spontaneously flow from a colder body to a hotter body and introduces the concept of entropy, which always increases in an isolated system.
热力学第一定律是对能量守恒的表述：ΔU = Q – W，其中 ΔU 为系统内能的变化量，Q 为加入系统的热量，W 为系统对外做的功。对于一个热力学循环，净输出功等于净输入热。第二定律指出，热量不能自发地从较冷的物体流向较热的物体，并引入了熵的概念——在孤立系统中，熵总是增加的。
The specific heat capacity c is the energy required to raise the temperature of 1 kg of a substance by 1 K: Q = mcΔθ, where Δθ is the temperature change. The specific latent heat L is the energy required to change the state of 1 kg of a substance without a change in temperature: Q = mL. For sensible heating and cooling, the quantity of heat transferred is proportional to the mass, the specific heat capacity, and the temperature change.
比热容 c 是将 1 kg 物质温度升高 1 K 所需的能量：Q = mcΔθ，其中 Δθ 为温度变化量。比潜热 L 是在温度不变的情况下，使 1 kg 物质改变物态所需的热量：Q = mL。对于显热加热和冷却过程，传递的热量与物质的质量、比热容以及温度变化量成正比。
Three primary modes of heat transfer govern thermal systems. Conduction, governed by Fourier’s Law, describes heat flow through a material: Q/t = kA(ΔT)/d, where k is the thermal conductivity, A is the cross-sectional area, ΔT is the temperature difference, and d is the thickness. Convection involves the transfer of heat by fluid motion, described by Newton’s Law of Cooling: Q/t = hA(T_s – T_f), where h is the convective heat transfer coefficient. Radiation is the emission of electromagnetic waves, governed by the Stefan-Boltzmann Law: P = εσAT⁴, where ε is emissivity and σ is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²K⁴).
三种主要的传热方式支配着热力系统。热传导遵循傅里叶定律，描述了热量通过材料的流动：Q/t = kA(ΔT)/d，其中 k 为导热系数，A 为横截面积，ΔT 为温差，d 为厚度。对流涉及通过流体运动传递热量，遵循牛顿冷却定律：Q/t = hA(T_s – T_f)，其中 h 为对流传热系数。辐射是电磁波的发射，遵循斯特藩-玻尔兹曼定律：P = εσAT⁴，其中 ε 为发射率，σ 为斯特藩-玻尔兹曼常数（5.67 × 10⁻⁸ W/m²K⁴）。
The ideal gas law links the macroscopic properties of a gas: PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant (8.31 J/mol·K), and T is the absolute temperature in kelvin. For a fixed mass of gas, the combined gas law states that (P₁V₁)/T₁ = (P₂V₂)/T₂, with Boyle’s Law (P ∝ 1/V at constant T), Charles’s Law (V ∝ T at constant P), and the Pressure Law (P ∝ T at constant V) as special cases.
理想气体定律将气体的宏观性质联系起来：PV = nRT，其中 P 为压强，V 为体积，n 为摩尔数，R 为普适气体常数（8.31 J/mol·K），T 为开尔文绝对温度。对于固定质量的气体，组合气体定律指出 (P₁V₁)/T₁ = (P₂V₂)/T₂，特殊情况包括玻意耳定律（温度恒定，P ∝ 1/V）、查理定律（压强恒定，V ∝ T）和压力定律（体积恒定，P ∝ T）。
8. Electrical Engineering Fundamentals | 电气工程基础
Ohm’s Law is the foundational relationship in circuit analysis: V = IR, where V is the potential difference in volts (V), I is the current in amperes (A), and R is the resistance in ohms (Ω). This law assumes the resistance is constant, which holds for ohmic conductors at constant temperature. Kirchhoff’s Current Law (KCL) states that the sum of currents entering a junction equals the sum of currents leaving it: ΣI_in = ΣI_out. Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of voltages around any closed loop is zero: ΣV = 0.
欧姆定律是电路分析中的基础关系式：V = IR，其中 V 为电位差（伏特 V），I 为电流（安培 A），R 为电阻（欧姆 Ω）。该定律假定电阻恒定，这对恒温条件下的欧姆导体是成立的。基尔霍夫电流定律（KCL）指出，流入一个节点的电流之和等于流出该节点的电流之和：ΣI_in = ΣI_out。基尔霍夫电压定律（KVL）指出，沿任意闭合回路，电压的代数和为零：ΣV = 0。
Resistors in series and parallel follow specific rules. For series resistors: R_total = R₁ + R₂ + R₃ + … The current is the same through each resistor, but the voltage divides proportionally to resistance. For parallel resistors: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + … The voltage is the same across each branch, but the current divides inversely proportional to resistance. The power dissipated in a resistor can be calculated using P = IV = I²R = V²/R.
串联和并联电阻遵循特定规则。对于串联电阻：R_total = R₁ + R₂ + R₃ + …，每个电阻中通过的电流相同，但电压按电阻比例分配。对于并联电阻：1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …，各支路电压相同，但电流按电阻反比例分配。电阻中耗散的功率可使用公式 P = IV = I²R = V²/R 进行计算。
Capacitance C is the ability of a component to store electric charge, defined as C = Q/V, where Q is the charge in coulombs (C) and V is the potential difference. The SI unit of capacitance is the farad (F). The energy stored in a capacitor is E = ½CV² = ½QV. For capacitors in parallel, C_total = C₁ + C₂ + C₃ + …, while for capacitors in series, 1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + …
电容 C 是元件储存电荷的能力，定义为 C = Q/V，其中 Q 为电荷量（库仑 C），V 为电位差。电容的国际单位制单位是法拉（F）。电容器储存的能量为 E = ½CV² = ½QV。对于并联电容：C_total = C₁ + C₂ + C₃ + …，而对于串联电容：1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + …
9. AC Circuits and Electromagnetism | 交流电路与电磁学
Alternating current (AC) varies sinusoidally with time: V(t) = V₀ sin(ωt) and I(t) = I₀ sin(ωt – φ), where V₀ and I₀ are the peak values, ω = 2πf is the angular frequency, and φ is the phase angle between voltage and current. The root mean square (RMS) value of an AC signal is V_rms = V₀/√2 and I_rms = I₀/√2, representing the equivalent DC heating effect.
交流电（AC）随时间呈正弦变化：V(t) = V₀ sin(ωt)，I(t) = I₀ sin(ωt – φ)，其中 V₀ 和 I₀ 为峰值，ω = 2πf 为角频率，φ 为电压与电流之间的相位角。交流信号的均方根（RMS）值为 V_rms = V₀/√2，I_rms = I₀/√2，其代表了等效的直流热效应。
In AC circuits, reactance describes the opposition to current flow from capacitors and inductors. Capacitive reactance is Xc = 1/(ωC) = 1/(2πfC), causing the current to lead the voltage by 90°. Inductive reactance is Xl = ωL = 2πfL, causing the current to lag the voltage by 90°. Impedance Z combines resistance and reactance: Z = √(R² + X²). The phase angle is given by tan φ = X/R.
在交流电路中，电抗描述了电容和电感对电流的阻碍作用。容抗为 Xc = 1/(ωC) = 1/(2πfC)，导致电流领先电压 90°。感抗为 Xl = ωL = 2πfL，导致电流滞后电压 90°。阻抗 Z 综合了电阻和电抗：Z = √(R² + X²)。相位角由公式 tan φ = X/R 给出。
Faraday’s Law of electromagnetic induction states that the induced EMF in a circuit equals the negative rate of change of magnetic flux linkage: ε = -N (dΦ/dt), where N is the number of turns and Φ is the magnetic flux. Lenz’s Law, indicated by the negative sign, states that the induced EMF opposes the change that caused it. The magnetic flux through an area A in a uniform magnetic field B is Φ = BA cos θ.
法拉第电磁感应定律指出，电路中感应的电动势等于磁通链变化率的负值：ε = -N (dΦ/dt)，其中 N 为匝数，Φ 为磁通量。由负号体现的楞次定律指出，感应电动势的方向反抗引起该感应电动势的变化。在均匀磁场 B 中，穿过面积 A 的磁通量为 Φ = BA cos θ。
10. Digital Electronics and Logic Gates | 数字电子与逻辑门
Digital systems use binary signals (0 and 1) to represent information. The fundamental logic gates are NOT (inverter), AND, OR, NAND, NOR, XOR, and XNOR. Boolean algebra provides the mathematical framework for analysing and simplifying digital circuits. Key identities include: A + 0 = A, A · 1 = A, A + ⏨A = 1, A · ⏨A = 0, and De Morgan’s Theorems: ⏨⏨⏨⏨(A · B) = ⏨A + ⏨B and ⏨⏨⏨⏨(A + B) = ⏨A · ⏨B.
数字系统使用二进制信号（0 和 1）来表示信息。基本的逻辑门包括 NOT（非门）、AND（与门）、OR（或门）、NAND（与非门）、NOR（或非门）、XOR（异或门）和 XNOR（同或门）。布尔代数为分析和简化数字电路提供了数学框架。关键恒等式包括：A + 0 = A，A · 1 = A，A + ⏨A = 1，A · ⏨A = 0，以及德摩根定理：⏨⏨⏨⏨(A · B) = ⏨A + ⏨B 和 ⏨⏨⏨⏨(A + B) = ⏨A · ⏨B。
Combinational logic circuits produce outputs that depend only on the current inputs. Common combinational circuits include multiplexers (data selectors), demultiplexers, encoders, and decoders. The truth table exhaustively lists all possible input combinations and their corresponding outputs. Karnaugh maps (K-maps) provide a graphical method for simplifying Boolean expressions and minimising the number of logic gates required.
组合逻辑电路产生的输出仅取决于当前输入。常见的组合电路包括多路复用器（数据选择器）、多路分配器、编码器和解码器。真值表详尽列出了所有可能的输入组合及其对应的输出。卡诺图（K-map）提供了一种图形化方法，用于简化布尔表达式，并最小化所需逻辑门的数量。
Sequential logic circuits, unlike combinational circuits, have memory because their outputs depend on both current inputs and the history of previous inputs. Flip-flops are the fundamental building blocks, with the D-type and JK flip-flops being the most common. These bistable devices store one bit of information and switch states in response to clock pulses. Counters and shift registers are constructed using interconnected flip-flops.
与组合电路不同，时序逻辑电路具有记忆功能，因为其输出既取决于当前输入，也取决于过去输入的历史。触发器是基本构建模块，最常见的是 D 型和 JK 型触发器。这些双稳态器件存储一位信息，并响应时钟脉冲进行状态切换。计数器和移位寄存器是由相互连接的触发器构成的。
11. Engineering Materials and Manufacturing Processes | 工程材料与制造工艺
Engineering materials are broadly classified into metals (ferrous and non-ferrous), polymers (thermoplastics and thermosets), ceramics, and composites. Ferrous metals, including mild steel (0.15–0.3% carbon), medium carbon steel (0.3–0.6% carbon), and high carbon steel (0.6–1.4% carbon), have increasing strength but decreasing ductility with higher carbon content. Alloying elements such as chromium, nickel, and molybdenum enhance specific properties like corrosion resistance and high-temperature strength.
工程材料大致分为金属（铁基和非铁基）、聚合物（热塑性和热固性）、陶瓷和复合材料。铁基金属包括低碳钢（含碳量 0.15–0.3%）、中碳钢（含碳量 0.3–0.6%）和高碳钢（含碳量 0.6–1.4%），随着含碳量的增加，强度增加但韧性降低。铬、镍、钼等合金元素能够增强特定性能，如耐腐蚀性和高温强度。
Heat treatment processes alter the microstructure and mechanical properties of metals. Annealing involves heating a metal to a specific temperature and then cooling it slowly to soften the material and relieve internal stresses. Quenching involves rapid cooling from a high temperature to produce a hard martensitic structure. Tempering is applied after quenching to reduce brittleness while retaining most of the hardness. Case hardening introduces carbon or nitrogen into the surface layer to create a hard, wear-resistant exterior with a tough core.
热处理工艺能够改变金属的微观结构和力学性能。退火是将金属加热到特定温度，然后缓慢冷却，以软化材料并消除内应力。淬火是从高温快速冷却，以产生坚硬的马氏体组织。回火在淬火后进行，旨在降低脆性，同时保留大部分硬度。表面硬化通过在表层渗入碳或氮，形成坚硬耐磨的外壳和韧性良好的芯部。
Manufacturing processes are selected based on the material, production volume, and required tolerances. Casting involves pouring molten metal into a mould; sand casting is economical for low volumes, while die casting suits high-volume production of non-ferrous metals. Forming processes such as rolling, forging, and extrusion shape metal through plastic deformation, often improving grain structure and strength. Machining processes including turning, milling, and drilling remove material to achieve precise dimensions and surface finishes.
制造工艺根据材料、产量和所需公差进行选择。铸造是将熔融金属倒入模具中；砂铸适合小批量生产且经济性好，而压铸则适用于有色金属的大批量生产。成形工艺如轧制、锻造和挤压，通过塑性变形使金属成形，通常能够改善晶粒结构并提高强度。机械加工工艺包括车削、铣削和钻削，通过去除材料来实现精确的尺寸和表面光洁度。
12. Control Systems and Signal Processing | 控制系统与信号处理
Control systems regulate the behaviour of dynamic systems to achieve desired outputs. An open-loop system applies a predetermined input without monitoring
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