📚 AS Eduqas Engineering: Quick Reference Formula & Theorem Handbook | AS Eduqas 工程：公式定理速查手册
This handbook provides a concise summary of the essential formulas, laws, and theorems you need to master for the AS Eduqas Engineering qualification. It covers the core principles from mechanics, material science, and electrical/electronic theory, organised for quick revision and problem-solving. Use it as your go-to reference when tackling calculations, analysing systems, or preparing for examinations.
本手册简明扼要地总结了 AS Eduqas 工程学科必须掌握的关键公式、定律与定理，涵盖力学、材料科学以及电气/电子理论的核心原理，并按快速复习和解题需求编排。在计算、系统分析或备考时，它会是您最得力的速查工具。
1. Linear Motion & Kinematics | 直线运动与运动学
The equations of uniformly accelerated motion describe the relationship between displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). These are valid only when acceleration is constant.
匀加速运动方程描述了位移 (s)、初速度 (u)、末速度 (v)、加速度 (a) 和时间 (t) 之间的关系，仅在加速度恒定时适用。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
s = ½ (u + v) t
When analysing motion, always define a positive direction. Quantities like velocity and acceleration are vectors, so their signs matter. For projectile motion, split the velocity into horizontal and vertical components and apply the equations independently to each direction, with the vertical acceleration being g = 9.81 m/s² downward.
分析运动时，务必先定义正方向。速度和加速度是矢量，其正负至关重要。对于抛体运动，将速度分解为水平和竖直分量，对两个方向独立应用运动方程，其中竖直加速度为 g = 9.81 m/s² 向下。
2. Newton’s Laws of Motion | 牛顿运动定律
Newton’s First Law states that an object remains at rest or moves with constant velocity unless acted upon by a net external force. Newton’s Second Law quantifies the relationship: the net force F (N) equals mass m (kg) multiplied by acceleration a (m/s²).
牛顿第一定律指出，物体将保持静止或匀速直线运动，除非受到净外力作用。牛顿第二定律定量描述了这一关系：净外力 F（牛）等于质量 m（千克）乘以加速度 a（米/秒²）。
F = m a
Newton’s Third Law states that if body A exerts a force on body B, then body B exerts an equal and opposite force on body A. These forces act on different bodies and must not cancel when analysing a single object. Free-body diagrams are essential tools to isolate forces acting on a system.
牛顿第三定律指出，若物体 A 对物体 B 施加力，则 B 同时对 A 施加大小相等、方向相反的力。这两个力作用在不同物体上，在分析单个物体时不可相互抵消。隔离受力图是分离系统所受各力的核心工具。
Momentum p (kg·m/s) is the product of mass and velocity. In an isolated system, total momentum is conserved. Impulse (N·s) equals change in momentum and also equals average force multiplied by contact time.
动量 p (kg·m/s) 是质量与速度的乘积。在孤立系统中，总动量守恒。冲量 (N·s) 等于动量的变化，也等于平均力乘以接触时间。
p = m v
F ∆t = ∆p = m v − m u
3. Work, Energy & Power | 功、能与功率
Work done W (J) by a constant force F (N) acting over a displacement s (m) in the direction of the force is given by W = F s. When the force is applied at an angle θ to the displacement, only the component in the direction of motion does work.
恒力 F（牛）在位移方向作用距离 s（米）时所做的功 W（焦）为 W = F s。当力与位移方向成 θ 角时，只有沿运动方向的分力做功。
W = F s cos θ
Kinetic energy Eₖ (J) is the energy an object possesses due to its motion. Gravitational potential energy Eₚ (J) is the energy an object has because of its height h (m) above a reference point.
动能 Eₖ（焦）是物体因运动而具有的能量。重力势能 Eₚ（焦）是物体因相对于参考点的高度 h（米）而具有的能量。
Eₖ = ½ m v²
Eₚ = m g h
The work-energy principle states that the net work done on an object equals its change in kinetic energy. Power P (W) is the rate of doing work or transferring energy. For a constant force moving at speed v, power can also be expressed as P = F v.
功能原理指出，对一个物体所做的净功等于其动能的变化量。功率 P（瓦）是做功或传递能量的速率。当以恒力 F 并以速度 v 运动时，功率也可表示为 P = F v。
P = W / t = E / t
P = F v (for constant force and speed)
4. Stress, Strain & Young’s Modulus | 应力、应变与杨氏模量
When a material is subjected to a tensile or compressive force F (N) over a cross-sectional area A (m²), the direct stress σ (Pa) is defined as force per unit area. Direct strain ε (dimensionless) is the extension x (m) per original length L (m).
材料在截面积为 A（m²）上承受拉伸或压缩力 F（N）时，正应力 σ（帕）定义为每单位面积上的力。正应变 ε（无量纲）是伸长量 x（m）除以原长 L（m）。
σ = F / A
ε = x / L
Young’s modulus E (Pa) is a measure of stiffness, defined as the ratio of stress to strain within the linear elastic region of the stress-strain curve. The formula only applies when Hooke’s law is obeyed.
杨氏模量 E（帕）是衡量材料刚度的指标，定义为在应力-应变曲线的线弹性区域内应力与应变之比。该公式仅在满足胡克定律时适用。
E = σ / ε = (F L) / (A x)
For many engineering materials, the limit of proportionality, elastic limit, yield point, and ultimate tensile strength are key values on the stress-strain graph. The area under a force-extension graph gives the work done (strain energy) on the material.
对许多工程材料而言，比例极限、弹性极限、屈服点和极限抗拉强度是应力-应变图上的关键点。力-伸长曲线下的面积表示对材料所做的功（应变能）。
5. Fundamentals of Electrical Circuits | 电路基础
Electric charge Q (C) is related to current I (A) and time t (s). Current is the rate of flow of charge. The potential difference V (V) between two points is the energy transferred per unit charge.
电荷 Q（库）与电流 I（安）和时间 t（秒）相关。电流是电荷流动的速率。两点间的电势差 V（伏）是每单位电荷所转移的能量。
I = Q / t
V = W / Q
Electrical power P (W) can be expressed in terms of voltage and current. Combining with Ohm’s law (V = I R) gives alternative forms useful for analysing resistive circuits.
电功率 P（瓦）可以用电压和电流表示。结合欧姆定律 (V = I R) 可得到分析电阻电路时常用的其他形式。
P = V I = I² R = V² / R
Resistance R (Ω) depends on the resistivity ρ (Ω·m) of the material, its length L (m), and cross-sectional area A (m²). Resistance increases with temperature for metals; this relationship can be modelled using the temperature coefficient of resistance α (°C⁻¹).
电阻 R（欧）取决于材料的电阻率 ρ（欧·米）、长度 L（米）和截面积 A（米²）。金属的电阻随温度升高而增大，该关系可用电阻温度系数 α（°C⁻¹）来建模。
R = ρ L / A
Rₜ = R₀ (1 + α ΔT)
6. Series & Parallel Circuits | 串联与并联电路
In a series circuit, the current is the same through all components. The total resistance is the sum of individual resistances, and the supply voltage equals the sum of the p.d.s across each component. The voltage divider rule is extremely useful for finding the output voltage across one resistor in a series chain.
在串联电路中，通过所有组件的电流相同。总电阻等于各电阻之和，电源电压等于各元件两端电势差之和。分压公式对求串联链中某一电阻的输出电压极为有用。
Rₛₑᵣᵢₑₛ = R₁ + R₂ + R₃ + …
Vₒᵤₜ = Vₛ × (R₂ / (R₁ + R₂))
In a parallel circuit, the voltage is the same across each branch. The total current is the sum of the branch currents. The reciprocal of total resistance equals the sum of reciprocals of individual resistances. For two resistors in parallel, use the product-over-sum shortcut.
在并联电路中，各支路电压相同。总电流为各支路电流之和。总电阻的倒数等于各电阻倒数之和。对于两个并联电阻，可使用积和商速算公式。
1/Rₚₐᵣₐₗₗₑₗ = 1/R₁ + 1/R₂ + 1/R₃ + …
R = (R₁ R₂) / (R₁ + R₂) [for two resistors]
Kirchhoff’s current law (KCL): the algebraic sum of currents entering any node is zero. Kirchhoff’s voltage law (KVL): the sum of emfs around any closed loop equals the sum of IR drops. These laws form the foundation for network analysis.
基尔霍夫电流定律 (KCL)：流入任一节点的电流代数和为零。基尔霍夫电压定律 (KVL)：沿任一闭合回路的电动势之和等于 IR 压降之和。这些定律是网络分析的基础。
7. Capacitors & RC Circuits | 电容器与 RC 电路
A capacitor stores charge Q (C) that is directly proportional to the applied voltage V (V), with capacitance C (F) as the constant of proportionality. The energy stored in a capacitor is given by the area under the charge-voltage graph.
电容器储存的电荷量 Q（库）与外加电压 V（伏）成正比，电容 C（法）为比例常数。电容器储存的能量等于电荷-电压曲线下的面积。
Q = C V
E = ½ C V² = ½ Q V
When a capacitor charges through a resistor, the time constant τ (s) = R C dictates how quickly the voltage rises to about 63% of the supply. During discharge, the voltage decays exponentially. The same time constant applies; after τ seconds, the voltage drops to 37% of its initial value. The equations below model charging and discharging behaviour.
当电容器通过电阻充电时，时间常数 τ（秒）= R C 决定了电压上升到约电源电压 63% 的快慢。放电时电压按指数衰减。同样的时间常数适用；经过 τ 秒后，电压降至初始值的 37%。以下方程可模拟充放电行为。
Charging: Vc = V₀ (1 − e^(−t/RC))
Discharging: Vc = V₀ e^(−t/RC)
(Note: In examinations, decay equations are often used in graphical or time-ratio problems; precise exponential calculations may be provided or required using data sheets.)
（注意：考试中衰减方程常用于图像或时间比例问题；精确的指数计算可能直接给出或需使用数据表。）
8. Diodes, Rectification & Transistors | 二极管、整流与晶体管
A silicon diode conducts when forward-biased (typically above ~0.7 V) and blocks current when reverse-biased. The ideal diode model simplifies circuit analysis. In a half-wave rectifier, only one half of the AC cycle is passed, while a full-wave bridge rectifier uses four diodes to utilise both half-cycles, producing a smoother DC output.
硅二极管正向偏置（通常高于 ~0.7 V）时导通，反向偏置时阻断电流。理想二极管模型可简化电路分析。在半波整流器中，只有交流周期的一半通过；全波桥式整流器使用四只二极管令两个半周均被利用，产生更平滑的直流输出。
A bipolar junction transistor (BJT) operates in three regions: cut-off, active, and saturation. In an NPN transistor, a small base current I_B controls a larger collector current I_C. The DC current gain h_FE is the ratio I_C / I_B. For switching applications, the transistor is driven to saturation where V_CE is very low (≈0.2 V).
双极结型晶体管 (BJT) 工作于三个区域：截止区、放大区和饱和区。在 NPN 晶体管中，微小的基极电流 I_B 控制较大的集电极电流 I_C。直流电流增益 h_FE 为 I_C / I_B 之比。在开关应用中，晶体管被驱至饱和，此时 V_CE 极低（约 0.2 V）。
I_C = h_FE × I_B
When using a transistor as a switch, calculate the base resistor R_B to provide sufficient base current, taking into account the base-emitter voltage V_BE (≈0.7 V for silicon): R_B = (V_input − V_BE) / I_B.
当晶体管用作开关时，需计算基极电阻 R_B 以提供足够的基极电流，并计入基射极电压 V_BE（硅管约 0.7 V）：R_B = (V_input − V_BE) / I_B。
9. Logic Gates & Boolean Algebra | 逻辑门与布尔代数
Digital systems use logic gates whose inputs and outputs are binary (0 or 1). The basic gates are AND, OR, NOT, NAND, NOR, XOR, and XNOR. Truth tables define the output for all possible input combinations.
数字系统使用输入输出为二进制（0 或 1）的逻辑门。基本门包括与门、或门、非门、与非门、或非门、异或门和同或门。真值表定义了所有可能输入组合下的输出。
| Gate | Boolean Expression | Description |
|---|---|---|
| AND | Q = A · B | Output 1 only if all inputs are 1 |
| OR | Q = A + B | Output 1 if at least one input is 1 |
| NOT | Q = A̅ (or ¬A) | Inverts the input |
| NAND | Q = ¬(A · B) | AND followed by NOT |
| NOR | Q = ¬(A + B) | OR followed by NOT |
| XOR | Q = A ⊕ B | Output 1 when inputs are different |
Boolean algebra simplifies logic expressions. Key laws include commutativity, associativity, distributivity, and De Morgan’s theorems: ¬(A · B) = ¬A + ¬B and ¬(A + B) = ¬A · ¬B. These are essential for minimising gate count in a circuit design.
布尔代数可用于化简逻辑表达式。核心定律包括交换律、结合律、分配律以及德·摩根定理：¬(A · B) = ¬A + ¬B 与 ¬(A + B) = ¬A · ¬B。这些对于减少电路设计中的门数量至关重要。
10. Operational Amplifiers (Op-Amps) | 运算放大器
An ideal op-amp has infinite open-loop gain, infinite input impedance, and zero output impedance. In linear circuits, negative feedback is used to produce stable, predictable gain. The two basic configurations are the inverting and non-inverting amplifier.
理想运放具有无穷大开环增益、无穷大输入阻抗和零输出阻抗。在线性电路中，负反馈用于产生稳定、可预测的增益。两种基本组态为反相放大器和同相放大器。
Inverting: Vₒᵤₜ = − (R_f / R_in) V_in
Non-inverting: Vₒᵤₜ = (1 + R_f / R₁) V_in
The summing amplifier produces an output proportional to the sum of several weighted input voltages. The comparator, often used without feedback, switches the output high or low depending on which input is larger. Op-amp circuits are essential building blocks in sensor signal conditioning and control systems.
求和放大器输出的电压与多个加权输入电压之和成正比。比较器通常无反馈，根据两个输入的大小将输出切换为高或低。运放电路是传感器信号调理和控制系统中的基本构建模块。
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