WM1AD formula sheet

Week 1: circuit fundamentals

Want	Formula	Guide
Charge	$Q = I t$	Electrical quantities
Ohm’s law	$V = I R$	Electrical quantities
Resistance of a conductor	$R = ρ L / A$	Electrical quantities
Power	$P = V I = I^{2} R = V^{2} / R$	Electrical quantities
Energy	$E = P t$ (J = W s; kWh = kW × h)	Electrical quantities
Efficiency	$η = P_{o u t} / P_{in}$ ; stages multiply	Electrical quantities
KCL / KVL	$\sum I = 0$ at a node · $\sum V = 0$ round a loop	Kirchhoff’s laws
Series / parallel	$R_{1} + R_{2}$ · $R_{1} R_{2} / (R_{1} + R_{2})$	Kirchhoff’s laws
Voltage divider	$V_{2} = V R_{2} / (R_{1} + R_{2})$	Kirchhoff’s laws
Power balance	$\sum I^{2} R = \sum V I$	Kirchhoff’s laws
Bridge balance	$R_{1} / R_{2} = R_{3} / R_{4}$	Wheatstone bridge and thermistors
Thermistor	$R = R_{0} exp [B (1/ T - 1/ T_{0})]$ , T in K	Wheatstone bridge and thermistors

Week 2: matrices, mesh analysis and transients

Want	Formula	Guide
2×2 determinant	$a d - b c$	Matrices and Cramer’s rule
Cramer’s rule	$x_{k} = Δ_{k} /Δ$	Matrices and Cramer’s rule
Mesh matrix	diagonal = total R round mesh; off-diagonal = −shared R	Mesh analysis
Time constant	$τ = R C$	RC transients
Charging	$u_{C} = V (1 - e^{- t / τ})$ , $i = (V / R) e^{- t / τ}$	RC transients
Discharging	$u_{C} = V_{0} e^{- t / τ}$	RC transients
Stored energy	$E = \frac{1}{2} C V^{2}$	RC transients
Capacitors	parallel add; series $1/ C = 1/ C_{1} + 1/ C_{2}$	RC transients

Week 3: power electronics

Want	Formula	Guide
Peak from RMS	$V_{p k} = 2 V_{R M S}$	AC values
Average	$V_{a v g} = 0.637 V_{p k} = 0.9 V_{R M S}$	AC values
Bridge heat	$P_{D (F B)} = 1.8 V_{F} I_{D, R M S}$	Bridge rectifiers
Capacitor peak	$V_{C, ma x} = V_{p k} - 2 V_{F}$	Bridge rectifiers
Ripple	$Δ V = I / (2 F_{g r i d} C)$	Bridge rectifiers
PWM average	$V_{a v g} = D V_{D D}$ , $D = t_{o n} / T$	PWM
Conduction loss	$P_{co n d} = R_{D S (o n)} I_{D}^{2}$	MOSFET power loss
Switching energy	$E_{O N} = \frac{V _{D D} I _{D}}{2} (t_{d (o n)} + t_{r})$ , $E_{O F F} = \frac{V _{D D} I _{D}}{2} (t_{d (o f f)} + t_{f})$	MOSFET power loss
Switching loss	$P_{S W} = (E_{O N} + E_{O F F}) F_{S W}$	MOSFET power loss
Junction temperature	$T_{J} = T_{amb} + P (R_{t h 1} + R_{t h 2} + R_{t h 3})$	Thermal design and heatsinks
Heatsink needed	$R_{t h 3} = (T_{J} - T_{amb}) / P - (R_{t h 1} + R_{t h 2})$	Thermal design and heatsinks
Safety margin	$100 - (T_{J} /175) \times 100$ , aim ≥ 30%	Thermal design and heatsinks
Buck / boost	$V_{o u t} = D V_{in}$ · $V_{o u t} = V_{in} / (1 - D)$	DC-DC converters
Buck L and C	$L = V_{o u t} (1 - D) / (F_{S W} Δ I_{L})$ · $C \geq Δ I_{L} / (8 F_{S W} Δ V)$	DC-DC converters
Three-phase drive	$I_{p ha se} = P_{o u t} / (3 V_{L} cos φ η)$ · $I_{D, R M S} = I_{p ha se} / 2$ · $V_{L} = V_{D C} / 2$	Motor drives

Week 4: motors

Want	Formula	Guide
Force and EMF on a conductor	$F = B I L$ · $e = B Lv$	Electric motors and DC motor types
DC motor model	$V = I_{a} R_{a} + K_{e} ω$ · $T = K_{t} I_{a}$ · $K_{t} = K_{e}$ (SI)	Electric motors and DC motor types
Stall current	$I_{s t a l l} = V / R_{a}$	Electric motors and DC motor types
Motor heat	$Δ T = P_{l oss} R_{θ}$	Electric motors and DC motor types
The chain	$E_{b} = V - I_{a} R_{a}$ → $ω = E_{b} / K_{e}$ → $n = 60 ω /2 π$ → $P_{o u t} = T ω$	DC motor performance and selection
Robot side	$n_{w} = n / G$ · $v = ω_{w} r$ · km/h $= 3.6 v$ · $T_{w} = T G$	DC motor performance and selection
Selection	utilisation $= P_{o u t} / P_{r a t e d}$ · reserve $= P_{r a t e d} - P_{o u t}$	DC motor performance and selection
Stepper resolution	$N = 360°/ step$ · $N_{μ} = N M$ · $θ_{μ} = step / M$	Stepper motors
Stepper rates	$f_{s t e p} = N S_{m} /60$ · $f_{μ} = f_{s t e p} M$ · $n_{o u t} = S_{m} / G$ · $T_{o u t} = T_{h} G$	Stepper motors
Stepper checks	$P_{p ha se} = R I^{2}$ · $τ_{e} = L / R$ · $S F = T_{h} / T_{l o a d}$ · $P = T_{h} ω_{m}$	Stepper motors
Drive sizing	$F = ma + C_{r r} m g + m g sin θ$ · $T_{m} = F r / (N_{w} G η_{g})$ · × SF	Sizing a drive motor

Numbers worth knowing

UK mains: 230 V RMS, 325 V peak, 50 Hz
RC: 63% at 1τ, ~100% at 5τ
Silicon: 175 °C absolute max; design to ≤ 120 °C
MOSFET below ~700 V; IGBT above ~1000 V
Hybrid stepper: 1.8°, 200 steps per revolution
Motor windings: above about 130 °C exceeds typical insulation ratings