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Scene 1: Starting the Linear DC Machine Magnetic Field Region (B ⊗ into page) R VB + − ON l Find v → eind = vBl Dashboard — Time Curves eind (Induced Voltage) VB t i (Current) VB/R t v (Speed) vnl t Accelerating… How It Works 1 Switch closes → current flows: i = VB/R (maximum) 2 Current in magnetic field → force on bar: Find = i(Bl) (maximum, then decreases) 3 Bar accelerates → eind = vBl increases → i decreases 4 Steady state: eind = VB, i = 0, Find = 0, bar at vnl = VB/Bl Key Equations VB = eind + iR eind = vBl Find = ilB Fnet = ma Physical Insight • Find starts at max, shrinks to 0 • eind grows: 0 → VB • i decays: VB/R → 0 • v grows: 0 → vnl At steady state: i = 0, F = 0

Starting the Linear DC Machine

When the switch closes, current flows through the bar. The magnetic field exerts force F = i(Bl), accelerating the bar. As it gains speed, induced voltage e_ind = vBl rises, reducing current until equilibrium at no-load speed v_nl = V_B/Bl.

F = i × (Bl)
e_ind = vBl
V_B = e_ind + iR
F_net = ma
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Scene 2: Linear DC Machine as a Motor Magnetic Field Region (B ⊗ into page) R VB + − eind ↓ l Fload Find vload → (slow) Motor Mode — Time Curves eind (Induced Voltage) VB t i (Current) max t v (Speed) vnl t vload Motor Mode (Loaded) Motor Operation: Step by Step 1External load Fload applied opposite to motion (← red) 2Bar slows down (fast → slow → steady) → eind decreases 3Current increases: i↑ = (VB − eind)/R 4Find grows (→ green) until Find = Fload at lower speed vload 5Battery drains: electrical → mechanical, Pconv = eind × i Power Conversion Electrical → Mechanical Pconv = eind × i = Find × vload Motor: eind < VB vload < vnl Current: Battery(+) → R → rail → bar ↓ → rail → Battery(−)

Motor Mode — Applying a Load

A load force F_load opposes the bar's motion, causing deceleration. Reduced speed lowers e_ind, increasing current. The induced force eventually balances the load at a lower speed v_load < v_nl. Electrical power converts to mechanical.

i = (V_B - e_ind)/R ↑
e_ind < V_B
P_conv = e_ind · i
v_load < v_nl
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Scene 3: Linear DC Machine as a Generator Magnetic Field Region (B ⊗ into page) R ⚡charging + − eind > VB l Fapp Find vapp → (fast) Generator Mode Dashboard eind (Induced Voltage) VB t i (Charging Current) t v (Speed) t Generator Mode Generator Operation: Step by Step 1External driving force Fapp applied in direction of motion (→ red) 2Bar accelerates (slow → fast) → eind = vBl increases above VB 3Current flows into battery: i = (eind − VB)/R 4Find grows (← green) until Find = Fapp 5Mechanical → Electrical: Pconv = eind × i (charging battery) Power Conversion Mechanical → Electrical Pconv = eind × i = Find × vapp

Generator Mode — Applying a Driving Force

An applied driving force F_app accelerates the bar beyond v_nl, making e_ind = vBl exceed V_B. Current flows into the battery, charging it. The induced force opposes until F_ind = F_app at higher speed v_app > v_nl. Mechanical power converts to electrical.

i = (e_ind - V_B)/R ↑
e_ind > V_B
P_conv = e_ind · i
v_app > v_nl