Electric Current Through Gases

When both plate voltage and grid voltage change in a triode:

$\Delta i_p = \frac{1}{r_p}\Delta V_p + g_m \Delta V_g$

Where:

• $\Delta i_p$ = change in plate current
• $\Delta V_p$ = change in plate voltage
• $\Delta V_g$ = change in grid voltage
• $r_p$ = dynamic plate resistance = $\left(\frac{\Delta V_p}{\Delta i_p}\right)_{\Delta V_g = 0}$
• $g_m$ = mutual conductance = $\left(\frac{\Delta i_p}{\Delta V_g}\right)_{\Delta V_p = 0}$

Analysis: This equation shows that plate current changes are the sum of two effects:

1. Effect of plate voltage change: $\frac{1}{r_p}\Delta V_p$
2. Effect of grid voltage change: $g_m \Delta V_g$

The equation represents a first-order approximation valid for small changes around the operating point.

Thomson's experimental design enabled precise e/m measurement through:

1. Controlled electron path: Electrons from the cathode (C) passed through a small aperture in the anode (A) creating a narrow, well-defined beam

2. Balanced force method:

   • Applied perpendicular electric field (E) causing upward deflection
   • Applied perpendicular magnetic field (B) causing downward deflection
   • When forces balanced exactly (eE = evB), the beam passed undeflected

3. Mathematical relationship:

   • When balanced: E/B = v (electron velocity)
   • From kinetic energy: ½mv² = eV (where V is accelerating voltage)
   • Substituting: e/m = E²/(2B²V)

This brilliant design isolated the ratio without needing to measure either quantity independently, as the deflection pattern provided all necessary data when the fields were precisely calibrated.

The charge can be calculated using the balance of forces equation:

$q = \frac{4\pi r^3(\rho_{oil} - \rho_{air})g}{3E}$

Where:

• q = charge on the droplet (C)
• r = radius of the droplet (m)
• ρₒᵢₗ = density of oil (kg/m³)
• ρₐᵢᵣ = density of air (kg/m³)
• g = gravitational acceleration (9.8 m/s²)
• E = electric field strength (V/m)

Process:

1. When stationary: Electric force (qE) = Weight (mg) - Buoyancy (B)
2. Weight = (4/3)πr³ρₒᵢₗg
3. Buoyancy = (4/3)πr³ρₐᵢᵣg
4. Solve for q using the measured variables

Note: The radius can be determined by measuring terminal velocity when the field is off and applying Stokes' law.

The principle relies on opposing forces reaching equilibrium:

• Electric force on electron: F₁ = eE (upward in standard setup)
• Magnetic force on electron: F₂ = evB (downward when properly oriented)
• At equilibrium: eE = evB

This balance occurs at a specific velocity v = E/B where the electron beam travels undeflected.

The genius of Thomson's approach is that measuring this balance point eliminates the need to directly measure electron velocity, which would be extremely difficult. Instead, measurable macroscopic quantities (E, B, V) can be used to calculate the microscopic property e/m.

This method demonstrated that all cathode rays have identical e/m values regardless of cathode material, proving electrons are universal constituents of matter.

• Space charge effect creates the S-shaped characteristic curve by:

  • Initially limiting electron flow (steep at low voltages)
  • Gradually diminishing as plate voltage increases (middle region)
  • Finally reaching saturation when all emitted electrons are collected (flat top)

• Dynamic resistance relationship to space charge:

  • High rp at low voltages: space charge dominates, inhibiting current flow
  • Medium rp in middle region: partial neutralization of space charge
  • High rp at saturation: limited by cathode emission, not plate voltage

• Mathematical expression: Langmuir-Child law governs the relationship where ip = kVp^(3/2)

  • The 3/2 power law directly impacts the changing slope (and thus rp)

• Practical application: Tube designers manipulate electrode spacing and geometry to optimize rp values for specific applications like amplifiers, rectifiers, or oscillators

• Example: Power rectifiers need low rp for efficiency, while voltage amplifiers benefit from higher rp values

Analysis of half-wave rectifier with 120V RMS AC input and 1kΩ load:

1. Calculate peak voltage:

   • Vpeak = VRMS × √2 = 120V × 1.414 = 169.7V

2. Key output parameters:

   • Average DC voltage: Vavg = Vpeak/π = 169.7V/π ≈ 54V
   • Average DC current: Iavg = Vavg/R = 54V/1000Ω = 54mA
   • Peak current: Ipeak = Vpeak/R = 169.7V/1000Ω ≈ 170mA
   • RMS current: IRMS = Ipeak/2 = 85mA

3. Power characteristics:

   • Power delivered to load: P = Iavg²R = (0.054A)² × 1000Ω ≈ 2.92W
   • Ripple factor: RF = 1.21 (without filtering)
   • Form factor: FF = 1.57

4. Efficiency: η = Pout/Pin = 40.6% (theoretical maximum for half-wave)

Note: This analysis assumes an ideal diode with zero forward voltage drop and infinite reverse resistance. For more precise calculations, include diode forward voltage drop (typically 0.7V for silicon).

Signal coupling effects:

• The input signal (vs) directly modulates the grid voltage: vs = ΔVg
• Signal amplitude determines how far the operating point moves along characteristic curve
• Small signals keep operation within linear region, maintaining faithful amplification
• Large signals cause:
  1. Distortion when operation enters nonlinear regions
  2. Clipping if the signal drives into cutoff or saturation regions
  3. Grid current if grid becomes positive relative to cathode
  4. Waveform asymmetry if operating point is not properly centered

To maintain fidelity:

• Signal amplitude must keep operation within the linear region
• Operating point should be centered to allow equal positive/negative swings
• Grid bias must be sufficient to prevent positive grid excursions

Example: A 1V peak signal with μ=20 produces up to 20V peak output, but only if operation stays within linear region.

The Crookes dark space appears when:

• The mean free path of electrons exceeds the dimensions of the tube
• Electrons travel from cathode to anode with insufficient collisions to cause ionization
• No ionization means no light emission, creating a dark region

Key relationships:

• Longer mean free path → larger dark space
• Higher vacuum (lower pressure) → longer mean free path
• If mean free path >> tube length, the entire tube becomes a Crookes dark space
• As pressure increases, mean free path decreases and dark space shrinks

Example: In a tube where electron mean free path is 20 cm but tube length is only 10 cm, electrons typically reach the anode without collisions, causing the entire tube to appear as a dark space.

Derivation of output voltage relationship:

1. Define output voltage as voltage across load resistor: v₀ = RLΔip

2. Apply plate circuit voltage relationship:

   • Vp = V - RLip (where V is battery voltage)
   • ΔVp = -RLΔip = -v₀

3. Use triode characteristic equation:

   • Δip = ΔVp/rp + gmΔVg
   • Δip = -v₀/rp + gmvₛ

4. Substitute into output equation:

   • v₀ = RL(-v₀/rp + gmvₛ)
   • v₀ = -RLv₀/rp + RLgmvₛ

5. Solve for v₀:

   • v₀(1 + RL/rp) = RLgmvₛ
   • v₀ = RLgmvₛ/(1 + RL/rp)
   • v₀ = RLgmrpvₛ/(rp + RL)
   • v₀ = RLμvₛ/(rp + RL) (since μ = gmrp)

Final relationship: v₀ = μvₛ/(1 + rp/RL)

A triode amplifier works by:

• Converting small grid voltage variations into larger plate current changes
• Key operational principle: A small change in grid voltage (input signal) produces a proportionally larger change in plate current
• Mathematical relationship: Δip = gm × ΔVg (where gm is transconductance)
• The amplified current flowing through a load resistor (RL) creates a voltage drop (vo = RL·Δip) that follows the input pattern but with increased amplitude
• This creates voltage gain: A = vo/vs
• For maximum linearity, the operating point must remain on the linear portion of the mutual characteristics curve

Example: A microphone signal of 0.01V can be amplified to several volts at the output, preserving the waveform pattern while increasing its magnitude.