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.

Electric field distribution explaining striations:

1. Electrons moving from cathode toward anode undergo cycles of:

   • Acceleration due to electric field (gaining energy)
   • Ionization of gas molecules (losing energy)
   • Deceleration due to collisions (energy too low for ionization)
   • Re-acceleration to gain sufficient energy again

2. These recurring cycles create alternating regions of:

   • Bright bands: Where electrons have sufficient energy to ionize and excite molecules, causing light emission
   • Dark bands: Where electrons have lost energy after ionization and haven't yet reaccelerated enough

3. The spacing between striations corresponds to the distance electrons must travel to regain ionization energy

4. This phenomenon is most pronounced at specific pressure ranges (~0.1mm Hg) where mean free path allows for complete cycles of energy gain and loss

Note: The succession of bright and dark bands serves as a visual representation of the spatially periodic changes in electron energy distribution.

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 experiment determines the elementary electric charge (e) by balancing electrostatic and gravitational forces on charged oil droplets
• Key principles:
  • Oil droplets acquire random electric charges through friction or ionization
  • When suspended between charged parallel plates, the droplets experience:
    • Gravitational force (downward): mg
    • Electrostatic force (upward): qE
    • Buoyancy and air resistance (when moving)
  • By adjusting the electric field to hold droplets stationary: qE = mg - buoyancy
  • By measuring many droplets, Millikan discovered charges were always integer multiples of e = 1.6×10⁻¹⁹ C
• This experiment provided conclusive evidence for the quantization of electric charge

Methodological challenges and solutions:

1. Determining droplet size:

   • Challenge: Cannot measure microscopic oil droplets directly
   • Solution: Used Stokes' law to calculate radius by measuring terminal velocity in air

2. Charge variability:

   • Challenge: Droplets acquired random, unpredictable charges
   • Solution: Observed many droplets and identified the greatest common divisor

3. Measurement precision:

   • Challenge: Needed extreme precision to detect quantization
   • Solution: Used telescope with calibrated reticle for accurate position tracking

4. Environmental factors:

   • Challenge: Air currents and temperature fluctuations affected measurements
   • Solution: Used sealed chamber with temperature control

5. Ionization control:

   • Challenge: Needed to change droplet charge during experiment
   • Solution: Used X-rays or radioactive sources to ionize air molecules

This methodology ultimately revealed that all measured charges were integer multiples of e = 1.6×10⁻¹⁹ C, confirming charge quantization.

A triode valve consists of three essential components arranged in a specific order:

1. Cathode (K): Located at the bottom/center, heated to emit electrons through thermionic emission
2. Grid (G): Located in the middle between cathode and plate, controls electron flow
3. Plate/Anode (P): Located at the top, collects electrons that pass through the grid

The electrons flow from cathode → through grid → to plate, with the grid acting as a control element that can either encourage or inhibit this flow based on its voltage.

The fundamental relationship μ = rp × gm affects triode performance as follows:

• In voltage amplifiers: Higher μ yields better voltage gain, approaching μ/(1+rp/RL)
• In power amplifiers: Lower rp allows more current delivery to load
• For impedance matching: rp should be close to load impedance for maximum power transfer
• For bandwidth considerations: Lower rp-RL combination produces wider bandwidth
• For input sensitivity: Higher gm provides better response to small input signals
• For output drive capability: Lower rp improves ability to drive low impedance loads

Circuit optimization involves balancing these parameters:

• Voltage amplifier stages: Maximize μ
• Power output stages: Minimize rp
• Wide bandwidth applications: Keep rp-RL product low
• High input sensitivity applications: Maximize gm

Example: A triode with high μ but also high rp may excel as a voltage amplifier but perform poorly as a power output stage.

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.

Conditions for an ideal triode amplifier:

• Operating point centered on perfectly linear region of mutual characteristics
• Small signal amplitude to stay within linear region
• Very high load resistance (RL >> rp) to approach theoretical maximum gain of μ
• Constant transconductance (gm) across the operating range
• Zero grid current under all operating conditions

Practical limitations:

• Non-linearity in actual triode characteristics, especially at extremes
• Limited linear region in real tubes restricts undistorted signal amplitude
• Power supply voltage limits achievable plate voltage swing
• Practical load resistances must be lower than optimal for sufficient current
• Internal capacitances limit high-frequency performance
• Cathode temperature variations affect emission characteristics
• Grid current at higher signal levels in real tubes
• Practical power requirements limit how high RL can be

Example: A triode with μ=100 might achieve gain of only 60-70 in practice due to these limitations.

Definition and Calculation

• Dynamic plate resistance (rp) is the effective resistance of a diode valve at a specific operating point
• Defined as the ratio of a small change in plate voltage to the resulting change in plate current
• Mathematical expressions:
  • rp = ΔVp/Δip
  • rp = (∂Vp/∂ip) at constant cathode temperature
• Calculation method:
  1. Identify operating point (Vp, ip) on the characteristic curve
  2. Apply small voltage change ΔVp and measure current change Δip
  3. Calculate the ratio to find rp

Variation Across Operating Regions

• Low voltage region:
  • High resistance
  • Space charge dominates, limiting current flow
  • Relatively flat slope on ip-Vp curve
• Middle region (typical operating region):
  • Moderate resistance
  • Space charge partially neutralized, allowing more efficient electron flow
  • Steeper slope on ip-Vp curve
  • Ideal for amplification applications
• Saturation region:
  • Very high resistance
  • Nearly horizontal curve
  • Limited by emission rate, not by applied voltage
  • Diode behaves as a high-impedance current source

Physical Significance

• Represents the inverse of the slope of the diode characteristic curve at the operating point
• Lower values indicate the diode is more responsive to voltage changes
• Critical parameter for designing amplifier circuits, calculating gain, and impedance matching

Example

A diode operating at 40V with 50mA current that changes to 60mA when voltage increases to 42V has rp = 2V/10mA = 200Ω