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Annex 1 (BE01) — Lesson BE.01.09 • TLO: 28XX-MAIN-1002 • ELOs: 28XX-MAIN-1002bp, 1002bt, 1002bx, 1002cb

Potentiometers & Rheostats

A three-terminal sliding-contact variable resistor can be wired into a circuit in two fundamentally different ways: as a Rheostat (using two terminals in series with a load to control current) or as a Potentiometer (using all three terminals across a load to control voltage). Because their circuit topologies differ, rotating the shaft from $0$ (CCW) to $100$ (CW) produces opposite total-resistance trends between the two configurations.

1. Variable Resistor Construction, Symbology & Tapers

Compositions & Power Ratings

Sliding-contact variable resistors are manufactured in two primary compositions:

1. Carbon Composition: Used for low-power applications (less than $2,\text{Watts}$) because carbon resistance values become unstable when subjected to high voltages and currents.
2. Wire-Wound: Used for high-current and high-power applications ($1{,}000,\text{Watts}$ or greater) where accurate, stable resistance values are required.

Three-Part Schematic Symbol

As introduced in DC Theory, every variable resistor schematic symbol contains three mandatory components:

1. The Resistive Element: Fixed total resistance between the two outer end terminals.
2. The Arrow (Wiper Arm): Movable contact connected to the rotary control shaft.
3. Positional Reference (CW or CCW): Indicates the direction the wiper arrow moves along the resistive element when the shaft is rotated Clockwise (CW) or Counter-Clockwise (CCW). (Note: Always inspect the schematic for the location of the CW/CCW marker—its placement can be at the top or bottom depending on the diagram.)

Resistance Taper: Linear (S) vs. Logarithmic (Z)

Taper defines the relationship between the mechanical angle of shaft rotation (expressed as a percentage from $0$ to $100$) and the resulting change in resistance:

2. Rheostat Configuration (Series Current Controller)

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Rheostat Characteristics (ELO 28XX-MAIN-1002bp)

• Terminal Connections: Uses two of the three leads (one fixed terminal and one variable/wiper terminal; the unused third terminal may be left open or tied to the wiper).
• Circuit Placement & Function: Connected in series with the load ($R_L$) and used as a current controller ($I_T = I_L = \frac{V_A}{R_1 + R_L}$).
• Construction: Typically wire-wound resistors with a limited ohmic range, built to handle high current in low-resistance circuits.

Rheostat Wiper Movement Analysis (With CW Increasing Series Resistance $R_1$)

In the standard BE.01.09 rheostat circuit (Figure 09-10: $V_A = 10,\text{V}$, Rheostat $R_1 = 8,\text{k}\Omega$, Series Load $R_L = 4,\text{k}\Omega$):

1. At $0$ (Full CCW — Minimum Rheostat Resistance, $R_1 = 0,\Omega$):
   • The wiper sits immediately adjacent to the fixed terminal feeding $R_L$, bypassing $R_1$ electronically ($R_1 = 0,\Omega$).
   • Total Resistance ($R_T$): At MINIMUM ($R_T = 0,\Omega + 4,\text{k}\Omega = \mathbf{4,\text{k}\Omega}$).
   • Circuit / Load Current ($I_T = I_L$): At MAXIMUM ($\frac{10,\text{V}}{4,\text{k}\Omega} = \mathbf{2.50,\text{mA}}$).
   • Load Voltage ($V_L$): At MAXIMUM ($V_L = V_A = \mathbf{10.00,\text{V}}$; $V_{R1} = 0,\text{V}$).
2. At $50$ CW (Midpoint, $R_1 = 4,\text{k}\Omega$):
   • Half of the rheostat’s resistance ($4,\text{k}\Omega$) is in series with $R_L$ ($4,\text{k}\Omega$).
   • $R_T = 8,\text{k}\Omega$, $I_T = 1.25,\text{mA}$, $V_{R1} = 5.00,\text{V}$, $V_L = 5.00,\text{V}$.
3. At $100$ (Full CW — Maximum Rheostat Resistance, $R_1 = 8,\text{k}\Omega$):
   • All $8,\text{k}\Omega$ of $R_1$ is placed in series with $R_L$ ($4,\text{k}\Omega$).
   • Total Resistance ($R_T$): At MAXIMUM ($R_T = 8,\text{k}\Omega + 4,\text{k}\Omega = \mathbf{12,\text{k}\Omega}$).
   • Circuit / Load Current ($I_T = I_L$): At MINIMUM ($\frac{10,\text{V}}{12,\text{k}\Omega} = \mathbf{833.33,\mu\text{A}}$).
   • Load Voltage ($V_L$): At MINIMUM ($V_L = 833.33,\mu\text{A} \times 4,\text{k}\Omega = \mathbf{3.33,\text{V}}$; $V_{R1} = \mathbf{6.67,\text{V}}$).

3. Potentiometer Configuration (Parallel Voltage Controller)

Potentiometer Characteristics (ELO 28XX-MAIN-1002bt)

• Terminal Connections: Uses all three terminals (two fixed end terminals and one variable/wiper terminal).
• Circuit Placement & Function: Connected in parallel with (across) the load ($R_L$) and used as a load voltage controller (variable voltage divider). Because the full resistive element remains connected across the source, potentiometers have limited current-handling capability compared to wire-wound rheostats.

Potentiometer Equivalent Circuits Across Wiper Rotation (0% CCW, 75% CW, 100% CW)

In the standard BE.01.09 potentiometer circuit (Figure 09-11: $V_A = 10,\text{V}$, Potentiometer $R_1 = 4,\text{k}\Omega$ with CW at top, Load $R_L = 4,\text{k}\Omega$ connected from wiper to bottom ground return):

1. At $0$ (Full CCW — Wiper at Bottom Ground Rail)

• Circuit Topology: The wiper is turned all the way down to ground. All $4,\text{k}\Omega$ of $R_1$ is in series with the source, while $0,\Omega$ (a direct short) is placed in parallel with $R_L$.
• Resulting Parameters:
  • Because $R_L$ is in parallel with a $0,\Omega$ short, $R_{EQ} = 0,\Omega$ (MIN).
  • Total Resistance ($R_T$): At MAXIMUM ($R_T = R_1 + 0,\Omega = \mathbf{4,\text{k}\Omega}$).
  • Total Current ($I_T$): At MINIMUM ($I_T = \frac{10,\text{V}}{4,\text{k}\Omega} = \mathbf{2.50,\text{mA}}$).
  • Load Voltage ($V_L$) & Load Current ($I_L$): At MINIMUM ($\mathbf{0,\text{V}}$ and $\mathbf{0,\text{A}}$), with $P_L = \mathbf{0,\text{W}}$.

2. At Partial Rotation (e.g., $75$ CW — Series-Parallel Loaded Divider)

• Circuit Topology: The wiper splits $R_1$ ($4,\text{k}\Omega$) into two sections:
  • Lower $75$ of $R_1$ ($R_{1\text{Parallel}} = 0.75 \times 4,\text{k}\Omega = 3,\text{k}\Omega$): In parallel with load $R_L$ ($4,\text{k}\Omega$).
  • Upper $25$ of $R_1$ ($R_{1\text{Series}} = 0.25 \times 4,\text{k}\Omega = 1,\text{k}\Omega$): In series with that parallel bank.
• 6-Step Solution at $75$ CW:
  1. $R_{EQ} = 3,\text{k}\Omega \parallel 4,\text{k}\Omega = \frac{3,\text{k} \times 4,\text{k}}{3,\text{k} + 4,\text{k}} = \mathbf{1.71,\text{k}\Omega}$
  2. $R_T = R_{1\text{Series}} + R_{EQ} = 1,\text{k}\Omega + 1.714,\text{k}\Omega = \mathbf{2.71,\text{k}\Omega}$
  3. $I_T = \frac{V_A}{R_T} = \frac{10,\text{V}}{2.714,\text{k}\Omega} = \mathbf{3.68,\text{mA}}$
  4. $V_L = V_{REQ} = I_T \times R_{EQ} = 3.684,\text{mA} \times 1.714,\text{k}\Omega = \mathbf{6.32,\text{V}}$ (Slide rounds intermediate to $6.31,\text{V}$)
  5. $P_L = \frac{V_L^2}{R_L} = \frac{(6.31,\text{V})^2}{4,\text{k}\Omega} = \mathbf{9.95,\text{mW}}$

3. At $100$ (Full CW — Wiper at Top Positive Rail)

• Circuit Topology: The wiper is turned all the way to the top rail. $0,\Omega$ of $R_1$ is in series, and all $4,\text{k}\Omega$ of $R_1$ is placed directly in parallel with $R_L$ ($4,\text{k}\Omega$).
• Resulting Parameters:
  • Total Resistance ($R_T$): At MINIMUM ($R_T = R_{EQ} = 4,\text{k}\Omega \parallel 4,\text{k}\Omega = \mathbf{2,\text{k}\Omega}$).
  • Total Current ($I_T$): At MAXIMUM ($I_T = \frac{10,\text{V}}{2,\text{k}\Omega} = \mathbf{5.00,\text{mA}}$).
  • Load Voltage ($V_L$): At MAXIMUM ($V_L = V_A = \mathbf{10.00,\text{V}}$).
  • Load Power ($P_L$): At MAXIMUM ($P_L = \frac{(10,\text{V})^2}{4,\text{k}\Omega} = \mathbf{25.00,\text{mW}}$).

4. Master Comparison: Rheostat vs. Potentiometer

5. Bench & Troubleshooting Application (BE.01.09 Lab)

1. Critical Breadboard Orientation Rule for the Blue Variable Resistor

Mandatory Lab Rule (BE.01.09 Student Lab, Page 3): During BE.01.09 and all subsequent labs, always insert the blue $10,\text{k}\Omega$ thumbwheel variable resistor into the breadboard with its three pins facing to the LEFT. Installing the component backwards (pins facing right) reverses the internal fixed end terminals relative to the thumbwheel dial, causing Clockwise (CW) rotation to decrease resistance instead of increasing it (producing inverted readings across every table in the lab).

2. Breadboard Potentiometer Circuit (Part II Equivalent Circuit Verification)

In Part II of the BE.01.09 lab, a series limiting resistor ($R_1 = 2,\text{k}\Omega$) feeds the top pin of the $10,\text{k}\Omega$ potentiometer ($R_2$), with the load resistor ($R_L = 10,\text{k}\Omega$) connected from the center wiper pin to ground:

• At $100$ CW: The $10,\text{k}\Omega$ pot is completely in parallel with the $10,\text{k}\Omega$ load ($R_{EQ} = 5,\text{k}\Omega$), in series with $R_1 = 2,\text{k}\Omega \implies R_T = \mathbf{7,\text{k}\Omega}$ (Minimum $R_T$, Maximum $V_L$).
• At $50$ CW: The upper $5,\text{k}\Omega$ of the pot adds in series with $R_1 = 2,\text{k}\Omega$ ($7,\text{k}\Omega$ series portion), while the lower $5,\text{k}\Omega$ of the pot is in parallel with $R_L = 10,\text{k}\Omega$ ($R_{EQ} = \frac{5,\text{k}\times 10,\text{k}}{15,\text{k}} = 3.33,\text{k}\Omega$) $\implies R_T = 7,\text{k}\Omega + 3.33,\text{k}\Omega = \mathbf{10.33,\text{k}\Omega}$.
• At $0$ CCW: The wiper grounds out $R_L$ ($R_{EQ} = 0,\Omega$, $V_L = 0,\text{V}$), leaving all $10,\text{k}\Omega$ of the pot in series with $R_1 = 2,\text{k}\Omega \implies R_T = \mathbf{12,\text{k}\Omega}$ (Maximum $R_T$, Minimum $I_T$).

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