Potential Dividers & Variable Resistance

Potential Divider Definition: A potential divider (or voltage divider) consists of two or more resistors connected in series across a voltage source. Its primary purpose is to provide a specific, lower voltage from a higher voltage supply by tapping into the connection point between the resistors.

Input and Output Voltages: The Input Voltage ($V_{in}$) is the total potential difference supplied to the entire series chain. The Output Voltage ($V_{out}$) is the potential difference measured across one specific resistor in that chain, typically the one connected to the negative terminal or ground.

Proportional Distribution: The fundamental concept is that in a series circuit, the total voltage is shared among components. The component with the higher resistance will always take a larger share of the total voltage, following the relationship $V \propto R$ when current is constant.

Ohm's Law Application: Since the resistors are in series, the same current ($I$) flows through both. According to Ohm's Law ($V = IR$), the voltage across each resistor is directly proportional to its resistance value.

The Potential Divider Formula: The output voltage across a specific resistor ($R_2$) in a two-resistor chain is calculated using the ratio of that resistor to the total resistance: $V_{out} = V_{in} \times \frac{R_2}{R_1 + R_2}$

Voltage Ratios: An alternative way to view the principle is through the ratio of voltages being equal to the ratio of resistances: $\frac{V_1}{V_2} = \frac{R_1}{R_2}$. This is particularly useful for quickly determining how a change in one resistor affects the distribution of potential difference.

Step 1: Identify the Series Chain: Ensure that the resistors are truly in series and that no significant current is being drawn from the $V_{out}$ terminals, as this would change the effective resistance of the bottom branch.

Step 2: Calculate Total Resistance: Sum the resistances ($R_{total} = R_1 + R_2 + ...$) to find the total opposition to current in the divider branch.

Step 3: Apply the Divider Ratio: Multiply the input voltage by the fraction representing the 'target' resistor over the total resistance. If you need to find the voltage across $R_1$, use $R_1$ in the numerator; for $R_2$, use $R_2$.

Designing for a Specific Output: To create a specific $V_{out}$, choose a ratio of resistors such that $\frac{R_2}{R_1} = \frac{V_{out}}{V_{in} - V_{out}}$. This allows for the selection of standard resistor values to achieve a desired signal level.

Potentiometers: A potentiometer is a three-terminal variable resistor. By moving a sliding contact (wiper), the ratio of resistance between the wiper and the two ends changes, providing a continuously adjustable $V_{out}$ from $0$ to $V_{in}$.

Light Dependent Resistors (LDR): In an LDR, resistance decreases as light intensity increases. When placed in a potential divider, the $V_{out}$ will change based on light levels, allowing the circuit to act as a light sensor for automated switches.

Thermistors (NTC): Negative Temperature Coefficient thermistors see a decrease in resistance as temperature rises. Using a thermistor as one of the resistors in a divider creates a temperature-sensitive voltage output, essential for digital thermometers or cooling system triggers.

The 'Share' Logic: Always perform a sanity check. If $R_2$ is much larger than $R_1$, $V_{out}$ should be very close to $V_{in}$. If $R_1$ and $R_2$ are equal, $V_{out}$ must be exactly half of $V_{in}$.

Direction of Change: In sensor questions, trace the logic step-by-step: (1) Physical change (e.g., more light) $\rightarrow$ (2) Resistance change (LDR $R$ drops) $\rightarrow$ (3) Voltage share change ($V_{out}$ drops if LDR is $R_2$).

Unit Consistency: Ensure all resistance values are in the same units (e.g., all in $k\Omega$ or all in $\Omega$) before plugging them into the ratio formula to avoid decimal errors.

Kirchhoff's Voltage Law: Remember that $V_{R1} + V_{R2}$ must always equal $V_{in}$. If you calculate $V_{out}$ across $R_2$, you can find the voltage across $R_1$ simply by subtracting $V_{out}$ from $V_{in}$.