Unit 2: Divider Quest

I. Orientation

Divider laws describe how voltage and current distribute among circuit elements connected in series or parallel. They are direct consequences of Ohm’s law, Kirchhoff’s Voltage Law (KVL), Kirchhoff’s Current Law (KCL), and the conservation of electrical energy and charge.

• Voltage division: In a series circuit, the source voltage divides between resistors in proportion to their resistance values.
• Current division: In a parallel circuit, the total current divides among branches inversely proportional to branch resistance.
• Ohm’s law: For a resistor, voltage, current, and resistance are related by (V = IR).
• Series convention: Resistors in series carry the same current, while their voltages may differ.
• Parallel convention: Resistors in parallel have the same voltage, while their branch currents may differ.
• Ideal assumptions: The source resistance, connecting-wire resistance, and measuring-instrument loading are neglected unless they are explicitly included.
• Polarity and direction: Voltage polarity and current direction must be selected consistently before applying either divider law.
• Validity condition: The voltage divider law applies to series paths; the current divider law applies to parallel branches connected across the same two nodes.

II. Voltage Divider Law — Series Distribution of Voltage

A. Validation and analysis of the Voltage Divider Law

The Voltage Divider Law states that the voltage across a series resistor equals the total applied voltage multiplied by the ratio of that resistor to the total series resistance.

Consider (n) series resistors (R_1, R_2, \ldots, R_n) connected to a source voltage (V_S). Since the same current flows through every series element, the total resistance is

R_T = R_1 + R_2 + ... + R_n

where (R_T) is total series resistance. Applying Ohm’s law to the complete circuit gives

I = V_S / R_T

where (I) is the common series current. The voltage across resistor (R_k) is therefore

V_k = I R_k
    = V_S (R_k / R_T)

where (V_k) is the voltage across (R_k), and (R_k) is the selected resistor.

• Law statement: For any resistor (R_k) in a series network,

  V_k = V_S [R_k / (R_1 + R_2 + ... + R_n)]

• KVL validation: Kirchhoff’s Voltage Law requires the sum of resistor voltage drops to equal the source voltage:

  V_1 + V_2 + ... + V_n = V_S

Substituting the divider expressions gives

  V_S [(R_1 + R_2 + ... + R_n) / R_T] = V_S

This confirms the law.

• Proportionality: If (R_2) is twice (R_1), then the voltage across (R_2) is twice the voltage across (R_1), provided both are in the same series path.
• Resistance effect: Increasing (R_k) increases its voltage share, but it also changes the total resistance and therefore changes the circuit current.
• Measurement validation: A voltmeter connected across (R_k) should read approximately (V_k). A difference may result from resistor tolerance, source variation, or voltmeter loading.
• Dimensional check: The ratio (R_k/R_T) is dimensionless, so (V_S(R_k/R_T)) has the unit volt.

Worked example

A (12\text{ V}) source supplies two series resistors, (R_1=2\text{ k}\Omega) and (R_2=4\text{ k}\Omega).

R_T = 2 kΩ + 4 kΩ = 6 kΩ
V_1 = 12 V (2/6) = 4 V
V_2 = 12 V (4/6) = 8 V

The drops add to (4\text{ V}+8\text{ V}=12\text{ V}), satisfying KVL. The common current is (I=12\text{ V}/6\text{ k}\Omega=2\text{ mA}), and (IR_1=4\text{ V}), (IR_2=8\text{ V}).

B. Applications and limitations

The voltage divider is used when a smaller voltage is required from a larger source, but its output must be analyzed with the load connected.

• Reference-voltage generation: Two resistors can produce a fraction of a supply voltage, such as (5\text{ V}) from a (12\text{ V}) source.
• Sensor interfaces: A thermistor or light-dependent resistor can occupy one divider position so that resistance changes become measurable voltage changes.
• Load-free output: With no external load, the output voltage across (R_2) is

  V_OUT = V_S [R_2 / (R_1 + R_2)]

where (V_{OUT}) is the unloaded output voltage.

• Loaded divider: If a load resistance (R_L) is connected across (R_2), the effective lower resistance becomes

  R_B = R_2 || R_L = (R_2 R_L) / (R_2 + R_L)

and the output becomes

  V_OUT = V_S [R_B / (R_1 + R_B)]

where (R_L) is load resistance and (R_B) is the parallel equivalent.

• Loading error: Because (R_L) is parallel with (R_2), (R_B) is smaller than (R_2), so the loaded output is lower than the unloaded value.
• Power limitation: Each resistor dissipates power according to

  P_k = I^2 R_k = V_k^2 / R_k

where (P_k) is resistor power in watts. A resistor’s power rating must exceed its calculated dissipation.

• Practical accuracy: Resistor tolerance, typically specified as (1\%) or (5\%), changes the actual voltage ratio. A divider is unsuitable as a regulated power supply because its output changes with load current.

III. Current Divider Law — Parallel Distribution of Current

A. Validation and analysis of the Current Divider Law

The Current Divider Law states that the current through a parallel branch is equal to the total current multiplied by the conductance of that branch divided by total conductance. For two branches, this means branch current is inversely proportional to branch resistance.

Consider two resistors (R_1) and (R_2) connected in parallel across the same voltage (V). Let (I_T) be the total current entering the parallel combination, and let (I_1) and (I_2) be the branch currents.

Because both resistors have the same voltage, Ohm’s law gives

I_1 = V / R_1
I_2 = V / R_2
I_T = I_1 + I_2

The equivalent resistance of the two parallel resistors is

R_P = (R_1 R_2) / (R_1 + R_2)

Using (V=I_T R_P) in (I_1=V/R_1) gives

I_1 = I_T [R_2 / (R_1 + R_2)]

Similarly,

I_2 = I_T [R_1 / (R_1 + R_2)]

• Two-branch law: The current through (R_1) is

  I_1 = I_T [R_2 / (R_1 + R_2)]

The opposite resistance (R_2) appears in the numerator because a lower (R_1) must receive the larger current.

• General conductance form: If (G_k=1/R_k) is the conductance of branch (k), then

  I_k = I_T [G_k / (G_1 + G_2 + ... + G_n)]

where (I_k) is branch current and (G_k) is measured in siemens.

• KCL validation: Kirchhoff’s Current Law requires

  I_T = I_1 + I_2

Substituting the two divider expressions produces

  I_T [(R_2 + R_1) / (R_1 + R_2)] = I_T

confirming current conservation.

• Inverse relationship: The branch with smaller resistance carries greater current. Equal resistors divide the current equally.
• Voltage condition: The law is valid because parallel branches share the same two-node voltage. It must not be applied directly to arbitrary resistors that do not have identical terminal connections.
• Direction convention: If a branch current is defined opposite to the actual direction, its calculated value appears negative; the magnitude indicates the physical current.

Worked example

A total current of (6\text{ mA}) enters two parallel resistors, (R_1=2\text{ k}\Omega) and (R_2=4\text{ k}\Omega).

I_1 = 6 mA [4 kΩ / (2 kΩ + 4 kΩ)] = 4 mA
I_2 = 6 mA [2 kΩ / (2 kΩ + 4 kΩ)] = 2 mA

The branch currents add to (4\text{ mA}+2\text{ mA}=6\text{ mA}). The common branch voltage is (I_1R_1=8\text{ V}), and (I_2R_2=8\text{ V}), confirming the parallel-voltage condition.

B. Applications and limitations

The current divider is useful for predicting branch currents, selecting component ratings, and designing parallel paths with controlled current sharing.

• Parallel-load analysis: When a known total current enters parallel resistors, the law avoids separately calculating the equivalent voltage before finding each branch current.
• Current sensing: A small shunt resistor can carry a known portion of current, producing a measurable voltage (V=IR).
• Bias and transistor circuits: Parallel resistive paths can establish current levels, although active devices may require a complete circuit model beyond the basic law.
• Equal sharing: Identical parallel resistors divide current equally. For (n) identical branches carrying (I_T),

  I_k = I_T / n

where (I_k) is the current in each branch.

• Unequal sharing: A (1\text{ k}\Omega) branch in parallel with a (9\text{ k}\Omega) branch carries nine times the current of the (9\text{ k}\Omega) branch, because both branches have the same voltage.
• Power rating: Branch power can be found from either

  P_k = I_k^2 R_k
  P_k = V^2 / R_k

where (P_k) is branch power and (V) is the common parallel voltage.

• Source and wire resistance: Real sources may have internal resistance, which changes the available total current. Connecting-wire resistance can also disturb the assumed branch relationships.
• Component tolerances: Parallel resistors with nominally equal values may not share current equally because their actual resistances differ. Temperature changes can further alter resistance and current.
• Open and short conditions: An open branch carries approximately zero current. An ideal short-circuit branch can draw extremely large current, limited in practice by source and conductor resistance; the simple divider calculation must then be interpreted with care.
• Measurement loading: An ammeter must be connected in series with the branch being measured. Connecting it directly across a parallel network can create an unintended low-resistance path and damage the instrument.