Kirchhoff's Laws

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KIRCHHOFF'S VOLTAGE LAW (KVL)

G.R. Kirchhoff extended Ohm's law in 1847.

Application of Kirchhoff's Voltage Law

V_S + V₁ + V₂ + V₃ + ... + V_n = 0

1. Assume a direction of current through the circuit.
2. Assign polarities to all resistors and sources.
3. Trace the loop. Any voltage (source or drop) whose negative terminal is reached first is a negative term and any voltage whose positive terminal is reached first is a positive term.
4. Write the equation as the loop is traced by writing each term using the appropriate sign negative or positive.
5. Solve for the unknown voltage.

Series-Aiding and -Opposing Sources

When multiple voltage sources exist in a circuit they can be series-aiding or series opposing. Sources that cause current to flow in the same direction are series aiding. Sources that cause current to flow in opposite directions are series opposing. In an opposing circuit the current flow direction is determined by the larger source.

VOLTAGE DIVIDERS

Voltage across a single resistor in a series circuit has the same proportionality to the total voltage as the resistor value is to the total resistance. This proportional method of determining a voltage drop is called the voltage-divider rule.

The voltage-divider rule allows voltage drops to be determined without solving for current in a circuit.

V_RX = ( R_X / R_T ) V_S

KIRCHHOFF'S CURRENT LAW (KCL)

Current divides mathematically in a parallel circuit. The sum of the currents entering and leaving any node in a circuit is equal to zero.

I₁ + I₂ + ... + I_n = 0

CURRENT DIVIDERS

You can determine individual branch currents in a parallel circuit if total current and resistance are known without solving for the source voltage.

The amount of current in one of two parallel branches is calculated by multiplying the total current by the opposite branch resistance divided by their resistive sum.

I₁ = ( R₂ / ( R₁ + R₂)) I_T

I₂ = ( R₁ / ( R₁ + R₂)) I_T

LOADED VOLTAGE DIVIDERS

When actual loads are placed across a series voltage divider circuit the circuit becomes a series-parallel circuit. The voltage and currents in the unloaded voltage divider change. To build a practical voltage divider the load requirements must be known.

There is some wasted energy or bleeder current in any voltage divider circuit. The bleeder current is the difference between the total of the individual load currents and the total circuit current.

Bleeder current is generally designed to be 10% to 25% of the total load current. If the bleeder current is too small slight variations in the load current can cause significant changes in the voltage. If the bleeder current is too high there is significant waste.

Designing a Loaded Voltage Divider

Calculate the total load current by adding each load current.

I_LT = I_L1 + I_L2 + I_L3 (Kirchhoff's current law)

Calculate the bleeder current by using a percentage of the total load current. (Generally 10% to 25%)

I_B = 10% of I_LT

Calculate the total current by adding the bleeder current to the total load current.

I_T = I_B + I_LT  (Kirchhoff's current law)

Calculate the voltage across each resistor based on each load voltage.

V₁ = V_L1 (Kirchhoff's voltage law)

V₂ = V_L2 - V_L1 (Kirchhoff's voltage law)

V₃ = V_L3 - V_L2 (Kirchhoff's voltage law)

Calculate the current flow for each resistor.

I_R1 = I_B

I_R2 = I_R1 + I_L1  (Kirchhoff's current law)

I_R3 = I_R2 + I_L2  (Kirchhoff's current law)

Calculate the resistor value for each resistor using ohm's law.

R₁ = V₁ / I_R1

R₂ = V₂ / I_R2

R₃ = V₃ / I_R3

Calculate the power for each resistor.

P₁ = I_R1 * V₁

P₂ = I_R2 * V₂

P₃ = I_R3 * V₃

For a safety factor use a resistor with at least double the power rating of the calculated resistor circuit power. Note that some standard resistor power ratings are 1/8 W (125 mW), 1/4 W (250 mW), 1/2 W (500 mW).

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