Kirchhoff's laws

Element laws such as Ohm's law describe each component on its own. To solve a circuit, we also need laws that describe how the components are connected. In 1845, the German physicist Gustav Kirchhoff, then a 21-year-old student, stated two such laws. One expresses conservation of charge, the other conservation of energy. Together with the element laws, they are all that is needed to solve any lumped circuit, however large. Every method in the rest of this course, including nodal analysis, mesh analysis and the circuit theorems, is a systematic way of applying them.

This chapter defines the vocabulary of circuit topology, states Kirchhoff's laws, and uses them to derive series and parallel combinations, voltage and current dividers, and the wye-delta transformation. It ends with circuits containing dependent sources, which need nothing new except one extra equation.

Nodes, branches and loops

Analysing a circuit starts with its topology, the pattern of connections, independent of what the elements are:

• A branch is a single two-terminal element, such as a resistor or a source.
• A node is a point where two or more branches connect. All the wire between elements counts as part of one node, because ideal wires have no resistance and therefore no voltage difference along them. A common beginner's error is to count two dots joined by a wire as two nodes.
• A loop is any closed path through the circuit that passes through no node more than once.
• A mesh is a loop that encloses no other branch: one of the "window panes" of a circuit drawn flat (a planar circuit).

For a connected circuit with \(b\) branches and \(n\) nodes, there are exactly \(b - n + 1\) independent loops, which is also the number of meshes. This relation tells us how many independent equations each method in Lesson 4 will produce.

Kirchhoff's current law (KCL)

Charge cannot accumulate at a node, because a node is just a junction of ideal wires with no ability to store charge. So whatever current flows in must flow out. Kirchhoff's current law states:

The algebraic sum of the currents entering any node is zero, with currents leaving counted as negative entering. Equivalently, the sum of currents entering equals the sum leaving. KCL also applies to any closed boundary drawn around part of a circuit, called a supernode or cut set: the net current crossing the boundary is zero. This generalized form is often a shortcut.

Kirchhoff's voltage law (KVL)

Voltage between two points is the energy per unit charge needed to move between them, and that does not depend on the path taken. Going around any closed loop and returning to the starting point, the net change in potential must be zero. Kirchhoff's voltage law states:

To apply KVL, choose a starting point and a direction around the loop, then add each element's voltage: count it as a drop (positive) if you enter the element at its + reference terminal, and as a rise (negative) if you enter at its − terminal. Any consistent convention works, provided it is used for every element in the loop.

A useful consequence: the voltage between any two nodes can be found by adding the element voltages along any path between them. Elements connected in parallel share the same two nodes, so they must have the same voltage.

Series and parallel combinations

Elements are in series when they carry the same current: they are connected end to end with nothing else attached at the nodes between them. Elements are in parallel when they share the same pair of nodes and so have the same voltage. Kirchhoff's laws turn these definitions into simple combination rules.

Resistors in series

For \(N\) resistors in series carrying current \(i\), KVL gives \(v = iR_1 + iR_2 + \dots + iR_N\), so the combination behaves as one resistor:

Resistors in parallel

For resistors in parallel with voltage \(v\), KCL gives \(i = v/R_1 + v/R_2 + \dots\), so conductances add:

For two resistors, \(R_{\text{eq}} = R_1R_2/(R_1 + R_2)\), "product over sum", written \(R_1 \parallel R_2\). The parallel combination is always smaller than the smallest resistor.

Sources combine too, with care. Voltage sources in series add (with signs); current sources in parallel add. Voltage sources of different values must never be put in parallel, and current sources of different values must never be put in series, because the circuit would violate KVL or KCL.

Voltage and current dividers

Two results occur so often that they are worth memorizing. In a series chain of resistors across voltage \(v\), the voltage across one of them is its share of the total resistance:

In a set of parallel resistors carrying total current \(i\), the current in one of them is its share of the total conductance; for two resistors,

Note the "opposite" resistor in the numerator: more current takes the path of less resistance.

The wye-delta transformation

Some networks contain resistors that are neither in series nor in parallel. The classic case is a bridge, where the resistors form a triangle (delta, Δ, also called pi) or a three-pointed star (wye, Y, also called tee). A delta and a wye connecting the same three terminals can be made exactly equivalent from the outside. For a delta with resistors \(R_{ab}\), \(R_{bc}\) and \(R_{ca}\) between terminals a, b and c, the equivalent wye has, from each terminal to the centre point:

Each wye resistor is the product of the two delta resistors touching that terminal, divided by the sum of all three. The reverse transformation, from wye to delta, is \(R_{ab} = (R_aR_b + R_bR_c + R_cR_a)/R_c\), and similarly for the others: the sum of the pairwise products divided by the opposite wye resistor. Power engineers use the same idea for wye- and delta-connected three-phase loads and transformers.

The Wheatstone bridge

A bridge whose two dividers produce the same voltage at their midpoints is balanced: no current flows in the middle branch, which can be removed or shorted without changing anything. The Wheatstone bridge exploits this to measure an unknown resistance precisely. With \(R_1\) and \(R_2\) forming one divider and \(R_3\) and the unknown \(R_x\) the other, arranged so that \(R_1\) faces \(R_3\) and \(R_2\) faces \(R_x\), the bridge balances when

For example, with \(R_1 = 100\) Ω, \(R_2 = 250\) Ω and \(R_3\) adjusted to 120 Ω for balance, \(R_x = 250 \times 120/100 = 300\) Ω. Strain gauges, pressure sensors and resistance temperature detectors (RTDs) are routinely read with bridge circuits, because a small change in one arm produces a small, easily amplified voltage across the middle.

Circuits with dependent sources

Dependent sources obey KCL and KVL like everything else. The only extra step is to write the controlling variable in terms of the unknowns.