Mesh analysis and superposition

Nodal analysis treats voltages as the unknowns. Mesh analysis is its counterpart: it treats loop currents as the unknowns and writes Kirchhoff's voltage law around each loop. For circuits with many series elements and few loops, mesh analysis gives fewer equations; for circuits with many parallel branches, nodal analysis usually wins. This lesson also introduces superposition, a property of every linear circuit that lets you find the effect of each source separately and add the results. Superposition explains how noise and offsets combine and is the foundation for the AC analysis in lessons 11 and 12.

The NS-4's power path is a natural example. During the day, the solar charger and the battery both connect to the main 12 V bus that feeds the regulator. Ines wants to know how the load current splits between them, and how much of the charger's output actually reaches the battery as charge, when the radar and radio are running.

Meshes and mesh currents

A planar circuit is one that can be drawn without crossing wires. Its meshes are the loops that contain no other loop inside them, the windows of the drawing. Mesh analysis assigns a current to each mesh, conventionally circulating clockwise. The actual current in any branch is then the sum of the mesh currents flowing through it: a branch on the outside of the drawing carries one mesh current, and a branch shared by two meshes carries the difference between them.

Mesh currents satisfy KCL automatically, because each one enters and leaves every node it passes through. Only KVL remains to be written, one equation per mesh. A circuit with b branches and n nodes has b − n + 1 meshes, which is often fewer than its n − 1 node voltages. Mesh analysis applies only to planar circuits; nodal analysis works on any circuit, which is one reason simulators use it.

Writing mesh equations

The procedure mirrors nodal analysis. Assign a clockwise current to every mesh. For each mesh, walk around it in the direction of its current and add up the voltage drops, using Ohm's law for each resistor with the net current through it, and add each source's voltage as a drop or a rise depending on its polarity. Set the sum to zero. The result is one linear equation per mesh, and for circuits with only resistors and independent voltage sources the equations take the form R i = v, where R is a symmetric resistance matrix that can be written by inspection.

The right-hand side holds the source voltages around each mesh, counted positive when a source would push current in the mesh's clockwise direction. As with nodal analysis, the pattern makes it possible to write the matrix directly from the schematic, which is the fastest way to set up a hand calculation and the easiest way to check one.

The mesh method found the currents with two equations. Nodal analysis would also need only one unknown, the bus voltage, because the circuit has just two nodes. With practice, you will look at a circuit and choose the method that gives the fewest equations: count the meshes and the non-reference nodes, and remember that voltage sources reduce the node count while current sources reduce the mesh count.

Current sources and supermeshes

A current source in a branch belonging to only one mesh fixes that mesh's current, removing an unknown. A current source in a branch shared by two meshes causes the same difficulty that a floating voltage source causes in nodal analysis: its voltage is unknown and cannot be written with Ohm's law. The cure is the supermesh. Combine the two meshes into one larger loop that avoids the current source, write KVL around the combined loop, and add the constraint that the difference of the two mesh currents equals the source current.

Linearity and superposition

A linear circuit is made of linear elements, such as resistors, capacitors, inductors and linear dependent sources, together with independent sources. Two properties follow from the linearity of the equations. Homogeneity says that scaling every source by a factor scales every voltage and current by the same factor. Additivity says that the response to several sources acting together is the sum of the responses to each acting alone. Together they form the superposition principle.

To apply superposition, turn on one independent source at a time and turn the others off. A voltage source that is turned off is replaced by a short circuit, because its voltage becomes zero; a current source that is turned off is replaced by an open circuit, because its current becomes zero. Dependent sources are never turned off, because they are part of the circuit's behavior rather than inputs. Solve each single-source circuit, then add the results with their signs.

Superposition applies to voltages and currents but not to power. Power is proportional to the square of current, and the square of a sum is not the sum of the squares. To find the power in an element, first add the currents or voltages from each source, then compute the power from the total.

Where superposition matters in digital hardware

Superposition is how engineers reason about signals that ride on top of each other. A logic signal traveling down a trace picks up a little crosstalk from its neighbor; the voltage at the receiver is the intended signal plus the crosstalk, each calculated separately. An analog sensor reading is the sensor's true voltage plus an offset from the amplifier plus noise from the switching regulator; the error budget adds the contributions. A 3.3 V rail carries its DC value plus ripple from the regulator plus spikes from chips switching. Treating each disturbance as a separate source and adding the effects is the core of signal-integrity and power-integrity analysis.

# Mesh analysis of the power bus, and a superposition check
def solve2(a, b, c, d, e, f):
    """Solve [[a, b], [c, d]] x = [e, f] by Cramer's rule."""
    det = a * d - b * c
    return (e * d - b * f) / det, (a * f - e * c) / det

Vch, Rch = 14.2, 0.4      # charger
Vb, Rb = 12.8, 0.1        # battery
RL = 12.0                 # load

i1, i2 = solve2(Rch + RL, -RL, -RL, RL + Rb, Vch, -Vb)
print("mesh currents: i1 = %.3f A, i2 = %.3f A" % (i1, i2))
print("load %.3f A, into battery %.3f A, bus %.2f V" % (i1 - i2, i2, RL * (i1 - i2)))

def par(a, b):
    return a * b / (a + b)

# superposition: one source at a time, the other shorted
i_load_1 = Vch / (Rch + par(RL, Rb)) * Rb / (Rb + RL)
i_load_2 = Vb / (Rb + par(RL, Rch)) * Rch / (Rch + RL)
print("charger alone %.3f A + battery alone %.3f A = %.3f A" % (i_load_1, i_load_2, i_load_1 + i_load_2))

# power does NOT superpose
p_true = (i1 - i2) ** 2 * RL
p_wrong = i_load_1 ** 2 * RL + i_load_2 ** 2 * RL
print("load power %.2f W; adding the separate powers would give %.2f W" % (p_true, p_wrong))

mesh currents: i1 = 3.017 A, i2 = 1.934 A
load 1.083 A, into battery 1.934 A, bus 12.99 V
charger alone 0.235 A + battery alone 0.848 A = 1.083 A
load power 14.07 W; adding the separate powers would give 9.29 W

Choosing a method

For hand analysis, count before you start. A circuit with five nodes and three meshes needs four nodal equations or three mesh equations. Subtract one equation for each voltage source tied to the reference node in nodal analysis, and one for each current source in an outer branch in mesh analysis. If the circuit is a ladder of series and parallel parts, reduction may be quickest of all. If the question asks about the effect of one particular source, superposition may answer it directly. Whatever method you choose, a second method or a power balance is the best check.