Circuit analysis techniques

Lesson 5 covered the rules for series, parallel and combination circuits and Kirchhoff's laws. Many practical circuits need a few more tools: dividers that are loaded by what they feed, bridges that measure small changes in resistance, sources that have their own internal resistance, circuits with more than one source, and three-wire systems with a shared neutral. This lesson develops these tools at the level an electrician needs to understand control circuits, sensors, batteries and the split-phase systems found in every home. The same ideas reappear in AC circuits (ELEC-210) and in engineering courses (ELE-202).

Voltage dividers under load

An unloaded voltage divider of two series resistors \(R_1\) and \(R_2\) across a supply \(V_s\) gives an output across \(R_2\) of:

When a load \(R_L\) is connected across the output, it is in parallel with \(R_2\), lowering the effective resistance of the lower leg and therefore the output voltage. The divider only holds its unloaded value if the load resistance is much larger than \(R_2\), typically at least ten times larger.

Current dividers

In a parallel circuit, current divides so that branches with lower resistance carry more current. For two branches, the current in branch 1 is:

Note that the other branch's resistance appears on top: the smaller \(R_1\) is, the larger its share. Current dividers explain how shunts work in ammeters, how parallel conductors share current (and why unequal lengths cause unequal sharing), and how ground fault current divides between paths.

Bridge circuits

A Wheatstone bridge consists of two voltage dividers side by side across the same supply, with a meter connected between their midpoints. When the ratios of the two dividers are equal, the midpoints are at the same voltage and the meter reads zero: the bridge is balanced.

Bridges detect very small changes in resistance, so they are used with strain gauges in load cells and pressure transmitters, with RTDs in temperature measurement, and in older precision resistance meters. A small change in one resistor unbalances the bridge and produces a small voltage proportional to the change.

Sources with internal resistance

Real sources are not ideal. A battery, generator or power supply behaves like an ideal voltage source in series with a small internal resistance \(r\). When current flows, some voltage is dropped inside the source, so the terminal voltage falls as load increases:

where \(E\) is the open-circuit voltage (sometimes called the electromotive force). Internal resistance explains why a weak battery shows nearly full voltage on a meter but collapses when a load is connected, why cranking a cold engine drops a car battery's voltage, and why a short circuit on a battery produces a large but limited current.

Maximum power transfer

A source delivers the most power to a load when the load resistance equals the source's internal resistance. At that point, however, half the power is wasted inside the source, so efficiency is only 50%. Power systems never operate this way; they keep the load resistance much higher than the source resistance for high efficiency. Matching matters in communications and audio, where getting the most signal power is more important than efficiency.

Circuits with more than one source

When a circuit has two or more sources, such as batteries in parallel or a battery being charged by a charger, Kirchhoff's laws (Lesson 5) still apply. One useful method is superposition: in a circuit of resistors, the current in any branch equals the sum of the currents caused by each source acting alone, with the other voltage sources replaced by short circuits (and current sources by open circuits).

Three-wire systems and the neutral

Homes in North America are supplied by a three-wire system: two "hot" conductors and a neutral, giving 120 V from each hot to neutral and 240 V between the hots. The same idea in DC, the Edison three-wire system, was used in the first distribution networks. With loads on each side, the neutral carries only the difference between the two side currents:

When the two sides are balanced, the neutral carries no current. If the neutral is broken (an "open neutral"), the two 120 V loads end up in series across 240 V, and the voltage divides according to their resistances: the side with the lighter load (higher resistance) gets more than 120 V and can be damaged.

Solving circuits systematically

For any DC circuit, a reliable approach is:

1. Draw the circuit neatly and label every component, source and node.

2. Simplify series and parallel groups where possible (Lesson 5).

3. For circuits that cannot be simplified, write Kirchhoff's current law at nodes or voltage law around loops, and solve the equations.

4. Check the answer: currents into each node should sum to zero, voltages around each loop should sum to zero, and total power supplied should equal total power used.

5. Ask whether the result makes sense physically.

Engineering courses (ELE-202) formalize these methods as nodal and mesh analysis and add theorems such as Thévenin's, which replaces any network seen from two terminals by a single source and series resistance, exactly the model of a source with internal resistance used above.