Chapter 04: Series Circuits and Kirchhoff’s Voltage Law
Series Circuits and Kirchhoff’s Voltage Law
String a few old-fashioned holiday lights together and you have built one of the simplest circuits there is—and learned its most famous weakness. When a single bulb burns out, the entire string goes dark. That all-or-nothing behavior is the signature of a series circuit: components connected one after another, end to end, so that there is only a single path for current to follow.
Series circuits are everywhere—batteries stacked in a flashlight, a fuse guarding a load, the volume control on a speaker. Because they have just one path, they obey two beautifully simple rules: the current is the same everywhere, and the voltages around the loop always add up. That second rule is Kirchhoff’s voltage law, one of the two great bookkeeping laws of circuit analysis. This chapter shows how to find the current, the voltage across each part, and the power in any series circuit—building directly on Ohm’s law and the power law from the last two chapters.
4.1 What Is a Series Circuit?
A series circuit connects its components in a single chain, so that current leaving one component has nowhere to go but into the next. There are no branches and no forks—just one continuous loop from the source, through every component in turn, and back again. Figure 4.1 shows a battery driving three resistors wired in series.
Because there is only one path, the charge that flows past any point must also flow past every other point—none of it can disappear or take a shortcut. This gives the defining property of a series circuit:
This is exactly what you would expect from the water analogy of Chapter 2: if a pump pushes water around a single pipe with several narrow sections in a row, the same flow must pass through every section—there is nowhere else for the water to go. The narrow sections (the resistors) each slow the flow, and, as we will see, the pressure drops across them add up to the pump’s pressure.
4.2 Adding Up Resistance
When resistors sit one after another in a series circuit, each one adds its own opposition to the same single path. The current has to fight its way through all of them in turn, so their resistances simply add together. The combined value is called the total (or equivalent) resistance, written RT:
RT = R1 + R2 + R3 + …
A string of resistors in series therefore always has a larger resistance than any one of them alone—adding more resistors makes the path harder, not easier. Figure 4.2 shows three resistors collapsing into a single equivalent resistor.
4.3 Current and Voltage Drops
Finding the current is only the first step. As that single current pushes through each resistor, it produces a voltage drop across that resistor—the amount of the source’s voltage “spent” there. Because the same current I flows through every resistor, each drop is just Ohm’s law applied to that resistor:
V1 = IR1, V2 = IR2, V3 = IR3
A useful consequence: since the current is shared, the largest resistor takes the largest share of the voltage. Resistance and voltage drop march in step.
4.4 Kirchhoff’s Voltage Law
In every example so far, the voltage drops have added up to the source voltage. This is not a coincidence but a law of nature—Kirchhoff’s voltage law (KVL), named for the physicist Gustav Kirchhoff. It can be stated very simply:
KVL is really a statement about energy. Each coulomb of charge picks up energy at the battery—VS joules per coulomb—and then spends a little of it as heat crossing each resistor. By the time the charge returns to the battery, it has spent exactly what it gained, no more and no less. Figure 4.3 traces this rise and fall in potential as you walk once around the loop.
KVL is most useful when one voltage in a loop is unknown. If you know the source and all but one of the drops, the missing drop is whatever it takes to make the total balance.
4.5 The Voltage Divider
One of the handiest uses of a series circuit is to produce a smaller voltage from a larger one. Two resistors in series split the source voltage between them in proportion to their resistances, so by tapping the connection between them you get a predictable fraction of the source. This arrangement is called a voltage divider, and it appears in sensors, audio controls, and countless other circuits.
Since the same current flows through both resistors, the voltage across the lower resistor R2 is its share of the total—the voltage-divider rule:
Vout = VS × R2 ⁄ (R1 + R2)
The fraction R2/(R1 + R2) is just that resistor’s slice of the total resistance. Figure 4.4 shows a divider that turns 9 V into 6 V.
Figure 4.4 A voltage divider. R₂ takes two-thirds of the total resistance, so it takes two-thirds of the voltage.
4.6 Power and Practical Series Circuits
Each resistor in a series circuit dissipates power according to the power law from Chapter 3, and the total power is simply the sum of the individual powers. It also equals the source voltage times the one circuit current:
PT = VSI = P1 + P2 + P3
Both routes must agree, which makes a handy check on your work.
Chapter Summary
- A series circuit has a single path for current—components connected end to end in one loop.
- The current is the same everywhere in a series circuit (the constant-current property).
- Resistances in series add: RT = R1 + R2 + …, which is always larger than any single resistor.
- The circuit current is I = VS/RT, and the drop across each resistor is V = IR. The largest resistor drops the most voltage.
- Kirchhoff’s voltage law: the voltage drops around a loop add up to the source voltage. It is a statement of energy conservation.
- A voltage divider splits the source voltage in proportion to the resistances: Vout = VS × R2/(R1 + R2).
- Total power is the sum of the individual powers and equals VSI. Stacking cells in series adds their voltages.
Key Terms
Review Problems
A. Concepts
A1. In a series circuit, which quantity is the same at every point?
A2. How do you find the total resistance of resistors connected in series?
A3. State Kirchhoff’s voltage law in your own words.
A4. In a series circuit, which resistor has the largest voltage drop—the largest or the smallest?
B. Total resistance
B1. 100 Ω + 220 Ω + 330 Ω in series.
B2. 1 kΩ + 2.2 kΩ + 4.7 kΩ in series.
B3. Three 47 Ω resistors in series.
B4. 1.5 kΩ + 680 Ω in series.
C. Current and voltage drops
C1. A 12 V source drives 20 Ω and 40 Ω in series. Find I, V₁, and V₂.
C2. A 9 V source drives 1 kΩ and 2 kΩ in series. Find I, V₁, and V₂.
C3. A 24 V source drives three 100 Ω resistors in series. Find I and the drop across each.
C4. A 5 V source drives 220 Ω and 330 Ω in series. Find I, V₁, and V₂.
D. Kirchhoff’s voltage law
D1. A 12 V source has drops of 3 V and 4 V across two of three series resistors. Find the third drop.
D2. Three series resistors drop 5 V, 7 V, and 3 V. Find the source voltage.
D3. A 9 V source drives four series resistors that drop 2 V, 2 V, and 2 V across three of them. Find the fourth drop.
E. Voltage dividers
E1. A 10 V supply feeds two equal 1 kΩ resistors. Find the voltage across one of them.
E2. A 12 V supply feeds a 1 kΩ (top) and a 3 kΩ (bottom). Find the voltage across the 3 kΩ.
E3. A 9 V supply feeds a 2 kΩ (top) and a 1 kΩ (bottom). Find the voltage across the 1 kΩ.
F. Power and applications
F1. A series circuit draws 0.2 A from a 12 V source. Find the total power.
F2. Two 1.5 V cells are stacked in series to power a bulb. What is the total voltage?
F3. A 9 V source drives 330 Ω and 470 Ω in series. Find I, V₁, V₂, and the total power.
