Analysis of Resistors in Series and Parallel

Catalog

I. Resistors in Series and Parallel Circuits

1. Resistors in Series Circuits

When resistors are connected end-to-end with no branches, they form a series circuit. The diagram below shows a series circuit with two resistors.

Resistors in Series Circuits

Key properties of series circuits:

(1) The current is the same through every component.

For n series resistors:

(2) The source voltage equals the sum of the voltage drops across each series resistor.

(3) The total (equivalent) resistance is the sum of all series resistances.

Replacing R1 and R2 by their equivalent R leaves current and node voltages unchanged.

(4) Voltage division and power in series:

Because the current is identical everywhere, each resistor’s voltage is proportional to its resistance, and the power dissipated by each is also proportional to its resistance.

Tip: Series resistors are commonly used as voltage dividers (e.g., to extend a voltmeter’s range). Always verify power ratings and add a safety margin.

2. Resistors in Parallel Circuits

Two or more resistors connected between the same two nodes (i.e., with the same voltage across them) form a parallel circuit.

Resistors in Parallel Circuits

Key properties of parallel circuits:

(1) Each branch has the same voltage.

(2) The total current equals the sum of branch currents.

(3) The inverse of the equivalent resistance equals the sum of inverses of branch resistances.

(4) Current division and power in parallel:

Because branch voltages are equal, branch current is inversely proportional to branch resistance. Branch power is also inversely proportional to resistance for a fixed branch voltage.

For two parallel resistors, the current division (shunt) formula is:

Note: Household lighting and appliance circuits are in parallel so that one device turning off or failing does not interrupt others.

II. Calculation of Series and Parallel Resistance

Complex resistor networks mix series and parallel groupings. Reduce them step-by-step using the same rules: combine series groups into sums and parallel groups using reciprocal sums. Remember: series elements share current; parallel elements share voltage.

1. Calculation Example 1

Find the total current drawn from a 12 V supply.

First, R2 and R3 are in series: R2 + R3 = 8 Ω + 4 Ω = 12 Ω. Replace them by 12 Ω (call this RA).

Now RA (12 Ω) is in parallel with R4 (12 Ω), giving R_comb = 6 Ω, then in series with R1 (6 Ω) for a total R_total = 12 Ω.

By Ohm’s law: I = V/R = 12 V / 12 Ω = 1 A.

Voltage drop on R1 is V_R1=I·R1=1·6=6 V, leaving 6 V across the parallel pair. So branch currents are I1=6/12=0.5 A and I2=6/12=0.5 A; the supply current is 1 A (0.5 A + 0.5 A), consistent.

2. Calculation Example 2

Find the equivalent resistance R_EQ for the ladder network.

From the rightmost side, combine step by step (images preserved for each step):

(Formula 4-1)

Thus, R_A + R7 = 4 + 8 = 12 Ω.

12 Ω in parallel with R6 gives R_B=4 Ω, then R_B + R5 = 4 + 4 = 8 Ω.

(Formula 4-2)

8 Ω in parallel with R4 yields R_C=? (per diagram’s numeric values), then R_C + R3 = 8 Ω (as shown in the original worked steps).

(Formula 4-3)

This 8 Ω is in parallel with R2, producing R_D=4 Ω, then R_D + R1 = 4 + 6 = 10 Ω, so R_EQ = 10 Ω.

For networks that cannot be reduced by series/parallel (e.g., bridges, T-pads), use Kirchhoff’s laws, nodal/mesh analysis, or Δ–Y (delta–wye) transforms (see the Practical Add-Ons section).

III. Fault Characteristics and Treatment

1. Features of Short Circuit and Open Circuit in Series Circuits

(1) Short Circuit Features

If one series element becomes shorted (≈0 Ω), the total resistance drops and the series current increases (limited by the remaining series resistance and the source). This may overheat other components and the source.

Short Circuit in a Series Circuit

• Only the remaining resistors limit current; total current increases.

• Voltage drops redistribute—less across the shorted element, more across the others (since current is larger).

• Overcurrent can damage both components and the power supply.

(2) Open Circuit Features

An open circuit (infinite resistance) anywhere in a series path stops current everywhere. No current means ideal resistors have zero voltage drop; practically, the full source voltage appears across the open break (or across any device that became open).

Open Circuit in a Series Circuit

2. Fault Analysis of Series Resistors

Illustrative summary for a two-resistor series string (R1–R2):

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3. Fault Detection of Series Resistors

Use a multimeter (power off for resistance checks; appropriate DC/AC ranges for live voltage checks). Measuring voltage drops across series elements quickly reveals opens (0 A) and abnormal divisions due to drifted values.

Structure of a Multimeter

(1) Open-Circuit Detection

In series, if any element is open, current is zero. With power on, you’ll typically see the full source voltage across the open location and ~0 V across intact resistors.

(2) Short-Circuit Detection

A short lowers total resistance and increases current. Compare measured total/branch currents and voltage drops against expected values. For safety, first measure resistance with power off if possible (to spot near-zero resistance where there should be some finite value).

4. Failure Testing of Parallel Resistors

(1) Open-Circuit Detection (Power Off)

Measure the total resistance of the parallel group. Normally, R<sub>TOTAL</sub> is less than the smallest branch resistor. If the measured value is higher than either branch value, one branch may be open. Branch currents (with power on) can confirm.

Detection of Open Circuit in Parallel Circuits

(2) Short-Circuit Detection (Power Off)

If measured total resistance is ~0 Ω, there is a shorted branch. With power on, that branch will carry disproportionally large current. Isolate branches to localize the fault.

IV. Equivalent Series Resistance (ESR)

ESR is the small, frequency-dependent resistance that appears in series with real capacitors (and inductors), arising from electrode/foil resistance, electrolyte, leads, and dielectric losses. Adding capacitors in series increases total ESR; adding them in parallel lowers ESR.

Because of ESR, a capacitor’s voltage can exhibit an instantaneous step equal to I × ESR at current transients, degrading filtering. Low-ESR capacitors improve ripple and transient response in power supplies, while some regulators actually require a minimum ESR range for loop stability.

Typical ESR magnitudes vary widely by technology, size, and frequency:

• MLCC (ceramic): very low ESR (milliohms to a few tens of milliohms at high frequency).

• Tantalum (MnO<sub>2</sub>): commonly ~0.1–2 Ω; polymer tantalum can reach tens of milliohms.

• Aluminum electrolytic: legacy types may be >0.1–1+ Ω; low-ESR aluminum parts can also reach tens of milliohms at certain frequencies.

ESR also interacts with ESL (equivalent series inductance). Older wound capacitors had higher ESL; modern constructions reduce ESL, but at high frequencies ESL can dominate and produce resonances.

ESL and ESR Cancellation for Capacitors

Ripple relation: for a ripple current I, ESR produces ripple voltage V<sub>ESR</sub> = ESR × I. Reducing ESR or current ripple reduces output ripple.

Typical ESR vs. Frequency (Tantalum)

Design notes (updated): check regulator datasheets for required ESR ranges; consider paralleling multiple capacitors (e.g., bulk electrolytic + MLCC) to lower net ESR/ESL; observe ripple current ratings and thermal derating.

V. Practical Add-Ons (2025 Update)

• Δ–Y (Delta–Wye) Transform: For bridge-like networks, convert between delta and wye to enable series/parallel reduction.

• Power Rating & Derating: Always ensure P=I²R=V²/R stays below the resistor’s rated power, with margin. Apply temperature derating per datasheet.

• Temperature Coefficient (TCR): Precision resistors (e.g., metal-film, thin-film) have low TCR (e.g., &plusmn;5–50 ppm/°C) and are preferred for accurate dividers.

• Tolerance & E-Series: Standard values follow E6/E12/E24… series. Match tolerance to application (e.g., 0.1% for precision sensing).

• Noise: Carbon composition resistors are noisier than metal-film/thin-film. For low-noise designs, prefer metal-film and keep resistor values moderate.

• Paralleling Resistors: In high-power designs, use equal values in parallel to share current; place them thermally symmetrically.

• Measurement Safety: Measure resistance with power removed and capacitors discharged. For live measurements, start with the highest meter range.

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