Exercises: Series, parallel and mixed circuits

Explain in your own words the difference between connecting two light bulbs in series and connecting them in parallel to the same battery. What happens in each case if one of the two bulbs blows?

Complete and explain these rules with a simple numerical example for each: (a) in a series circuit, the current is the same through all components and the voltages...; (b) in a parallel circuit, the voltage is the same across all branches and the currents...

Two resistors of $10\ \Omega$ and $15\ \Omega$ are connected in series. Knowing that in series the resistances add up ($R_\text{eq} = R_1 + R_2$), calculate the equivalent resistance and explain what "equivalent resistance" means.

Explain what a node (or junction) is in an electrical circuit and how it relates to parallel circuits.

The lamps and sockets in a home are connected in parallel to the mains. Give two practical reasons for this choice and explain what would happen if they were connected in series.

Hint. Think about the voltage each appliance receives and about what happens to the others when one of them is switched off, burns out or is disconnected.

Explain in your own words what the equivalent resistance of a set of resistors is and why calculating it simplifies the analysis of a circuit.

Three identical resistors of $9\ \Omega$ are connected in series. Calculate the equivalent resistance and explain the rule you used. What would the equivalent resistance be with four identical resistors of $9\ \Omega$ in series?

Two identical light bulbs connected in series shine less brightly than a single bulb connected to the same battery. Explain why, referring to the total resistance of the circuit and the current flowing in each case.

Two resistors of $12\ \Omega$ and $6\ \Omega$ are connected in parallel. Knowing that $\frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$, calculate the equivalent resistance and verify that the result is smaller than the smaller of the two resistors, explaining why this always happens in parallel.

A series circuit with a $12\ \text{V}$ battery has two resistors: $R_1 = 10\ \Omega$ and $R_2 = 14\ \Omega$. (a) Calculate the equivalent resistance and the current flowing through the circuit. (b) Calculate the voltage drop across each resistor ($V_1 = I \cdot R_1$ and $V_2 = I \cdot R_2$). (c) Verify that $V_1 + V_2$ equals the battery voltage.

The circuit in the diagram is a mixed circuit: resistor $R_1 = 4\ \Omega$ is in series with the parallel combination formed by $R_2 = 12\ \Omega$ and $R_3 = 6\ \Omega$, all powered by a $12\ \text{V}$ battery, as shown in the diagram.

(a) Calculate the equivalent resistance of the parallel section. (b) Calculate the equivalent resistance of the entire circuit. (c) Calculate the total current supplied by the battery, stating its unit.

A string of 20 identical light bulbs connected in series is plugged into $220\ \text{V}$. (a) Calculate the voltage across each bulb if it is distributed equally. (b) If each bulb has a resistance of $110\ \Omega$, calculate the total resistance of the string and the current flowing through it. (c) Explain what happens to the set if a single bulb blows and why locating the fault is so tedious.

Two resistors of $10\ \Omega$ and $15\ \Omega$ are connected in parallel to a $12\ \text{V}$ battery. (a) Calculate the equivalent resistance. (b) Calculate the current flowing through each branch. (c) Calculate the total current and verify that it equals the sum of the branch currents.

Three resistors of $12\ \Omega$, $6\ \Omega$ and $4\ \Omega$ are connected in parallel. Calculate the equivalent resistance using the expression $\frac{1}{R_\text{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ and explain why the result is smaller than the smallest of the three.

The circuit in the diagram has a $12\ \text{V}$ battery and two resistors in series, $R_1 = 4\ \Omega$ and $R_2 = 8\ \Omega$, as shown in the diagram.

(a) Calculate the total resistance of the circuit. (b) Calculate the current flowing through the circuit. (c) Calculate the voltage drop across $R_2$, which is the unknown marked in the diagram, stating its unit.

A student claims: "in a parallel circuit, the current is the same through all branches, just like in series." Correct the error by stating the correct rules for series and parallel circuits (what is shared and what is divided in each) and support your explanation with a numerical example using two different branches.

In a circuit powered by a $20\ \text{V}$ DC source, a resistor $R_1 = 4\ \Omega$ is connected in series with a parallel block made up of two resistors, $R_2 = 10\ \Omega$ and $R_3 = 15\ \Omega$. (a) During a control test, an operator measures with a voltmeter a voltage drop across $R_1$ of $20\ \text{V}$ instead of the theoretical design value ($8\ \text{V}$). Reasonably deduce what fault has occurred in the parallel block (has a branch short-circuited or opened?). (b) If under normal conditions resistor $R_3$ were to suffer an internal break and remain an open circuit, predict how the total current supplied by the source and the power dissipated in $R_1$ would vary, calculating their new numerical values. (c) Justify what auxiliary resistor should be placed in parallel with the original block ($R_2 \parallel R_3$) so that the voltage drop across $R_1$ increases to $12\ \text{V}$.

A traditional Christmas garland consists of 50 identical incandescent bulbs connected in series to $230\ \text{V}$. Each bulb is designed to consume $1\ \text{W}$ at its rated operating voltage ($4,6\ \text{V}$). (a) An apprentice proposes connecting those same 50 bulbs in parallel to $230\ \text{V}$ so that "if one burns out, the others stay lit". Quantitatively justify using Ohm's and Joule's laws what would immediately happen to the circuit. (b) In the series arrangement, each bulb has a thermal shunt mechanism that short-circuits when the filament burns out. If 5 bulbs burn out and become short-circuited, calculate the new voltage supported by each of the remaining 45 bulbs and predict the percentage increase in power they experience, evaluating the chain-reaction effect on their lifespan. (c) Explain why the industry technically replaced this system with low-voltage series branches powered in parallel via a transformer.

You must supply three independent light points of power ratings $40\ \text{W}$, $60\ \text{W}$, and $100\ \text{W}$ (all for $230\ \text{V}$) in an industrial warehouse located $80\ \text{m}$ away from the distribution board, using a two-wire copper line with a cross-section of $1,5\ \text{mm}^2$ (resistivity $\rho = 0,0175\ \Omega\cdot\text{mm}^2/\text{m}$). (a) Justify the type of connection you must choose between the three lamps to ensure their independent operation at their rated voltage, and calculate the total resistance of the line cable (outward and return). (b) With all three lamps switched on simultaneously, calculate the voltage drop in volts produced in the line and check whether the regulatory requirement of not exceeding a $3\,\%$ voltage drop for lighting is met. (c) A classmate claims that connecting the three lamps in series would save copper while maintaining the same lighting level. Quantitatively refute their claim by calculating the actual voltage each lamp would receive in series.

Two electric heaters of $18\ \Omega$ each are available, which must be powered from a $230\ \text{V}$ mains supply via a wiring line whose total conductor resistance is $R_L = 2\ \Omega$. Two connection options are considered: connecting the two heaters in parallel or connecting them in series. (a) For the parallel arrangement, calculate the actual voltage reaching the heaters, the total useful power they deliver, and the electrical efficiency of the line ($\eta = P_{\text{useful}} / P_{\text{total}}$). (b) Calculate the same parameters for the series arrangement. (c) Interpret the results: although the parallel arrangement yields much more thermal power, analyse which of the two arrangements generates greater Joule-effect losses in the line and justify why transmission efficiency is higher in series.

By mistake, a user connects two incandescent bulbs in series to the $230\ \text{V}$ mains that were designed to operate individually at $230\ \text{V}$, but with different rated powers: one of $P_1 = 25\ \text{W}$ and another of $P_2 = 100\ \text{W}$ (assume the resistance of their filaments remains constant). (a) The user believes that the $100\ \text{W}$ bulb will shine four times brighter than the $25\ \text{W}$ one. Analytically refute their claim by calculating the actual power dissipated by each bulb in this series arrangement. (b) Predict which of the two bulbs will face a greater thermal risk of damage and which will barely emit any light, justifying the result based on the voltage each one takes on. (c) If the bulb that heats up the most burns out and remains as an open circuit, predict what voltage a voltmeter connected across each lamp holder will measure.

A test bench supplies $20\ \text{V}$ to three parallel loads of $R_1 = 5\ \Omega$, $R_2 = 10\ \Omega$, and $R_3 = 20\ \Omega$. The main line is protected by a main fuse of $8\ \text{A}$, and the conductors of each branch withstand a maximum admissible current of $5\ \text{A}$. (a) Calculate the current in each branch and the total current. Justify whether the installation operates within safety margins. (b) Due to a mechanical overload, resistor $R_2$ degrades and abruptly drops to $2,5\ \Omega$. Predict which element will fail first: will the main fuse blow or will the line of the faulty branch burn out? Justify it with calculations. (c) Design a modification replacing the main fuse with individual protections in each branch, selecting standard commercial ratings ($1\ \text{A}$, $2\ \text{A}$, $3\ \text{A}$, $5\ \text{A}$, $6\ \text{A}$) so as to ensure selectivity (so that a fault in one branch does not cut off power to the other two).

To condition a sensor's signal for an analog-to-digital converter that accepts a maximum of $4,0\ \text{V}$, a voltage divider is designed from a $12,0\ \text{V}$ source using two resistors, $R_1$ (connected to positive) and $R_2$ (connected to ground, where the output voltage $V_s$ is obtained). (a) Calculate the theoretical ratio between $R_1$ and $R_2$ in an open circuit (no-load) to obtain $V_s = 4,0\ \text{V}$ and propose values setting $R_2 = 10\ \text{k}\Omega$. (b) When connecting the circuit to the sensor input, it exhibits an internal input resistance of $R_{\text{in}} = 20\ \text{k}\Omega$ in parallel with $R_2$. Calculate the actual voltage $V'_s$ it receives and the percentage error caused by the loading effect compared to the intended $4,0\ \text{V}$. (c) Redesign the value of $R_1$ so that, maintaining $R_2 = 10\ \text{k}\Omega$ and with the $20\ \text{k}\Omega$ sensor connected, the terminal voltage is exactly $4,0\ \text{V}$.

A $24\ \text{V}$ DC outdoor decorative luminaire made of white LED diodes is planned. Each LED has a forward operating voltage of $V_F = 3,0\ \text{V}$ at its nominal current of $I_F = 20\ \text{mA}$. (a) A designer proposes connecting 8 LEDs in series directly to the $24\ \text{V}$ without bias resistors to maximize efficiency. Justify why this solution is unfeasible in practice by analyzing the behavior of the diode's $V$-$I$ curve and thermal variations. (b) Design the recommended configuration by arranging the LEDs into identical parallel branches: determine how many series diodes per branch allow a safety voltage drop across the resistor of at least $3\ \text{V}$ and calculate the value of this branch resistor $R_s$. (c) If the final luminaire consists of 10 parallel branches like those designed in (b), calculate the total power consumed and predict what will happen to the current of the remaining branches if one LED in the first branch burns out in an open circuit.