Subject: Electrical Engineering · Type: Assignment · Level: Undergraduate · ~1901 words · Harvard referencing
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1. Introduction
The analysis of direct-current (DC) resistive networks is a cornerstone of electrical engineering education, because the tools developed for it underpin almost everything that follows, from power distribution to amplifier biasing and sensor interfacing. When a circuit contains more than one source of electromotive force (EMF), the simple series–parallel reduction rules are no longer sufficient on their own, and a systematic method is required. Kirchhoff’s current and voltage laws provide exactly such a method: they express the conservation of charge and energy in a form that yields a set of simultaneous linear equations for the unknown currents (Hughes et al., 2016).
This assignment analyses a defined two-source resistive network as a worked example. Specific resistor values and source EMFs have been chosen so that every quantity can be computed explicitly and cross-checked by more than one route. The branch currents and node voltage are first found using Kirchhoff’s laws, then confirmed by mesh (loop) analysis. A Thevenin equivalent circuit is then derived for a chosen load resistor, and the maximum power transfer theorem is applied to establish the load value that would extract the greatest power. The methods are entirely general and transfer directly to any linear resistive network.
2. Theoretical basis
Two laws formulated by Gustav Kirchhoff govern the behaviour of any lumped-element circuit (Boylestad, 2016):
1. Kirchhoff’s Current Law (KCL). The algebraic sum of the currents entering any node (junction) is zero. This is a statement of the conservation of electric charge: charge cannot accumulate at a point in a conductor, so whatever flows in must flow out. 2. Kirchhoff’s Voltage Law (KVL). The algebraic sum of the potential differences around any closed loop is zero. This follows from the conservation of energy: a unit charge taken around a closed path and returned to its starting point experiences no net change in potential energy.
Two further results are used in the later sections. Thevenin’s theorem states that any linear network of sources and resistances, viewed from a chosen pair of terminals, can be replaced by a single voltage source V<sub>Th</sub> in series with a single resistance R<sub>Th</sub>, without changing the behaviour of anything connected across those terminals (Bird, 2017). The maximum power transfer theorem states that a source of fixed internal (Thevenin) resistance delivers the greatest power to a load when the load resistance equals that internal resistance (Boylestad, 2016).
The analysis assumes ideal lumped components: the EMF sources have zero internal resistance, the resistors are linear and constant, and the connecting wires are perfect conductors. Under these assumptions the network is linear, so superposition holds and the governing equations are linear and uniquely solvable.
3. Circuit scenario (worked example)
Consider a DC network built around a single upper node, labelled a, and a common return rail taken as the reference (0 V), labelled b. Three parallel branches connect a to b:
- Branch 1 contains an EMF source E₁ = 12 V in series with a resistor R₁ = 4 Ω. The positive terminal of E₁ faces node a, so the source tends to drive current upward, from b to a, through this branch. Let the branch current be I₁.
- Branch 2 contains an EMF source E₂ = 6 V in series with a resistor R₂ = 6 Ω, again oriented to drive current from b to a. Let the branch current be I₂.
- Branch 3 is a single load resistor R₃ = 3 Ω connecting a back down to b. Let the current through it be I₃, taken as flowing from a to b.
Physically, the two sources both push current up into node a, and that combined current returns to the rail through the load R₃. The objective is to determine I₁, I₂ and I₃, the node voltage V<sub>a</sub>, the Thevenin equivalent seen by R₃, and the load value that maximises power transfer.
4. Branch-current analysis using Kirchhoff’s laws
4.1 Applying KCL
Node a is the only independent junction. Currents I₁ and I₂ enter it from below and I₃ leaves it toward the rail, so KCL at node a gives
$$I_1 + I_2 – I_3 = 0. \qquad (1)$$
4.2 Applying KVL
Two independent loops are available. The first is the loop formed by branch 1 and branch 3. Starting at the rail b, rising through E₁ (a potential gain), falling across R₁ and then falling across R₃ back to b, KVL requires
$$E_1 = I_1 R_1 + I_3 R_3. \qquad (2)$$
The second loop is formed by branch 2 and branch 3:
$$E_2 = I_2 R_2 + I_3 R_3. \qquad (3)$$
4.3 Solving the simultaneous equations
Substituting the numerical values, equations (1)–(3) become
$$I_1 + I_2 – I_3 = 0,$$ $$4 I_1 + 3 I_3 = 12,$$ $$6 I_2 + 3 I_3 = 6.$$
This is a set of three linear equations in three unknowns. Writing it in matrix form and solving (the solution was computed and cross-checked numerically) gives
$$I_1 = \tfrac{5}{3} = 1.667\ \text{A}, \qquad I_2 = \tfrac{1}{9} = 0.111\ \text{A}, \qquad I_3 = \tfrac{16}{9} = 1.778\ \text{A}.$$
The node voltage follows directly from the load branch:
$$V_a = I_3 R_3 = 1.778 \times 3 = \tfrac{16}{3} = 5.333\ \text{V}.$$
All three currents are positive, confirming that the assumed directions were correct: both sources genuinely feed current into node a, and the load returns it to the rail.
4.4 Verification of the laws
The results should satisfy the three governing equations exactly. Checking KCL: I₁ + I₂ = 1.667 + 0.111 = 1.778 A = I₃, as required. Checking KVL on loop 1: I₁R₁ + I₃R₃ = (1.667)(4) + (1.778)(3) = 6.667 + 5.333 = 12.0 V = E₁. Checking KVL on loop 2: I₂R₂ + I₃R₃ = (0.111)(6) + (1.778)(3) = 0.667 + 5.333 = 6.0 V = E₂. All three checks close to the last digit.
5. Cross-check by mesh analysis
An independent confirmation is obtained using mesh (loop) analysis, which reduces the number of unknowns by working with circulating loop currents rather than individual branch currents (Irwin and Nelms, 2015). Defining a left mesh current I<sub>A</sub> (circulating through branches 1 and 3) and a right mesh current I<sub>B</sub> (circulating through branches 2 and 3), and noting that the shared load carries the sum of the two, the mesh equations are
$$E_1 = I_A (R_1 + R_3) + I_B R_3,$$ $$E_2 = I_A R_3 + I_B (R_2 + R_3).$$
Inserting the values gives
$$\begin{aligned} 7 I_A + 3 I_B &= 12,\\ 3 I_A + 9 I_B &= 6. \end{aligned}$$
Solving yields I<sub>A</sub> = 1.667 A and I<sub>B</sub> = 0.111 A, so that the load current is I<sub>A</sub> + I<sub>B</sub> = 1.778 A. These match the branch-current results exactly, which provides strong confidence in the solution.
A further audit is available through the power balance. The total power supplied by the two sources must equal the total power dissipated in the resistors:
$$P_\text{supplied} = E_1 I_1 + E_2 I_2 = (12)(1.667) + (6)(0.111) = 20.667\ \text{W},$$ $$P_\text{dissipated} = I_1^2 R_1 + I_2^2 R_2 + I_3^2 R_3 = 11.111 + 0.074 + 9.481 = 20.667\ \text{W}.$$
The two figures agree, confirming that energy is conserved and that no arithmetic slip has occurred.
6. Thevenin equivalent for the load R₃
It is often convenient to characterise the network as seen by a single component; this is precisely what Thevenin’s theorem provides. Here the load of interest is R₃, so the theorem is applied to the remaining network at the terminals a–b with R₃ removed.
6.1 Open-circuit (Thevenin) voltage
With R₃ disconnected, branches 1 and 2 form a single series loop between the two source EMFs. The circulating current is driven by the difference between the sources:
$$I_\text{loop} = \frac{E_1 – E_2}{R_1 + R_2} = \frac{12 – 6}{4 + 6} = 0.600\ \text{A}.$$
The open-circuit voltage at the terminals is the potential of node a relative to the rail. Evaluating it along branch 1,
$$V_\text{Th} = E_1 – I_\text{loop} R_1 = 12 – (0.600)(4) = 9.600\ \text{V}.$$
Evaluating along branch 2 as an independent check gives the same value, V<sub>Th</sub> = E₂ + I<sub>loop</sub>R₂ = 6 + (0.600)(6) = 9.600 V, confirming the result.
6.2 Thevenin resistance
The Thevenin resistance is the resistance seen from the terminals when every ideal EMF source is replaced by a short circuit (its internal resistance). With E₁ and E₂ shorted, R₁ and R₂ appear in parallel between a and b:
$$R_\text{Th} = R_1 \parallel R_2 = \frac{R_1 R_2}{R_1 + R_2} = \frac{4 \times 6}{4 + 6} = 2.400\ \Omega.$$
6.3 Reconnecting the load
The original network, from the point of view of R₃, is now a 9.600 V source in series with 2.400 Ω. Reconnecting the 3 Ω load gives
$$I_3 = \frac{V_\text{Th}}{R_\text{Th} + R_3} = \frac{9.600}{2.400 + 3.000} = \frac{9.600}{5.400} = 1.778\ \text{A}.$$
This is identical to the value obtained from Kirchhoff’s laws in Section 4, which validates the Thevenin reduction. The great advantage of the equivalent circuit is now apparent: the effect of any load value can be found from a single division, without re-solving the whole network each time.
7. Maximum power transfer
Because the network delivering power to the load has been reduced to a fixed source and a fixed internal resistance, the maximum power transfer theorem can be applied directly. The power delivered to a load R<sub>L</sub> is
$$P_L = \left(\frac{V_\text{Th}}{R_\text{Th} + R_L}\right)^2 R_L,$$
and this is greatest when R<sub>L</sub> = R<sub>Th</sub>. For the present network the optimum load is therefore 2.400 Ω, and the corresponding maximum power is
$$P_\text{max} = \frac{V_\text{Th}^2}{4 R_\text{Th}} = \frac{9.600^2}{4 \times 2.400} = \frac{92.16}{9.600} = 9.600\ \text{W}.$$
The actual load in the worked example, R₃ = 3 Ω, is close to but not equal to the optimum, and the power it receives is
$$P_{R_3} = I_3^2 R_3 = (1.778)^2 \times 3 = 9.481\ \text{W}.$$
As expected, this is slightly below the theoretical maximum of 9.600 W, confirming that 3 Ω is a near-optimal but not perfectly matched load. It is worth noting that at the matched condition the efficiency is only 50 per cent, because an equal amount of power is dissipated inside R<sub>Th</sub>. Maximum power transfer is therefore desirable in signal and communications contexts, where extracting the most power matters, but not in power delivery, where efficiency is the priority (Bird, 2017).
8. Discussion
The worked example demonstrates the complementary strengths of the two analytical approaches. Kirchhoff’s laws are completely general and expose the full internal state of the circuit: every branch current and node voltage is obtained at once. Their cost is that the size of the simultaneous system grows with the number of nodes and loops, which becomes laborious by hand for larger networks. Mesh analysis mitigates this by reducing the number of equations, and its agreement here provides an independent check on the arithmetic.
Thevenin’s theorem serves a different purpose. Rather than describing the whole circuit, it condenses everything behind a chosen pair of terminals into two numbers, V<sub>Th</sub> and R<sub>Th</sub>. This is invaluable when a single component, such as a load, a meter or a protection element, is to be studied across a range of values, and it leads naturally into the maximum power transfer result. The fact that the load current computed from the Thevenin model matched the Kirchhoff solution exactly is not a coincidence but a demonstration that the theorem is an exact consequence of the underlying laws.
The maximum power transfer result also carries an important practical lesson that is easily overlooked in a purely numerical treatment. Because only half of the power supplied by the equivalent source reaches the load at the matched condition, deliberately matching a load is worthwhile only where the priority is to extract the largest possible signal, such as in audio, instrumentation and radio-frequency front ends. In a power-engineering context the opposite design choice is made: the source resistance is kept as small as possible relative to the load, so that efficiency approaches unity and dissipation inside the supply is minimised. The same two numbers, V<sub>Th</sub> and R<sub>Th</sub>, therefore inform two quite different engineering objectives, and recognising which objective applies is part of the analysis rather than an afterthought.
A limitation to acknowledge is that all of the methods used here assume linearity. Real sources have internal resistance, and real resistors have tolerances and temperature coefficients, so a measured circuit would deviate slightly from the idealised figures. Nevertheless, for the linear resistive networks that dominate introductory analysis, the agreement between the three routes taken in this assignment gives high confidence in the results.
9. Conclusion
A two-source DC network was analysed by three independent routes. Kirchhoff’s current and voltage laws gave branch currents of I₁ = 1.667 A, I₂ = 0.111 A and I₃ = 1.778 A, with a node voltage of 5.333 V; mesh analysis reproduced these exactly, and a power balance of 20.667 W confirmed energy conservation. Reducing the network to a Thevenin equivalent of 9.600 V in series with 2.400 Ω yielded the same load current of 1.778 A, and the maximum power transfer theorem showed that a matched load of 2.400 Ω would receive 9.600 W, against 9.481 W actually delivered to the 3 Ω load. The consistency across all methods illustrates both the reliability of the classical circuit theorems and the value of cross-checking a solution by more than one technique.
References
Bird, J. (2017) Electrical Circuit Theory and Technology. 6th edn. London: Routledge.
Boylestad, R.L. (2016) Introductory Circuit Analysis. 13th edn. Harlow: Pearson Education.
Hughes, E., Hiley, J., Brown, K. and Smith, I.M. (2016) Hughes Electrical and Electronic Technology. 12th edn. Harlow: Pearson Education.
Irwin, J.D. and Nelms, R.M. (2015) Basic Engineering Circuit Analysis. 11th edn. Hoboken, NJ: John Wiley & Sons.
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