# Circuit reasoning: loading changes both voltage and power — Study worksheet [Study guides](https://unireadinglist.com/guides/circuit-reasoning/) [Course topic: Circuits and Electronics](https://unireadinglist.com/subjects/engineering/#course-circuits-electronics) Sources checked: 2026-11-04 Study route and exercise: University Reading List editorial project. Ideal linear DC resistive analysis: loading, Thevenin equivalence, power balance and a global maximum over positive load resistance. Fiore means DC version 1.0.14; Alexander–Sadiku the selected seventh-edition ISE print; Storey the 2017 sixth-edition paperback. Sources and resistors are fixed and ideal. No component availability, ratings, thermal behavior or construction performance is inferred. ## Choose your starting point - [ ] I need a free first DC course Use Fiore for units, network equations and circuit theorems before its separate AC continuation. - [ ] I want a broad circuit-analysis spine Use Alexander–Sadiku for a systematic DC-to-AC sequence, without buying platform access by accident. - [ ] I want the measurement/system context Use Storey to connect circuit loading with measuring instruments, sensors and functional blocks. ## Compare the three books ### DC Electrical Circuit Analysis: A Practical Approach James M. Fiore · 1.0.14 · 2025 Free first DC analysis Choose it when: School algebra, simultaneous equations and unit conversion are your starting tools. Use another route when: You immediately need phasor or frequency-domain analysis; Fiore’s AC text is a separate continuation. Selected reading: Series-parallel circuits, network theorems and nodal/mesh analysis in the selected DC version. Limits and trade-offs: The full free text has noncommercial/sharealike terms and selected odd-numbered answers, not a complete universal answer key. #### Try before choosing Solve the loaded divider below directly, then account for all resistor powers. Check your work: The node is 6 V and supplied 72 mW equals 36+18+18 mW. Your notes: ________________________________ ISBN: 9781654515478 [Official free reading](https://www2.mvcc.edu/users/faculty/jfiore/books/DCElectricalCircuitAnalysis.pdf) [Book record and access options](https://unireadinglist.com/subjects/engineering/dc-electrical-circuit-analysis-fiore/) ### Fundamentals of Electric Circuits Charles K. Alexander; Matthew N. O. Sadiku · 7 (International Student Edition, Print) · 2021 Comprehensive circuit spine Choose it when: You want DC foundations followed by dynamics, phasors and advanced analysis. Use another route when: You need only an electronics overview or assume a print ISBN includes Connect assignments. Selected reading: Chapters 1–4 for concepts, laws, analysis and theorems; later 7 for first-order circuits and 9 for phasors. Limits and trade-offs: The selected ©2021 ISE print, ebook and Connect are distinct products; the early DC route needs less mathematics than later sections. #### Try before choosing Deactivate the ideal voltage source and find the output resistance of the original divider. Check your work: The source becomes a short, so 1 kΩ and 2 kΩ are parallel: Rth=2/3 kΩ, not 3 kΩ. Your notes: ________________________________ ISBN: 9781260570793 [Book record and access options](https://unireadinglist.com/subjects/engineering/fundamentals-of-electric-circuits/) ### Electronics: A Systems Approach Neil Storey · 6 · 2017 Systems and measurement route Choose it when: You want to see why an instrument or next block can change a circuit’s output. Use another route when: You need only a dense circuit-theorem reference without device/system context. Selected reading: Measurement of Voltages and Currents, Resistance and DC Circuits, then Electronic Systems and Sensors. Limits and trade-offs: Broader electronics context trades some focus for functional understanding; companion resources are not a complete free book. #### Try before choosing Treat a 2 kΩ input resistance as a voltmeter across the 2 kΩ lower divider resistor. Check your work: The input is a load, not an ideal observer: it changes an 8 V open-circuit output to 6 V. Your notes: ________________________________ ISBN: 9781292114064 [Book record and access options](https://unireadinglist.com/subjects/engineering/electronics-a-systems-approach-storey/) ## Check your preparation ### Can I draw the actual topology? Your notes: ________________________________ ### Can I keep units and power signs consistent? Your notes: ________________________________ ### What does a terminal equivalent preserve? Your notes: ________________________________ ## A six-step study plan - [ ] Draw and predict Selected reading: Fiore quantities and series-parallel material. What to do: Draw the unloaded and loaded circuit and predict whether output voltage rises or falls. Check your work: A finite passive shunt load reduces this divider’s output. Your notes: ________________________________ - [ ] Find the terminal equivalent Selected reading: Network-theorem material. What to do: Compute open-circuit voltage and resistance seen with the ideal source deactivated. Check your work: Vth=8 V; Rth=2/3 kΩ. Your notes: ________________________________ - [ ] Solve by two methods Selected reading: Thevenin and nodal analysis. What to do: Compute the loaded voltage using the equivalent and an original-node equation. Check your work: Both give 6 V for RL=2 kΩ. Your notes: ________________________________ - [ ] Close the original power balance Selected reading: DC power and Kirchhoff laws. What to do: Find each branch current and all original-circuit resistor powers. Check your work: Supplied 72 mW equals total absorbed power. Your notes: ________________________________ - [ ] Prove a global load optimum Selected reading: Network theorems and the follow-up. What to do: Maximize load power for all RL\>0 using a square inequality. Check your work: Equality requires RL=Rth; endpoint intuition or a plot alone is not the proof. Your notes: ________________________________ - [ ] Audit the efficiency claim Selected reading: Original circuit and terminal equivalent. What to do: Calculate actual source power at the load-power optimum and compare with the equivalent’s internal balance. Check your work: The original source efficiency is 25%; a terminal equivalent does not preserve its internal power bookkeeping. Your notes: ________________________________ ## Worked exercise: A divider with a finite load An ideal 12 V source feeds Rs=1 kΩ in series with an output node. R2=2 kΩ connects the node to ground. Add RL=2 kΩ from the same node to ground. Find the Thevenin equivalent, loaded voltage, branch currents and the original circuit’s power balance. Assumptions: Fixed ideal source and positive ideal linear resistors; steady DC. Ground is the source’s negative terminal. The load is in parallel with R2. Powers below are source delivery or resistor absorption, as labeled. Your notes: ________________________________ ## Change an assumption: Maximum load power is not a 50% claim about the original source Keep the original 12 V source, 1 kΩ series resistor and 2 kΩ lower resistor fixed, but allow RL to vary over all positive real resistances. Prove the global maximum load power, then compute the actual original-source efficiency PL/Psource at that load. Your notes: ________________________________ ## Before moving on - Solve a loaded divider directly and by terminal equivalence. - Prove the global load-power maximum for positive resistance. - Distinguish original-source power from equivalent-source bookkeeping. ## Answer and interpretation ### Can I draw the actual topology? A 12 V ideal source feeds Rs=1 kΩ, then a node. R2=2 kΩ and RL both connect from that node to ground; they are parallel. ### Can I keep units and power signs consistent? V/kΩ gives mA. Voltage times mA gives mW. State source power as delivered and resistor powers as absorbed. ### What does a terminal equivalent preserve? For this linear network it preserves voltage/current behavior at the load port. It does not preserve the original internal branch currents or dissipation pattern. ### A divider with a finite load 1. Without the load, Vth=12·2/(1+2)=8 V. Short the deactivated ideal voltage source: Rth=1||2=2/3 kΩ. 2. With RL=2 kΩ, Vout=8·2/(2+2/3)=6 V. Directly, R2||RL=1 kΩ and the 1 kΩ/1 kΩ divider also gives 6 V. 3. Source/Rs current is (12−6)/1 kΩ=6 mA; R2 and RL each carry 6/2 kΩ=3 mA. The node balance is 6=3+3 mA. 4. Delivered source power is 12·6=72 mW. Rs absorbs 6²·1=36 mW; R2 and RL each absorb 6·3=18 mW. Total 36+18+18=72 mW. Vth=8 V, Rth=2/3 kΩ, loaded output 6 V; source current 6 mA and each lower branch 3 mA. Load power is 18 mW; the original power balance closes at 72 mW. Test the result: Using the unloaded 8 V at the loaded node violates the original-node equation. The ideal source is shorted only while finding Rth, not in the operating circuit. ### Change an assumption: Maximum load power is not a 50% claim about the original source 1. Terminal equivalence gives PL=Vth² RL/(Rth+RL)². Since (RL−Rth)²≥0, (RL+Rth)²≥4RL Rth, so PL≤Vth²/(4Rth), with equality only at RL=Rth. 2. Thus RL=2/3 kΩ and PL=64/\[4(2/3)\] mW=24 mW. Output voltage is 4 V and load current is 6 mA. 3. Return to the original network: source current=(12−4)/1 kΩ=8 mA, R2 current=2 mA, and source delivery=96 mW. Rs absorbs 64 mW, R2 absorbs 8 mW and the load absorbs 24 mW. 4. Actual source efficiency at this load-power optimum is 24/96=25%. The equivalent 8 V source would deliver 48 mW, split equally between its Rth and load; that 50% balance belongs to the equivalent, not the original internal network. The unique global load-power optimum over RL\>0 is RL=2/3 kΩ, giving 24 mW. The actual original-source efficiency there is 25%. Test the result: Original powers satisfy 64+8+24=96 mW. As RL→0 or RL→infinity, load power tends to zero. The theorem maximizes load power, not efficiency, and does not require an actual stock resistor to equal the mathematical optimum. ## Common mistakes ### Ignoring the next stage An input resistance is part of the circuit. Use its loaded topology. ### Adding resistors after deactivation After shorting the ideal voltage source, inspect connections; the two divider resistors are parallel at the output. ### Using equivalent internal power as physical power Return to the original branches when calculating actual supply current or losses. ## Sources and edition checks - [Author: DC Electrical Circuit Analysis](https://www2.mvcc.edu/users/faculty/jfiore/books/DCElectricalCircuitAnalysis.pdf) The author PDF verifies DC version, full free scope, basic preparation, network-theorem route and selected-answer coverage. - [McGraw Hill: selected seventh-edition ISE print](https://www.mheducation.com.sg/ise-fundamentals-of-electric-circuits-9781260570793-asia-group) McGraw Hill distinguishes the selected ©2021 ISE print from ebook/Connect products and lists the early DC chapters. - [Pearson: selected sixth-edition paperback](https://www.pearson.com/en-ca/subject-catalog/p/electronics-a-systems-approach/P200000005502/9781292114064) Pearson identifies the 2017 sixth-edition print and measurement, resistance, systems and sensor contents. These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover. [How we curate](https://unireadinglist.com/editorial/)