First one-loop circuits
An ideal DC source and resistors form the first complete circuit problems: reference, law, solution, power and check.
- Solve a source-resistor loop.
- Check power balance.
- Interpret the sign of current from the chosen arrow.
- Predict how current changes when V or R changes.
Can we solve a complete electrical circuit before learning Kirchhoff's laws?
Yes, in the simplest possible topology: one ideal voltage source connected directly across one resistor. Both elements share the same two nodes, so the source fixes the resistor voltage and Ohm's law fixes the current.
05.1Read the topology before calculating
Figure — First resistive loop
Topology-only view: the ideal source and resistor connect between the same two nodes, VOUT and GND.
The topology-only figure shows that the source and resistor connect between the same VOUT and GND nodes. The ideal source imposes 9 V between those nodes. Therefore the resistor terminal-voltage magnitude is also 9 V.
The voltage/current reference convention on the resistor has already been introduced separately in Figure 1.1; we do not crowd the full circuit with duplicate arrows.
05.2Solve the operating point
The pair (VR,I)=(9 V,3 mA) is the DC operating point of the resistor in this circuit.
Independent check: 3 mA through 3 kΩ gives 9 V, so the units and magnitude close consistently.
05.3Power must balance between source and load
The resistor absorbs
With absorbed power counted positive, the ideal source must deliver the same amount:
This is a powerful independent check: if a simple ideal source–resistor circuit appears to create or destroy net power, the sign bookkeeping is wrong.
05.4Predict changes before touching a calculator
- Double source voltage with R fixed ⇒ I doubles and resistor power quadruples.
- Double R with source voltage fixed ⇒ I halves and resistor power halves.
- Reverse source polarity while keeping reference arrows fixed ⇒ signed V and I reverse; resistor absorbed power remains positive.
- Let R grow very large ⇒ I tends to zero.
- Let R become very small in the ideal model ⇒ current becomes very large, exposing why real sources need current limits/internal resistance.
These predictions are more valuable than memorizing isolated numbers because they survive changes in the example values.
05.5Bring a real resistor into the calculation
Suppose the nominal 3.0 kΩ resistor has ±5% tolerance. Then
With an ideal 9.00 V source, the current can therefore range approximately from
The “3 mA” answer is the nominal model result; a physical build naturally occupies a range even before meter uncertainty and source tolerance are considered.
05.6Independent problems and bridge to Kirchhoff
- A 12 V ideal source is connected across 4.7 kΩ. Find I and resistor power.
- If the source voltage in the 9 V/3 kΩ example falls to 8.1 V, find I and P.
- A 5 V source drives 100 Ω. Estimate current and resistor power; explain why a real source's current capability now matters.
- Explain why this one-source/one-resistor circuit can be solved without yet using KCL or KVL explicitly.
Solutions
1. I≈2.553 mA; P≈30.64 mW.
2. I=2.70 mA; P=21.87 mW.
3. I=50 mA; P=250 mW. A real source may not sustain 50 mA at 5 V without voltage droop, limiting or heating.
4. The ideal source and resistor share the same two nodes, so the source directly determines the resistor voltage; Ohm's law then determines current.
DC-02 now adds the conservation laws needed when a circuit has several branches and several unknown voltages. That is where the same element laws become a network-analysis method.