Independent ideal sources & source models

Let an ideal voltage source have its positive terminal at node A and negative terminal at node B. Its defining terminal law is simply

VS is specified independently of the current through the source. The source model does not give an additional equation such as I=V/R because no finite internal resistance belongs to the ideal voltage-source element.

That does not mean the current is arbitrary after the whole circuit is assembled. Kirchhoff's laws and the connected elements determine the current needed to satisfy the fixed source voltage.

With VS=10 V and RL=5 kΩ, the resistor shares the same terminal voltage as the ideal source:

If RL becomes 1 kΩ, the ideal source still imposes 10 V and the required current becomes 10 mA. If RL→∞, the branch approaches an open circuit and IL→0.

If RL→0 while VS remains nonzero,

This is a limiting statement. A perfect short directly across a nonzero ideal voltage source would simultaneously demand V=0 and V=VS, which is an inconsistent set of ideal constraints. Real sources avoid this mathematical pathology through current limits, internal impedance, protection, heating and other nonideal physics.

For an ideal current source, the defining law is the current itself. If its arrow defines positive current from terminal n toward terminal p, then

IS is prescribed independently of the voltage across the source. The surrounding network must develop whatever voltage is required to accommodate that current within the ideal model.

In Figure 1.3 the source injects 2 mA into VOUT. With only one resistor path back to ground, Kirchhoff's current law forces 2 mA through the resistor.

For IS=2 mA and RL=5 kΩ, Ohm's law on the load gives

If RL becomes 1 kΩ, the same ideal current needs only 2 V. If RL→0, VL→0 while the current remains 2 mA.

If RL→∞ while IS remains nonzero,

Again this is a limit. A perfect open circuit would demand branch current I=0 while a nonzero ideal current source simultaneously demands I=IS. Those constraints are inconsistent. A real current source instead reaches its finite compliance-voltage limit and can no longer maintain the commanded current.

Choose terminal voltage V on the horizontal axis and terminal current I on the vertical axis. Then an ideal independent voltage source satisfies

for whatever current the external ideal network determines. Its I–V locus is therefore a vertical line at V=VS.

An ideal independent current source satisfies

for whatever terminal voltage the external ideal network determines. Its locus is a horizontal line at I=IS.

These are constitutive/terminal laws, just as V=RI is the terminal law of an ideal resistor. The difference is which relationship each element imposes between its terminal variables.

Power sign depends on the chosen voltage polarity and current reference. Under the passive-sign convention, current is positive when it enters the positive-voltage terminal and

means positive P is absorbed power. If current instead leaves the positive terminal, the element delivers power and its absorbed power is negative.

For the 10 V voltage-source example, the 5 kΩ resistor absorbs

The source current leaves the source's positive terminal, so current entering that terminal is −2 mA:

The source delivers 20 mW. But if another part of a circuit forced positive current into the source's positive terminal, the same ideal voltage source would absorb power. Rechargeable-battery charging is a familiar real-world reminder that an element called a source need not deliver power at every instant, although an ideal voltage-source model is not itself a battery model.

The purpose of an ideal model is not to claim that infinite current or voltage exists in hardware. It isolates one dominant behavior so circuit equations stay transparent. Engineering judgment enters when we ask whether the required current, voltage, power and frequency range stay inside the real source's specifications.

Later, source resistance and terminal-equivalent models will quantify some nonideal behavior. Do not silently add those effects before the model says they are present.

Linear-circuit analysis often asks us to set an independent source value to zero. Apply the element's own defining law literally.

For an independent ideal voltage source:

Two terminals constrained to the same voltage are represented by an ideal short circuit.

For an independent ideal current source:

A branch constrained to zero current is represented by an ideal open circuit.

Chapter 03 will use this operation carefully in superposition. Dependent sources, introduced in chapter 02, are not simply turned off during superposition because their values remain tied to circuit variables.

1. Identify the source type. Voltage or current? Independent or, later, dependent?
2. Read the reference. For voltage, identify +/− terminals; for current, identify the arrow.
3. Write the defining source equation first. V=VS or I=IS.
4. Let the network determine the conjugate variable. Use Ohm's law and Kirchhoff's laws to find source current or source voltage.
5. Check extreme demands. Does the ideal solution require implausibly large current, voltage or power?
6. Compute signed power with an explicit convention. Do not assume a source must deliver.
7. Only then compare with a real source specification. Current limit and compliance are model-validity questions, not corrections to be guessed.

1. An ideal 12 V source drives a 3 kΩ resistor. Find load current and resistor power.
2. The same ideal 12 V source drives 300 Ω. What changes? Which quantity is still fixed by the source model?
3. An ideal 1.5 mA source drives 4 kΩ. Find terminal voltage and resistor power.
4. The same current source drives 40 kΩ. What voltage is required? Why might a real current source fail to realize the ideal answer?
5. A 5 V ideal source carries 2 mA entering its positive terminal. Does the source absorb or deliver power?
6. A 2 mA ideal current source points upward from ground into a node whose voltage is −3 V relative to ground. Using the upper node as the positive voltage reference, does the source absorb or deliver power?
7. What replaces a 0 V independent ideal voltage source in a linear-analysis diagram? What replaces a 0 A independent ideal current source?
8. Why is a nonzero ideal voltage source directly shorted, or a nonzero ideal current source directly opened, better described as inconsistent ideal constraints than as an ordinary finite circuit operating point?

Chapter 02 will keep the source idea but remove independence: the source value will be controlled by another voltage or current elsewhere in the circuit. That is the language used to model amplification and coupling before introducing any specific transistor technology.