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Ohm's Law, Branch Relationships, and Sources

1.1 Chapter Summary and Polarity Reference

1.1.1 Chapter Summary

Ohm's law describes the relationship between voltage, current, and resistance for resistive circuits. This chapter describes Ohm's law and the related circuit elements, resistors, current sources, and voltage sources. This chapter also covers the more general idea of branch relationships, the relation between the voltage across a circuit element and the current through a circuit element, for resistors, voltage sources, and current sources.

1.1.1.1 Ohm's Law

Figure 1.2 Resistor schematic symbol with passive sign convention.

The voltage across a resistor, shown in Figure 1.2, equals the current through the resistor times the resistance of the resistor.

The current through a resistor equals the voltage across the resistor divided by the resistance of the resistor.

1.1.1.2 Branch Relationships

The branch relationship is the equation describing the relationship between current through a circuit element (in a branch of a circuit) and the voltage across the circuit element. For example, Ohm's law V = IR or I = V/R is the branch relationship of a resistor. Figure 1.3, with a simple square shape, is a generic, non‐specific circuit element.

Figure 1.3 Passive sign convention on a generic fictitious schematic symbol.

1.1.2 Polarity Reference

In Figure 1.4a, 1 V divided by 5 Ω equals 200 mA (0.2 A).

Figure 1.4a 1 volt DC voltage source across a 5 ohm resistor.

In each of these two direct current (DC) examples, Figures 1.4a and 1.4b, the current in the loop is 200 mA DC, going down through the resistor from + to −, and the voltage across the 5 Ω resistor is 1 V DC.

Figure 1.4b 200 mA DC current source driving a 5 ohm resistor.

In Figure 1.4b, 0.2 A multiplied by 5 Ω equals 1 V.

1.1.2.1 DC Voltage Source Polarity Example

The voltage source in Figure 1.4a applies voltage across the resistor, and the + (plus) and − (minus) signs on the resistor indicate the polarity of the voltage.

Figure 1.5 shows a circuit simulation schematic corresponding to the circuit in Figure 1.4a. The voltage source symbol in Figure 1.5 is specific to DC voltage sources.

Figure 1.5 Circuit simulation schematic with avoltage source and a resistor.

There is a current meter in the right‐hand leg of the circuit, just below the resistor. The downward arrow in the current meter symbol indicates a reference direction pointing down; the current meter considers clockwise flow of current in the circuit to be positive.

For the DC circuit simulation, there is a table of results in Figure 1.5. VR, measured at the top of the circuit, is positive 1 V, and the current Pr1.I, measured by the current meter, is 200 mA, verifying that current flows downward through the resistor.

1.1.2.2 DC Current Source Polarity Example

The arrow, pointing up in the current source in Figure 1.4b, indicates the direction of a positive current from the source. Applying Ohm's law, multiplying the current times the resistance, tells us the voltage across the resistor, both the amount and the polarity of the voltage. The + (plus) and − (minus) signs on the resistor indicate the polarity of the voltage.

Figure 1.6 shows a circuit simulation schematic corresponding to the circuit in Figure 1.4b. The downward arrow in the current meter symbol indicates a reference direction pointing down; the current meter considers clockwise flow of current in the circuit to be positive.

Figure 1.6 Circuit simulation schematic with a current source and a resistor.

In Figure 1.6, the current source arrow points up, telling us that current flows up and out of the current source, and then down through the resistor. As expected from Ohm's law, 200 mA multiplied by 5 Ω yields 1 mV across the resistor. This 1 V result appears in the table under V1.V. The current direction of the current source and the current meter are the same, and the measured current, under Pr1.I, is positive 200 mA.

At the top of the schematic in Figure 1.6, there is a V1 marker, indicating a voltage measurement. This voltage is referenced to ground, so it is equivalent to the way schematics show a + (plus) sign above the resistor and a − (minus) sign below the resistor.

In Figure 1.7 the current source arrow points down, telling us that current flows down and out of the current source, and then up through the resistor, the opposite direction from the current source in Figure 1.6.

Figure 1.7 Circuit simulation schematic with a DC current source, pointed down, and a resistor.

This makes the measured voltage V2.V −1 V, as it is referenced from the top V2 voltage reference.

As expected from Ohm's law, −0.2 A multiplied by 5 Ω yields – 1 V across the resistor. This result appears under V2.V in units of volts.

In Figure 1.8, the current source arrow indicates counterclockwise circulation of current around the loop. The meter, pointing the same direction in Figure 1.8 as it does in Figure 1.7, indicates opposite circulation direction from the current source and correspondingly the current measured under Pr2.1 is negative 0.2 A.

Figure 1.8 Schematic with a current source, pointed up, and a resistor.

1.1.2.3 Reference Polarity versus Physical Current Flow Direction

It is important to distinguish between actual current flow direction and reference polarity. If we reversed the direction of the current meter in Figure 1.7, the simulation would indicate a positive current value for Pr2.1, but the current still flows counterclockwise in the circuit.

The arrow pointing down, in the current source in Figure 1.8, indicates positive current going down from the current source and then up through the resistor. Using the same + (plus) and − (minus) signs on the resistor as a reference polarity for the resistor, we would get a negative voltage.

In Figure 1.7 the reference direction for the current is down, indicated by an arrow, in both schematics (Figures 1.7 and 1.8). The current polarity will be measured relative to this counter clockwise reference direction.

1.2 Branch Relationships and I–V Characteristics

1.2.1 Circuit Element Branch Relationships

1.2.1.1 Ohm's Law is a Resistor's Branch Relationship

Ohm's law V = IR, describing the behavior of a resistor, is the branch relationship for a resistor. Figure 1.9 shows how voltage V can be expressed as a function of I (voltage as a dependent variable) or current can be expressed as a function of V (current as the dependent variable).

Figure 1.9 Resistor schematic diagram and ohm's law expressions where current or voltage are dependent.

1.2.1.2 Capacitor and Inductor...