Voltage Sources | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

Introduction

A voltage source is an active two-terminal element that provides a specified voltage between its terminals. An ideal voltage source maintains the same terminal voltage regardless of the current drawn from it. In practice, however, real voltage sources show a fall in terminal voltage as the load current increases because of internal resistance or impedance.

• Elements in electrical circuits are classed as active or passive. Active elements (for example, batteries, generators, operational amplifiers) can supply energy to a circuit. Passive elements (for example, resistors, capacitors, inductors) cannot generate energy and only store or dissipate it.
• Electrical sources supply electrical energy and are of two basic types: voltage sources and current sources. Each may be either direct (DC) or alternating (AC). For example, a battery is a DC voltage source while the mains supply is an AC voltage source.
• Electrical sources often convert one form of energy into electrical energy (chemical → electrical in a battery; mechanical → electrical in a generator). Sources can both deliver and absorb power.
• The I-V characteristic (current-voltage characteristic) of a source describes how its terminal voltage varies with current and is a key property used in circuit analysis and modelling.

Electrical Sources

Electrical sources may be classified as independent (ideal) or dependent (controlled). An independent source has a value fixed by the source; a dependent source has a value that depends on a voltage or current elsewhere in the circuit. Sources can be time-invariant (DC) or time-varying (AC).

• In circuit analysis, sources are often idealised. An ideal voltage source can deliver any current required while keeping its voltage fixed. Real sources, however, always have some internal resistance or impedance which affects their behaviour under load.
• For a current source, the associated non-ideal element is usually modelled in parallel; for a voltage source it is modelled in series.

The Voltage Source

A voltage source establishes a potential difference between two terminals and thereby can cause current to flow when the terminals are connected through a circuit. The voltage at the terminals of a source when no external current flows is often called the open-circuit voltage or the source EMF; the measured voltage under load is the terminal voltage.

• An ideal voltage source maintains a fixed voltage v across its terminals for any current i through it. Its I-V plot is a vertical line at the fixed voltage value (i may be any value theoretically).
• An ideal voltage source is an independent source: its voltage does not depend on currents or voltages elsewhere in the circuit.
• A dependent (controlled) voltage source produces a terminal voltage that is proportional to some other circuit variable (a voltage or a current). These dependent sources are commonly used to model amplifiers, transistors and op-amp behaviour and are drawn with a diamond symbol.

Connecting Voltage Sources Together

• Ideal voltage sources may be connected in series or in parallel with usual circuit rules: series voltages algebraically add; parallel identical voltages have the same value.
• Unequal ideal voltage sources must not be directly connected in parallel because that creates a contradiction (different fixed voltages at the same nodes) and in practice leads to large circulating currents or an undefined result.

Voltage Source in Parallel

• Two identical ideal voltage sources may be paralleled to supply the same terminal voltage while sharing current. For example, two 10 V ideal sources in parallel still provide 10 V between the terminals. In most analyses, one equivalent source of the same voltage is used instead.
• Paralleling ideal sources of different voltages is not permitted: the nodes would be forced to two different voltages simultaneously, which is physically inconsistent.

Badly Connected Voltage Sources

• Voltage sources of different voltages can be included in the same circuit provided there are other circuit elements between them so that Kirchoff's Voltage Law (KVL) is satisfied around loops and no direct conflict of potential exists.
• Series connections of voltage sources are always allowed; the net source voltage is the algebraic sum of the individual voltages. Two common cases are series-aiding and series-opposing connections.

Voltage Source in Series

• Series-aiding: sources are connected so their polarities add (for example + of one to - of next) and the net voltage is the sum. Example: two series aiding sources of 10 V and 5 V give VS = 10 + 5 = 15 V.
• Series-opposing: sources are connected so their polarities subtract. The net voltage is the algebraic difference; for 10 V and 5 V in series-opposing, VS = 10 - 5 = 5 V, with polarity determined by the larger source. If equal and opposing, the net voltage is zero and no net current flows unless there are other elements to drive it.

Voltage Source Example

Problem statement: Two series aiding ideal voltage sources of 6 V and 9 V respectively are connected together to supply a load resistance of 100 Ω. Calculate the source voltage VS, the load current IR and the power P dissipated by the resistor. Draw the circuit.

Sol.

VS is the algebraic sum of series aiding sources.

VS = 6 V + 9 V

VS = 15 V

The load current IR is given by Ohm's law.

IR = VS / R

IR = 15 V / 100 Ω

IR = 0.15 A

The power dissipated in the resistor is given by P = I^2 R.

P = (0.15 A)^2 × 100 Ω

P = 2.25 W

Practical Voltage Source

• Real voltage sources are not ideal; they have an internal resistance (or impedance) which causes the terminal voltage to drop with increasing load current.
• An ideal voltage source has zero internal resistance (RS = 0). A practical source can be modelled as an ideal source VS in series with RS. The same load current flows through RS and causes a voltage drop i·RS.
• Applying KVL around the loop gives the terminal (output) voltage VOUT as: VOUT = VS - i·RS.
• The I-V characteristic of a practical voltage source is therefore a straight line with slope -RS when plotted V against i, intercepting the voltage axis at VS when i = 0.

• This series model is identical in form to a Thevenin equivalent: any linear circuit of sources and resistances can be replaced by a single voltage source in series with a single resistance.
• When RS is small, the practical source approximates an ideal source. As RS increases the available terminal voltage under heavy load is reduced significantly.

Practical Voltage Source Characteristics

• Ideal sources give a vertical I-V characteristic for a fixed voltage; practical sources give a line with negative slope due to i·RS. The I-V plot of a real battery is close to ideal because RS is often small but measurable.
• Voltage regulation is a measure of how much the terminal voltage falls from no-load to full-load. One common definition (expressed as a percentage) is:

Voltage regulation (%) = (Vno-load - Vfull-load) / Vfull-load × 100%

• Good regulation means small change in terminal voltage between no load and full load. Regulation depends on internal resistance and the load current range.

Practical Voltage Source Example

Problem statement: A battery is modelled as an ideal voltage source in series with an internal resistance. The measured terminal voltages are VOUT1 = 130 V at I1 = 10 A, and VOUT2 = 100 V at I2 = 25 A. Find the ideal source voltage VS and the internal resistance RS. Draw the I-V characteristics.

Sol.

Write the two measured terminal voltage equations.

VS - 10·RS = 130

VS - 25·RS = 100

Subtract the second equation from the first to eliminate VS.

(VS - 10·RS) - (VS - 25·RS) = 130 - 100

15·RS = 30

RS = 30 / 15

RS = 2 Ω

Substitute RS back into either original equation to find VS.

VS - 10·(2 Ω) = 130

VS - 20 = 130

VS = 150 V

The calculated values are VS = 150 V and RS = 2 Ω. The I-V characteristic is a straight line with intercept 150 V at i = 0 and slope -RS = -2 Ω; the two measured points (10 A, 130 V) and (25 A, 100 V) lie on this line.

Dependent Voltage Source

A dependent (controlled) voltage source is an ideal source whose output voltage is a function of some other circuit variable. Dependent voltage sources are used to model active devices such as transistors and operational amplifiers.

• A Voltage-Controlled Voltage Source (VCVS) produces an output voltage proportional to a controlling voltage elsewhere: VOUT = μ·VIN, where μ is a dimensionless gain.
• A Current-Controlled Voltage Source (CCVS) produces an output voltage proportional to a controlling current elsewhere: VOUT = ρ·IIN, where ρ (rho) has units of ohms (V/A) and is sometimes called a transresistance.
• Dependent sources are drawn with a diamond-shaped symbol in circuit diagrams and are treated as ideal elements when used in linear analysis.

Summary

• Voltage sources can be ideal independent sources or dependent (controlled) sources. Independent sources supply a fixed voltage (DC or AC) independent of other circuit variables; dependent sources produce a voltage proportional to another voltage or current in the circuit.
• Ideal voltage sources have zero internal series resistance and maintain constant terminal voltage for any current; practical sources have a small internal resistance RS in series, so terminal voltage falls as load current increases. The practical model is VS in series with RS (Thevenin form).
• Series connected sources add algebraically (series-aiding or series-opposing). Parallel connection of ideal sources is only valid when the voltages are equal.
• The I-V characteristic of an ideal voltage source is an idealised straight line (constant voltage). The I-V characteristic of a practical source is a line with slope -RS. Voltage regulation quantifies how much the terminal voltage changes from no-load to full-load.
• Dependent voltage sources (VCVS and CCVS) are essential models for amplifying devices; the VCVS gain μ is dimensionless and the CCVS transresistance ρ has units of ohms.
• Voltage sources can both deliver and absorb power depending on the circuit conditions; they are widely used in circuit analysis, modelling and practical electrical and electronic systems.