5. Kirchhoff's Laws

1.Node - A point where two or more circuit elements join 2. essential node- A node where three or more circuit elements join 3. path - A trace of adjoining basic elements with no

elements included more than once 4.branch - A path that connects two nodes 5. essential branch - A path which connects two essential nodes without

passing through an essential node 6.

loop - A path whose last node is the same as the starting node 7. Mesh- A loop that does not enclose any other loops

8.planar circuit - A circuit that can be drawn on a plane with no crossing branches

Kirchhoff's current law (KCL)

The algebraic sum of all the currents at any node in a circuit equals zero.

To use Kirchhoff's current law, an algebraic sign corresponding to a reference direction must be assigned to every current at the node. Assigning a positive sign to a current leaving a node requires assigning a negative sign to a current entering a node. Conversely, giving a negative sign to a current leaving a node requires giving a positive sign to a current entering a node.

Kirchhoffs voltage law (KVL) •

The algebraic sum of all the voltages around any closed path in a circuitequals zero.

To use Kirchhoffs voltage law, we must assign an algebraic sign (reference direction) to each voltage in the loop. As we trace a closed path, a voltage will appear either as a rise or a drop in the tracing direction. Assigning a positive sign to a voltage rise requires assigning a negative sign to a voltage drop. Conversely, giving a negative sign to a voltage rise requires giving a positive sign to a voltage drop.

6. Sinusoidal current. Period, frequency, phase angle, amplitude, square root of the mean value

Thus far, we have focused on circuits with constant sources; in this chapter we are now ready to consider circuits energized by time-varying voltage or current sources. In particular, we are interested in sources in which the value of the voltage or current varies sinusoidally.

Sinusoidal sources and their effect on circuit behavior form an important area of study for several reasons.

First, the generation, transmission, distribution, and consumption of electric energy occur under essentially sinusoidal steady-state conditions.

Second, an understanding of sinusoidal behavior makes it possible to predict the behavior of circuits with nonsinusoidal sources.

Third, steady-state sinusoidal behavior often simplifies the design of electrical systems.

A sinusoidal voltage source (independent or dependent) produces a voltage that varies sinusoidally with time. A sinusoidal current source (independent or dependent) produces a current that varies sinusoidally with time. In reviewing the sinusoidal function, we use a voltage source, but our observations also apply to current sources.

Another important characteristic of the sinusoidal voltage (or current) is its rms value. The rms value of a periodic function is defined as the square root of the mean value of the squared function. Hence, if v = Vm cos (ω t + φ ), the rms value of v is

Hence the rms value of v is

Phase difference of the sine function

For instance, given the general expressions of sinusoidal voltage and current as

the phase difference between voltage and current is

If voltage leads current, or current lags voltage

If, current leads voltage, or voltage lags current

7. Resistors, inductors and capacitors in sinusoidal AC circuits

From Ohm's law, if the current in a resistor varies sinusoidally with time — that is, if —the voltage at the terminals of the resistor, is .

The signals are said to be in phase because they both reach corresponding values on their respective curves at the same time

Relationship between phasor voltage and phasor current for a resistor V = Rl,

We derive the relationship between the phasor current and phasor voltage at the terminals of an inductor by assuming a sinusoidal current and using Ldi/dt to establish the corresponding voltage. Thus, for , the expression for the voltage is

The voltage leads the current by 90°, or, equivalently, the current lags behind the voltage by 90°.

Relationship between phasor voltage and phasor current for an inductor V=jωLI

Relationship between phasor voltage and phasor current for a capacitor V=I/jωC

8. Impedance and admittance

In the previous chapter, we had learned that the phasor forms of relationship between voltage and current for resistor, inductor and capacitor in an AC circuit are as follows:

The above equations can be changed to a ratio of voltage and current

The ratio of voltage and current is the impedance of an AC circuit, and it can be generally expressed as Z = V/I. This equation is also known as Ohm’s law of AC circuits. The physical meaning of the impedance is that it is a measure of the opposition to AC current in an AC circuit. It is similar to the concept of resistance in DC circuits, so the impedance is also measured in ohms.

The admittance of resistor, inductor and capacitor are as follows:

Since the impedance is a vector quantity, it can be expressed in both polar form and rectangular form (complex number) as follows:

The rectangular form is the sum of the real part and the imaginary part, where the real part of the complex is the resistance R, and the imaginary part is the reactance X. The reactance is the difference of inductive reactive and capacitive reactance, i.e. X =XL – XC.

The relationship between R, X and Z in the expression of the impedance is a right triangle, and can be described using the Pythagoras’ theorem. This can be illustrated in Figure 9.1(a).

The admittance is also a complex number; it can be expressed in both polar and rectangular forms as follows:

The real part of the complex is the conductance G, and the imaginary part is called the susceptance B.

9. Power in AC circuits

10.Transformer

A transformer is an electrical device formed by two coils that are wound on a common core. A transformer uses the principle of mutual inductance to convert AC electrical energy from input to output. The first coil is called primary winding, and the second coil connected to the load ZL is called secondary winding.

Figure 11.4 Implified transformer circuits (a) air-core; (b) iron-core

Conversion of the voltage, current and impedance: The expressions of the transformer’s turns ratio indicate that a transformer can be used to convert voltage, current and impedance

Power transmis