Acoustics

Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

3.2. Physical and mathematical meanings of circuit elements

The circuit elements we shall use in forming a schematic diagram are those of electrical circuit theory. These elements and their mathematical meaning are tabulated in Table 3.1 and should be learned at this time. There are generators of two types. There are five types of circuit elements: resistance, capacitance, inductance, transformation, and gyration. There are three generic quantities: (1) the drop across the circuit element; (2) the flow through the circuit element; and (3) the magnitude of the circuit element [7].

Attention should be paid to the fact that the quantity

\tilde{a}

$\tilde{a}$

is neither restricted to voltage

\tilde{e}

$\tilde{e}$

nor

\tilde{b}

$\tilde{b}$

to electrical current

\tilde{i}

$\tilde{i}$

. In some problems

\tilde{a}

$\tilde{a}$

will represent force

\tilde{f}

$\tilde{f}$

or velocity

\tilde{u}

$\tilde{u}$

or pressure

\tilde{p}

$\tilde{p}$

or volume velocity

\tilde{U}

$\tilde{U}$

. In those cases

\tilde{b}

$\tilde{b}$

will represent, respectively, velocity

\tilde{u}

$\tilde{u}$

or force

\tilde{f}

$\tilde{f}$

or volume velocity

\tilde{U}

$\tilde{U}$

or pressure

\tilde{p}

$\tilde{p}$

. Similarly, the quantity c might be any appropriate quantity such as mass, compliance, inductance, resistance, etc. The physical meaning of the circuit elements c depends on the way in which the quantities

\tilde{a}

$\tilde{a}$

and

\tilde{b}

$\tilde{b}$

are chosen, with the restriction that

\frac{{\tilde{a}}^{*}}{\sqrt{2}} \cdot \frac{\tilde{b}}{\sqrt{2}}

$\frac{{\tilde{a}}^{*}}{\sqrt{2}} \cdot \frac{\tilde{b}}{\sqrt{2}}$

has the dimension of power in all cases. The complete array of alternatives is shown in Table 3.2.

An important idea to fix in your mind is that the mathematical operations associated with a given symbol are invariant. If the element is of the inductance type, for example, the drop

\tilde{a}

$\tilde{a}$

across it is equal to the time derivative of the flow

\tilde{b}

$\tilde{b}$

through it multiplied by its size c. Note that this rule is not always followed in electrical circuit theory because conductance and resistance there are often indiscriminately written beside the symbol for a resistance-type element. The invariant operations to be associated with each symbol are shown in columns 3 and 4 of Table 3.1.

Table 3.1

Mathematical and physical significance of symbols
Symbol	Name	Meaning
Symbol	Name	Transient	Steady-state
	Constant drop generator	The drop $\tilde{a}$ $\tilde{a}$ is independent of what is connected to the generator. Its internal impedance is zero so that if one of any number of generators in a circuit is switched off, it is replaced by a short circuit. The arrow points to the positive terminal of the generator.
	Constant flow generator	The flow $\tilde{b}$ $\tilde{b}$ is independent of what is connected to the generator. Its internal impedance is infinity so that if one of any number of generators in a circuit is switched off, it is replaced by an open circuit. The arrows point in the direction of positive flow.
	Resistance-type element	a = bc	$\tilde{a} = \tilde{b} c$ $\tilde{a} = \tilde{b} c$
	Capacitance-type element	$a = \frac{1}{c} \int b d t$ $a = \frac{1}{c} \int b d t$	$\tilde{a} = \frac{\tilde{b}}{j ω c}$ $\tilde{a} = \frac{\tilde{b}}{j ω c}$
	Inductance-type element	$a = c \frac{d b}{d t}$ $a = c \frac{d b}{d t}$	$\tilde{a} = j ω c \tilde{b}$ $\tilde{a} = j ω c \tilde{b}$
	Transformation-type element	$\begin{matrix} a = c g \\ b = \frac{d}{c} \\ \frac{a}{b} = c^{2} \frac{g}{d} \end{matrix}$ $\begin{matrix} a = c g \\ b = \frac{d}{c} \\ \frac{a}{b} = c^{2} \frac{g}{d} \end{matrix}$	$\begin{matrix} \tilde{a} = c \tilde{g} \\ \tilde{b} = \frac{\tilde{d}}{c} \\ \frac{\tilde{a}}{\tilde{b}} = c^{2} \frac{\tilde{g}}{\tilde{d}} \end{matrix}$ $\begin{matrix} \tilde{a} = c \tilde{g} \\ \tilde{b} = \frac{\tilde{d}}{c} \\ \frac{\tilde{a}}{\tilde{b}} = c^{2} \frac{\tilde{g}}{\tilde{d}} \end{matrix}$
	Gyration-type element	$\begin{matrix} c_{1} a = d \\ b = c_{2} g \\ \frac{a}{b} = \frac{1}{c_{1} c_{2}} \frac{d}{g} \end{matrix}$ $\begin{matrix} c_{1} a = d \\ b = c_{2} g \\ \frac{a}{b} = \frac{1}{c_{1} c_{2}} \frac{d}{g} \end{matrix}$	$\begin{matrix} c_{1} \tilde{a} = \tilde{d} \\ \tilde{b} = c_{2} \tilde{g} \\ \frac{\tilde{a}}{\tilde{b}} = \frac{1}{c_{1} c_{2}} \frac{\tilde{d}}{\tilde{g}} \end{matrix}$ $\begin{matrix} c_{1} \tilde{a} = \tilde{d} \\ \tilde{b} = c_{2} \tilde{g} \\ \frac{\tilde{a}}{\tilde{b}} = \frac{1}{c_{1} c_{2}} \frac{\tilde{d}}{\tilde{g}} \end{matrix}$

Table 3.2

Values for a, b, and c in electrical, mechanical, and acoustical circuits
Element	Electrical	Mechanical		Acoustical
Element	Electrical	Admittance analogy	Impedance analogy	Impedance analogy	Admittance analogy
$\tilde{a}$ $\tilde{a}$	$\tilde{e}$ $\tilde{e}$	$\tilde{u}$ $\tilde{u}$	$\tilde{f}$ $\tilde{f}$	$\tilde{p}$ $\tilde{p}$	$\tilde{U}$ $\tilde{U}$
$\tilde{b}$ $\tilde{b}$	$\tilde{i}$ $\tilde{i}$	$\tilde{f}$ $\tilde{f}$	$\tilde{u}$ $\tilde{u}$	$\tilde{U}$ $\tilde{U}$	$\tilde{p}$ $\tilde{p}$
	c = R _E	$c = \frac{1}{R_{M}} = Y_{M}$ $c = \frac{1}{R_{M}} = Y_{M}$	c = R _M	c = R _A	$c = \frac{1}{R_{A}} = Y_{A}$ $c = \frac{1}{R_{A}} = Y_{A}$
	c = C _E	c = M _M	c = C _M	c = C _A	c = M _A
	c = L	c = C _M	c = M _M	c = M _A	c = C _A
	$c = Z_{E} = \frac{\tilde{e}}{\tilde{i}}$ $c = Z_{E} = \frac{\tilde{e}}{\tilde{i}}$	$\begin{matrix} c = Y_{M} = \frac{\tilde{u}}{\tilde{f}} \\ = \frac{1}{Z_{M}} \end{matrix}$ $\begin{matrix} c = Y_{M} = \frac{\tilde{u}}{\tilde{f}} \\ = \frac{1}{Z_{M}} \end{matrix}$	$\begin{matrix} c = Z_{M} = \frac{\tilde{f}}{\tilde{u}} \\ = \frac{1}{Y_{M}} \end{matrix}$ $\begin{matrix} c = Z_{M} = \frac{\tilde{f}}{\tilde{u}} \\ = \frac{1}{Y_{M}} \end{matrix}$	$\begin{matrix} c = Z_{A} = \frac{\tilde{p}}{\tilde{U}} \\ = \frac{1}{Y_{A}} \end{matrix}$ $\begin{matrix} c = Z_{A} = \frac{\tilde{p}}{\tilde{U}} \\ = \frac{1}{Y_{A}} \end{matrix}$	$\begin{matrix} c = Y_{M} = \frac{\tilde{U}}{\tilde{p}} \\ = \frac{1}{Z_{A}} \end{matrix}$ $\begin{matrix} c = Y_{M} = \frac{\tilde{U}}{\tilde{p}} \\ = \frac{1}{Z_{A}} \end{matrix}$

An infinite impedance generator is a flow generator in the impedance analogy and a drop generator in the admittance analogy. Conversely, a zero impedance generator is a drop generator in the impedance analogy and a flow generator in the admittance analogy. A drop generator “hates” short circuits for obvious reasons. A flow generator “hates” open circuits because when the flow is blocked, the drop rises to infinity. In fact a flow generator can be approximated by a very large drop generator with a very large series resistance whose value is the drop divided by the desired flow.

The transformation element is ideal in that it neither creates nor dissipates power. Hence, the dot product

{\tilde{a}}^{*} \cdot \tilde{b}

${\tilde{a}}^{*} \cdot \tilde{b}$

on the primary side is always equal to

{\tilde{g}}^{*} \cdot \tilde{d}

${\tilde{g}}^{*} \cdot \tilde{d}$

on the secondary side. It is also reversible, unlike, for example, an amplifier. If the transformation ratio is c:1, as illustrated in Table 3.1, then you divide the drop

\tilde{a}

$\tilde{a}$

on the primary side to obtain the drop

\tilde{g}

$\tilde{g}$

on the secondary side. Conversely, if the transformation ratio is 1:c, then you multiply the drop

\tilde{a}

$\tilde{a}$

on the primary side to obtain the drop

\tilde{g}

$\tilde{g}$

on the secondary side. Of course, to conserve power, the opposite operation is performed on the flow so that it increases by the same ratio that the drop decreases or vice versa.

The gyration element is used to convert an admittance-type circuit to an impedance-type one or vice versa. This means that the flow

\tilde{d}

$\tilde{d}$

on the secondary side is equal to the drop

\tilde{a}

$\tilde{a}$

on the primary side multiplied by the forward mutual conductance c ₁. Likewise, the flow

\tilde{b}

$\tilde{b}$

on the primary side is equal to the drop

\tilde{g}

$\tilde{g}$

on the secondary side multiplied by reverse mutual conductance c ₂. The forward and reverse mutual conductances c ₁ and c ₂, respectively, may have different values in which case energy is either consumed (as in an amplifier) or dissipated. In this chapter, it will be used exclusively as an energy conserving element in passive transducers, in which case c ₁ = c ₂ = c.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for Acoustics

Create new playlist

Sign In

Sign Up

3.2. Physical and mathematical meanings of circuit elements

Table of Contents for
Acoustics