## Two-Port Circuits – Parameters – Examples

A two-port network is an electrical network with two different ports for input and output.

A port is a pair of terminals through which electrical current can enter and exit. As an example of a one-port circuit are the passive elements of a network: resistors, inductors and capacitors. A one-port network is represented by diagram of Figure 1:

Figure 1

On the other hand, a four-terminal circuit, such as those made up of operational amplifiers, transistors or transformers, is considered a two-port network, which can be represented by the following diagram in Figure 2:

Figure 2

The study of two-port networks is justified because it allows treating or modeling complex circuits as a “black box”, that is, as a box where we do not know in detail what is inside. A signal feeds this “box” through one of its ports (input port); the signal is processed by the linear network of Figure 2 and is then delivered to a load by the other port (output port), as exemplified in Figure 3:

Figure 3

The characterization of a two-port network is made by relating the quantities present at its terminals: V1, V2, I1, I2.

`Restrictions.`

The complex network model as a two port network has certain restrictions:

• There can be no energy stored within the circuit.
• There can be no independent sources within the circuit; Dependent sources, however, are allowed.
• The current entering the port (input or output) must be equal to the current leaving the port (input or output).

The equations that relate the quantities V1, V2, I1, I2 present at the input and output ports of a two-port network are called parameters.

`Impedance parameters.`

To derive the impedance parameters, we supply the two-port network with a voltage source (which can be the Thevenin voltage supplied by the circuit connected at the input port) or by a current source (which can be the Norton provided by the circuit connected at the input port) as shown in Figure 4 a) and b):

Figure 4

From either of these two configurations, we can express the relationships between voltages and currents as:

Equations (1) allow to represent the model for a network of two ports, the “black box”, in matrix form:

The Z terms are called impedance parameters. To evaluate these parameters, we run the following tests. The value of the parameters can be evaluated by setting I1=0 A (input port in open circuit), or I2=0 A (output port in open circuit). In summary:

According to the table of equations (2), we can evaluate Z11 and Z12 by connecting a voltage source V1 (or a current source I1) to port 1 with port 2 in open circuit, as in Figure 5:

Figure 5

Then, from the circuit of Figure 5, by means of circuit analysis, we determine the value of I1 und V2, and then obtain the parameters Z11 und Z21 using equations (3):

Similarly, parameters Z12 und Z22 are obtained by the following experiment, as in Figure 6:

Figure 6

Parameters Z12 und Z22 using equations (4):

`Example.`

Determine the Z parameters in Figure 7:

Figure 7

To determine Z11 und Z21, a voltage source V1 is applied to the input port and the output port is left open, as in Figure 8a). To determine Z12 and Z22, a voltage source V2 is applied to the input port and the output port is left open, as in Figure 8 b).

Figure 7

We determine the Z parameters in Figure 7 using:

Therefore, the matrix of the impedance parameters is:

Sources:

2. Análisis de Redes – Van Valkenburg,
3. Fundamentos_de_circuitos_electricos_5ta
4. Fundamentos_de_Señales_y_Sistemas_usando la Web y Matlab

Literature review by:

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

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Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

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## Relaciones fasoriales de los elementos de un circuito eléctrico

La resistencia, el inductor y el capacitor en circuitos de corriente alterna, requieren de un método de estudio particular. El siguiente método permite transformar la relación tensión-corriente del dominio del tiempo al dominio de la frecuencia (dominio fasorial), de los elementos pasivos de una red: resistencia, inductor y capacitor.

`Resistor o resistencia`

Supongamos que la corriente ir(t) que pasa a través de un resistor r, tiene la siguiente expresión matemática:

De acuerdo a lo discutido en Representación Fasorial de voltajes y corrientes – Fasores, en notación fasorial polar, ir(t)  puede ser escrita como:

De acuerdo con la Ley de Ohm, la tensión a través del resistor está dada por:

La ecuación (1) podemos expresarla mediante notación fasorial de la siguiente manera:

La relación entre el voltaje y la corriente en un resistor se puede apreciar en la Figura (1) tanto en el dominio del tiempo como en el dominio de la frecuencia:

La ecuación (2) indica que el voltaje y la corriente en un resistor tienen la misma fase, es decir, están en fase, lo que se puede apreciar en el diagrama fasorial de la Figura (2):

`Inductor o inductancia`

Supongamos que la corriente il(t) que pasa a través de un inductor L, tiene la siguiente expresión matemática y expresión fasorial exponencial:

De acuerdo con la Ley de Ohm, la tensión a través del inductor está dada por:

Debido a que:

La ecuación (3) se transforma en:

La ecuación (4) podemos expresarla mediante notación fasorial de la siguiente manera:

Debido a que:

Podemos reescribir la ecuación (5):

La relación entre el voltaje y la corriente en un resistor se puede apreciar en la Figura (3) tanto en el dominio del tiempo como en el dominio de la frecuencia:

La ecuación (5) indica que el voltaje se adelanta 90 grados con respecto a la corriente. En ingeniería eléctrica por convención se prefiere decir que la corriente se atrasa con respecto a el voltaje, lo que se puede apreciar en el diagrama fasorial de la Figura (4):

`Capacitor o capacitancia`

Supongamos que la corriente vc(t) que pasa a través de un capacitor c, tiene la siguiente expresión matemáticany expresión fasorial exponencial:

De acuerdo con la Ley de Ohm, la tensión a través del capacitor está dada por:

La ecuación (6) podemos expresarla mediante notación fasorial de la siguiente manera:

Podemos reescribir la ecuación (7):

Es decir:

La relación entre el voltaje y la corriente en un capacitor se puede apreciar en la Figura (5) tanto en el dominio del tiempo como en el dominio de la frecuencia:

La ecuación (6) indica que el voltaje se atrasa 90 grados con respecto a la corriente. En ingeniería eléctrica por convención se prefiere decir que la corriente se adelanta con respecto al voltaje, lo que se puede apreciar en el diagrama fasorial de la Figura (6):

En resumen:

SIGUIENTE:

Fuentes:

2. Análisis de Redes – Van Valkenburg,
3. Fundamentos_de_circuitos_electricos_5ta
4. Elementos básicos del circuito eléctrico
5. Capacitores e Inductores – Circuitos y asociaciones
6. Divisor de tensión y divisor de corriente

Revisión literaria hecha por:

Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer

Se hacen trabajos, se resuelven ejercicios!!

WhatsApp:  +34633129287  Atención Inmediata!!

Copywriting, Content Marketing, Tesis, Monografías, Paper Académicos, White Papers (Español – Inglés)

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV Caracas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: España. +34633129287

Caracas, Quito, Guayaquil, Jaén.

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## UNDERDAMPED SECOND-ORDER SYSTEM

We define two physically meaningful specifications for second-order systems: Natural Frequency (Wn) and Damping Ratio (ζ).

`Introduction`

Now that we have become familiar with second-order systems and their responses, we generalize the discussion and establish quantitative specifications defined in such a way that the response of a second-order system can be described to a designer without the need for sketching the response. We define two physically meaningful specifications for second-order systems. These quantities can be used to describe the characteristics of the second-order transient response just as time constants describe the first-order system response.

Natural Frequency, Wn

The natural frequency of a second-order system is the frequency of oscillation of the system without damping. For example, the frequency of oscillation of a series RLC circuit with the resistance shorted would be the natural frequency.

Damping Ratio,

We have already seen that a second-order system’s underdamped step response is characterized by damped oscillations. Our definition is derived from the need to quantitatively describe this damped oscillations regardless of the time scale.Thus, a system whose transient response goes through three cycles in a millisecond before reaching the steady state would have the same measure as a system that went through three cycles in a millennium before reaching the steady state. For example, the underdamped curve in Figure 4.10 has an associated measure that defines its shape. This measure remains the same even if we change the time base from seconds to microseconds or to millennia.

A viable definition for this quantity is one that compares the exponential decay frequency of the envelope to the natural frequency. This ratio is constant regardless of the time scale of the response. Also, the reciprocal, which is proportional to the ratio of the natural period to the exponential time constant, remains the same regardless of the time base.

We define the damping ratio, , to be:

Consider the general system:

Without damping, the poles would be on the jw-axis, and the response would be an undamped sinusoid. For the poles to be purely imaginary, a = 0. Hence:

Assuming an underdamped system, the complex poles have a real part, , equal to -a/2. The magnitude of this value is then the exponential decay frequency described in Section 4.4. Hence,

from which

Our general second-order transfer function finally looks like this:

Now that we have defined and Wn, let us relate these quantities to the pole location. Solving for the poles of the transfer function in Eq. (4.22) yields:

From Eq. (4.24) we see that the various cases of second-order response:

Underdamped Second-Order System

Now that we have generalized the second-order transfer function in terms of and Wn, let us analyze the step response of an underdamped second-order system.

Not only will this response be found in terms of and Wn, but more specifications
indigenous to the underdamped case will be defined. The underdamped second order system, a common model for physical problems, displays unique behavior that
must be itemized; a detailed description of the underdamped response is necessary
for both analysis and design. Our first objective is to define transient specifications
associated with underdamped responses. Next we relate these specifications to the
pole location, drawing an association between pole location and the form of the
underdamped second-order response. Finally, we tie the pole location to system
parameters, thus closing the loop: Desired response generates required system
components.

Let us begin by finding the step response for the general second-order system of Eq. (4.22). The transform of the response, C(s), is the transform of the input times the transfer function, or:

where it is assumed that < 1 (the underdamped case). Expanding by partial fractions, using the methods described, yields:

Taking the inverse Laplace transform, which is left as an exercise for the student, produces:

where:

A plot of this response appears in Figure 4.13 for various values of , plotted along a time axis normalized to the natural frequency.

We now see the relationship between the value of and the type of response obtained: The lower the value of , the more oscillatory the response.

The natural frequency is a time-axis scale factor and does not affect the nature of the response other than to scale it in time.

Other parameters associated with the underdamped response are rise time, peak time, percent overshoot, and settling time. These specifications are defined as follows (see also Figure 4.14):

1. Rise time, Tr. The time required for the waveform to go from 0.1 of the final value to 0.9 of the final value.
2. Peak time, TP. The time required to reach the first, or maximum, peak.
3. Percent overshoot, %OS. The amount that the waveform overshoots the steady-state, or final value at the peak time, expressed as a percentage of the steady-state value.
4. Settling time, Ts. The time required for the transient’s damped oscillations to reach and stay within 2% of the steady-state value.

All definitions are also valid for systems of order higher than 2, although analytical expressions for these parameters cannot be found unless the response of the higher-order system can be approximated as a second-order system.

Rise time, peak time, and settling time yield information about the speed of the transient response. This information can help a designer determine if the speed and the nature of the response do or do not degrade the performance of the system.

For example, the speed of an entire computer system depends on the time it takes for a hard drive head to reach steady state and read data; passenger comfort depends in part on the suspension system of a car and the number of oscillations it goes through after hitting a bump.

Evaluation of Tp

Tp is found by differentiating c(t) in Eq. (4.28) and finding the first zero crossing after t = 0.

Evaluation of %OS.

From Figure 4.14 the percent overshoot, %OS, is given by:

Evaluation of Ts

In order to find the settling time, we must find the time for which c(t) in Eq. (4.28) reaches and stays within ₎±2% of the steady-state value, C final.

Evaluation of Tr

A precise analytical relationship between rise time and damping ratio cannot be found. However, using a computer and Eq. (4.28), the rise time can be found. Let us look at an example.

We now have expressions that relate peak time, percent overshoot, and settling time to the natural frequency and the damping ratio. Now let us relate these quantities to the location of the poles that generate these characteristics. The pole plot for a general, underdamped second-order system is reproduced in Figure 4.17.

Now, comparing Eqs. (4.34) and (4.42) with the pole location, we evaluate peak time and settling time in terms of the pole location. Thus:

where is the imaginary part of the pole and is called the damped frequency of oscillation, and is the magnitude of the real part of the pole and is the exponential damping frequency part.

At this point, we can understand the significance of Figure 4.18 by examining the actual step response of comparative systems. Depicted in Figure 4.19(a) are the step responses as the poles are moved in a vertical direction, keeping the real part the same. As the poles move in a vertical direction, the frequency increases, but the envelope remains the same since the real part of the pole is not changing.

Let us move the poles to the right or left. Since the imaginary part is now constant, movement of the poles yields the responses of Figure 4.19(b). Here the frequency is constant over the range of variation of the real part. As the poles move to the left, the response damps out more rapidly.

Moving the poles along a constant radial line yields the responses shown in Figure 4.19(c). Here the percent overshoot remains the same. Notice also that the responses look exactly alike, except for their speed. The farther the poles are from the origin, the more rapid the response.

Sources:

1. Control Systems Engineering, Nise
2. Sistemas de Control Automatico Benjamin C Kuo
3. Modern_Control_Engineering, Ogata 4t

Literature Review by: Larry Francis Obando – Technical Specialist

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, Caracas.

Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, Valle de Sartenejas.

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: Jaén – España: Tlf. 633129287

Contact: Caracas, Quito, Guayaquil, Jaén, Villafranca de Ordizia- Telf. +34633129287

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Sources:

Control Systems Engineering, Norman Nise

1. Introduction Chapter 4 pp 162 (162)
2. Poles and Zeros 4.1 pp 162 –
3. First Order System 4.3 pp 165-168
4. Second Order System 4.4 pp 168-177
5. Underdamped Second-Order System 4.6 pp 177-186
1. Modern_Control_Engineering__4t
1. Introduction Chapter 5 pp 219 (232)
2. First Order Systems 221 (234)-224
3. Second Order System pp 224 (237)-234

Literature Review, Martes 14 noviembre 2017, 05:07 am – Caracas, Quito, Guayaquil.

## EL CAPACITOR. Relación corriente-voltaje.

Formalmente, la Capacitancia es la razón entre la carga de una placa del capacitor y la diferencia de tensión entre las dos placas:

`Relación corriente-voltaje del capacitor`

Para obtener la relación de corriente-tensión del capacitor, primero es necesario estudiar la relación entre la carga q y la corriente i. Dicha relación viene dada por la ecuación:

Para encontrar la carga q de las placas en el tiempo t se integra sobre todo el tiempo anterior:

Utilizando el hecho de que q=Cv, obtenemos la relación corriente-tensión del capacitor (suponiendo un capacitor lineal, es decir, que no depende del valor de la tensión v en el tiempo):

O sea:

Otra forma de presentar este resultado es mediante la fórmula:

Utilizando esta última ecuación, podemos graficar la relación corriente-voltaje del capacitor de la manera siguiente:

Recomiendo leer la siguiente guía: Capacitores e Inductores – Circuitos y asociaciones

Preliminares

Por tanto se concluye que la intensidad del campo eléctrico en cualquier punto a una distancia r de una carga puntual de Q coulombs, será directamente proporcional a la magnitud de la carga e inversamente proporcional al cuadrado de la distancia a la carga.

Capacitancia

Al instante en que el interruptor se cierra, se extraen los electrones de la placa superior y se depositan sobre la placa inferior debido a la batería, dando por resultado una carga neta positiva sobre la placa superior del capacitor y una carga negativa sobre la placa inferior…Cuando el voltaje en el capacitor es igual al de la batería, cesa la transferencia de electrones y la placa tendrá una carga neta Q=CV=CE

En este punto el capacitor asumirá las características de un circuito abierto: una caída de voltaje en las placas sin flujo de carga entre las placas.

El voltaje en un capacitor no puede cambiar de forma instantánea.

De hecho, la capacitancia en una red es también una medida de cuanto se opondrá ésta a un cambio en el voltaje de la red. Mientras mayor sea la capacitancia, mayor será la constante de tiempo y mayor el tiempo que le tomará cargar hasta su valor final

Ejemplo 2.2 (Fuente:3) La Figura 2.3 muestra un sistema compuesto por una resistencia y un capacitor, y cuyos valores son representados respectivamente por R y C. Además, la figura muestra que el sistema eléctrico es excitado por una señal x(t) = u(t) y su respuesta es medida a través de la tensión sobre el capacitor, donde u(t) representa la función escalón unitario:

El modelo matemático asociado al sistema representado por la Figura 2.3 puede obtenerse empleando elementales ecuación de redes eléctricas:

Entonces, al comparar el modelo matemático definido por la Ecuación (2.12) con el modelo obtenido, se tiene que el coeficiente a0 y la señal de excitación son:

,

Al aplicar la solución expresada por medio de la Ecuación (2.21), se puede afirmar que:

Al operar la Ecuación (2.26) se tiene que la respuesta del sistema es dada por:

Note que:

por cuanto el elemento de memoria representado por el capacitor no permite cambios bruscos y por tal motivo y(0-) = y(0) = y(0+). Además, para buscar una respuesta a la pregunta debe tomarse en cuenta que la excitación tiene un valor de cero y ella ha permanecido en cero desde mucho tiempo atrás, es decir, desde menos infinito, obviamente y(0) = 0.

Recomiendo leer la siguiente guía: Capacitores e Inductores – Circuitos y asociaciones

Fuentes:

1. El Parámetro Capacitancia p 20
2. Análisis de Redes – Van Valkenburg,
1. El Parámetro Capacitancia p 20
3. Análisis de Sistemas Lineales – Prof. Ebert Brea
1. Análisis de Sistemas en el Dominio Continuo pp 29 –
4. Fundamentos_de_circuitos_electricos_5ta

SIGUIENTE:

Escrito por Prof. Larry Francis Obando – Technical Specialist – Educational Content Writer – Twitter: @dademuch

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Mentoring Académico / Emprendedores / Empresarial

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Escuela de Ingeniería Electrónica de la Universidad Simón Bolívar, USB Valle de Sartenejas.

Escuela de Ingeniería Eléctrica de la Universidad Central de Venezuela, UCV CCs

Escuela de Turismo de la Universidad Simón Bolívar, Núcleo Litoral.

Contacto: Jaén – España: Tlf. 633129287

Caracas, Quito, Guayaquil, Lima, México, Bogotá, Cochabamba, Santiago.

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