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Active Components and Circuits

All circuits we have discussed so far are only composed of passive components (resistors, capacitors and inductors) driven by current and/or voltage sources. Later we will consider circuits containing active components such as bipolar junction transistors (BJT), field-effect transistors (FET), operational amplifiers (Op-Amps) containing many transistors, and voltage amplifiers. These active components can be considered as controlled voltage or current sources as functions (typically linear) of the input voltage or current.

For the purpose of describing the overall function and performance of such active components, instead of its internal structure and implementation which may be very complicated, an active component can be modeled by the following three parameters:

2portExamples.png

AmplifierModel.png

Example 1: Consider the circuit below containing an active component, a voltage amplifier, model by the three parameters $r_{in}$, $r_{out}$ and $A$, dirven by either a current source $i_s=v_s/R_s$ or a voltage $v_s=i_sR_s$ with internal resistance $R_s$:

voltageamplifierex1.gif voltageamplifierex2.gif


\begin{displaymath}
v_{in}=i_s (R_s  \vert\vert  r_{in})=i_s \frac{R_s r_{in}}{R_s+r_{in}}
=v_s \frac{r_{in}}{R_s+r_{in}}
\end{displaymath}


\begin{displaymath}
v_{out}=A v_{in} \frac{R_L}{R_L+r_{out}}
=A i_s \frac{R_s r_...
...t}}
=A v_s \frac{r_{in}}{R_s+r_{in}} \frac{R_L}{R_L+r_{out}}
\end{displaymath}

A circuit containing an active circuit such as a transistor or a voltage amplifier can be modeled as a two-port network with input port between terminals A and B and output port between terminals C and D. This 2-port network can also be described in terms of the three parameters, the open-circuit voltage gain, the input resistance and output resistance:

Note that a non-ideal source with internal resistance $R_s$ is used in the definition of $R_{out}$ as it is affected by $R_s$, while an ideal source with $R_s=0$ is assumed in the definition of $R_{in}$ and $A_{oc}$. In case the source is not ideal with $R_s\ne 0$, we will use the voltage $v_{AB}$ appearing across the input port as the input voltage.

The performance of such a circuit containing active components can also be described by terms such as ``voltage gain'' $G_V$, ``current gain'' $G_I$, and ``power gain'' $G_P$, which can be found based on the specific circuits. They are defined as below:

Example 3:

voltageamplifierex3.gif

Find $R_{in}$, $R_{out}$, and $A_{oc}$ of this two-port network containing $R_1$ and $R_2$ as well as the amplifier modeled by $r_{in}$, $r_{out}$ and $A$.

This 2-port network modeled as a voltage amplifier with $R_{in}$, $R_{out}$ and $A_{oc}$ can be used in more complicated circuits.

Example 4:

A 2-port network with a voltage aplifier on the left can be modeled by the circuit on the right. Find the parameters $R_{in}$, $R_{out}$ and $A_{oc}$ of the two-port network with the voltage amplifier embedded.

voltageamplifierex4.gif

In summary, the resistor $R_1$ shared by both the input and output loops serves as a negative feedback:

\begin{displaymath}v_s\uparrow \rightarrow i_{in}, v_{out}\uparrow \rightarrow v_1\uparrow
\rightarrow i_{in}, v_{out}\downarrow \end{displaymath}

As the result, the voltage gain $A_{oc}$ is reduced but both the input and output resistances are improved, i.e., $R_{in}$ is increased and the $R_{out}$ is reduced.

Example 5: (Homework)

voltageamplifierex5.gif

A voltage amplifier, denoted by the inner box (solid line) with three internal parameters $r_{in}$, $r_{out}$ and $A$, is used as a component in a two-port network, denoted by the outer box (dashed line). Its open-circuit output voltage is $Av_{in}$. Find the following three parameters of the two-port network.

Note that all output $v_{out}$ between C and D of the output port is fed back to the input port between A and B: $v_s=v_{in}+v_{out}$ or $v_{in}=v_s-v_{out}$, i.e., it is a negative feedback.

Then simplify the three results above by making reasonable approximations based on the assumptions that $A»1$, $r_{in}»r_{out}$.

Answer

Example 6: (Homework)

voltageamplifierex6.gif

Two amplifiers with parameters $A_1$, $r_{i1}$, $r_{o1}$ and $A_2$, $r_{i2}$, $r_{o2}$, respectively, can be connected in cascade as shown in the figure. Given a voltage source $v_s$ in series with an internal resistance $R_s$, find the output voltage. To maximize the output $v_{out}$, how would you change the values of the six parameters?

Find the power gain $G_p$ of the system.

Answer

Example 7: (Homework)

The input and output resistances $R_{in}$ and $R_{out}$, as well as the voltage gain $A_{oc}$ of a two-port network can be obtained experimentally. First, connect an ideal voltage source $v_s$ (a new battery with very low internal resistance) in series with a resistor $R_s$, and then connect load $R_L$ of two different resistances to the output port. Now the three parameters can be derived from the known values of $v_s$, $R_s$ and the two measurements of the load voltage $v_{out}$, corresponding to the two resistance values used.

Assume $v_s=1.5V$, $R_s=5 k\Omega$, and the input voltage is measured to be $v_{in}=1.25 V$; also, assume the two different load resistors used are $R_1=150 \Omega$ and $R_2=200 \Omega$ respectively, with the two corresponding output voltage $v_1=18.75V$ and $v_2=20$. Find $R_{in}$, $R_{out}$ and $A_{oc}$.

Answer


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Next: About this document ... Up: Chapter 2: Circuit Principles Previous: Two-Port Networks
Ruye Wang 2016-09-21