next up previous
Next: About this document ... Up: Chapter 2: Circuit Principles Previous: Two-Port Networks

Active Circuits

So far all circuits we have discussed are composed of passive components (resistors, capacitors and inductors) driven by current and voltage sources. In the future we will be considering active components such as bipolar junction transistors (BJT) and field-effect transistors (FET), operational amplifiers (Op-Amps), as well as more sophisticated circuits such as voltage amplification circuits. These active components (as simple as single transistors and as complicated as some Op-Amps) can be considered as controlled current or voltage sources that generate current or voltage as a function (typically linear) of the input current or voltage.

For the purpose of describing the overall function and performance of such components and circuits (instead of its internal structure and implementation), a general model can be used with the following three parameters:


Example 1:


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

Ideally when $r_{in}=\infty$ and $r_{out}=0$, $v_{out}=A i_s R_s$.

Example 2:


\begin{displaymath}v_{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 v_s \frac{r_{in}}{R_s+r_{in}} \frac{R_L}{R_L+r_{out}} \end{displaymath}

The terms ``voltage gain'' $G_V$, ``current gain'' $G_I$, and ``power gain'' $G_P$ need to be specifically defined for different circuit configurations. In this case, they can be defined as below: The alternative definitions of these voltage, current, and power gains may be used, depending on the specific applications.

A two-port network (of which an active circuit such as a voltage amplifier may be a component) 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.

Example 3:


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.


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)


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 $A»1$, $r_{in}»r_{out}$.


Example 6: (Homework)


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.


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 $A_{oc}$.


next up previous
Next: About this document ... Up: Chapter 2: Circuit Principles Previous: Two-Port Networks
Ruye Wang 2016-06-20