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.
Example 1: Consider the circuit below containing an active component, a voltage amplifier, model by the three parameters , and , dirven by either a current source or a voltage with internal resistance :
Find , , and of this two-port network containing and as well as the amplifier modeled by , and the open-circuit voltage gain .
A 2-port network with a voltage aplifier modeled by , and voltage gain on the left can be modeled by the circuit on the right. Find the parameters , and of the two-port network with the voltage amplifier embedded.
We first find the short-circuit current
at the output port. Assume a voltage source with internal resistance
is applied to the input port while the output port is short-circuited.
Applying KVL to the two loops of the circuit, we get:
Example 4: (Homework)
The transistor emitter follower and the op-amp buffer shown below are very important circuits which find wide applications in practice. These two circuits can be similarly modeled based on the individual models of the transistor and the an op-amp (the inner dashed boxes), also shown in the figure. Note that the two models are equivalent (the outter dashed boxes), as the non-ideal current and voltage in the models can be converted to each other.
The parameter of the transistor model is its current gain, and the parameter of the op-amp model is its voltage gain, both of them are much greater than 1. And for the op-amp, we also have .
We can now find the three parameters of the model of the two circuits:
Example 5: (Homework)
Two amplifiers with parameters , , and , , , respectively, can be connected in cascade as shown in the figure. Given a voltage source in series with an internal resistance , find the output voltage. To maximize the output , how would you change the values of the six parameters?
Find the power gain of the system.
Example 6: (Homework)
The input and output resistances and , as well as the voltage gain of a two-port network can be obtained experimentally. First, connect an ideal voltage source (a new battery with very low internal resistance) in series with a resistor , and then connect load of two different resistances to the output port. Now the three parameters can be derived from the known values of , and the two measurements of the load voltage , corresponding to the two resistance values used.
Assume , , and the input voltage is measured to be ; also, assume the two different load resistors used are and respectively, with the two corresponding output voltage and . Find , and .