Amplifier Circuits

Ideal Amplifiers

A port is a pair of terminals of a network (circuit). Across the port is a voltage, v, and through it flows a current, i, as shown below.

Amplifiers have two ports, input and output. An electrical waveform is a voltage or current as a function of time. A waveform to be amplified is applied to the input port and another waveform appears at the output port that is larger than the input waveform. Input and output quantities can be either voltages or currents, resulting in four basic kinds of amplifiers:

Amplifier Type	Input Quantity	Output Quantity
Voltage, A_v	voltage, v_i	voltage, v_o
Current, A_i	current, i_i	current, i_o
Transresistance, R_m	current, i_i	voltage, v_o
Transconduct-ance, G_m	voltage, v_i	current, i_o

In the table under amplifier type is the expression for amplification or gain (or transfer function), which is the output quantity divided by the input quantity. In general, A = x_o/x_i, where x is either a voltage or current.

An ideal input port is not affected by input source resistance nor is an ideal output port affected by output load resistance. The general amplifier is shown below:

The combination source/resistance symbol is a generalized source: either a Thevenin or Norton equivalent circuit. The amplifier has an input resistance R_in and output resistance, R_out. The input source, x_i (where x_i is v_i or i_i), has resistance R_i. It forms a divider (voltage or current) with R_in so that x_i ¹ x_in. Similarly, output resistance R_out forms a divider with output port load resistance R_L so that the output x_out = K× x_in ¹ x_o. The amplification of x_in by K results in x_out that is K times larger. K is the gain, and it scales x_in. (Gain less than 1 is called attenuation.) If source or load resistance is unknown or varies with K, then error in the overall amount of gain results. An accurate (or at least unchanging) gain is required for calibrated sensor circuits, so that the transducer output is multiplied by a known (and constant) amount.

An example of a voltage amplifier is shown below:

The overall voltage gain is:

The first factor is the input voltage divider attenuation, the second is the amplifier voltage gain and the third is the output voltage divider attenuation. For the ideal voltage amplifier, A_v = K. This is achieved when R_in approaches infinity (open-circuit input) and R_o = 0. The ideal port resistances are given in the following table:

Port Type	Ideal Resistance
Voltage input	infinite (open)
Current input	zero
Voltage output	zero
Current output	infinite (open)

In practice, good amplifier design approaches the ideal so that input and output loading does not affect the overall amplifier gain accuracy.

Transistor Configurations

Transistors have three terminals connected to input and output circuit loops. One-transistor amplifiers are two-port networks; one of the three transistor terminals must be shared by both input and output ports as the common terminal. This results in three possibilities. The first is the common emitter (CE) amplifier. The emitter is common to both input and output, as shown below.

The emitter is part of both input and output loops. It is the common terminal of the transistor that is connected to both an input and output port terminal. With a series emitter resistor R_E the emitter terminal is still common to both loops. The output loop current is shown flowing from the power supply (+V_CC), through R_L and the BJT, through R_E to ground, which is connected to the negative terminal of the supply. The closure of the output loop from ground to +V_CC implies flow through the voltage-source, +V_CC.

The common-base (CB) configuration is shown below:

The common-collector (CE) configuration, also known as the emitter-follower, is shown below.

Common-Emitter Amplifier

The CE amplifier was analyzed in the Transistors chapter. The voltage gain was found by the transresistance approach: a ratio of output (load) resistance and transresistance, the resistance across which the input voltage develops the common (emitter) current. Not all of the emitter current gets to the collector. Some is lost to the base, and the a factor accounts for this in the voltage-gain equation:

Because a » 1, the voltage gain is a ratio of resistances. The input voltage v_i is applied across r_M, producing i_E = v_i/r_M. Then i_C ( = a× i_E) gets through to the collector and develops a voltage of v_o = - i_C× R_L at the output. By solving these equations for A_v, the above gain equation results.

The input resistance of the CE is v_i/i_i = v_i/i_B or

The resistance of the input loop is the base resistance in series with the resistance in the emitter-side of the circuit, referred to the base by the b transform. (See Transistors for details.)

At the output node, the BJT transistor model shows a current source (infinite resistance) in parallel with load resistance R_L. The output resistance is therefore R_L.

The CE amplifier has relatively high input resistance due to the b-transform effect at the base. It is better as a voltage-input port. Its output resistance is relatively low if the load resistor is not made too large.

The current gain of the CE is i_o/i_i = i_C/i_B = b. Its input-loop transresistance used to calculate gain is r_M, but the overall amplifier transresistance is R_m = v_o/i_i = A_v× r_in and its transconductance is the inverse of the transresistance, or G_m = 1/R_m.

Common-Base Amplifier

The CB amplifier input source is in the emitter loop so that emitter current flows through it. This current is b + 1 times larger than the base current. Consequently, the CB input resistance is relatively low and would make a better current-input than voltage-input port. Its input resistance is

or typically about R_E. Its output resistance is the same as the CE, or R_L. The CB voltage gain is

Unlike the CE, it is non-inverting (no negative sign). The CB current gain is a, or slightly less than one.

Compared to the CE, the CB input resistance is lower by (b + 1) and is therefore better as a current-input port than the CE. According to the ideal-port table, the CB most closely approaches an ideal current amplifier, though its current gain is slightly less than one!

Common-Collector Amplifier

The CC or emitter-follower has the same input resistance as the CE but its output resistance is

or typically about r_e, a relatively small resistance of around a few ohms. With high input resistance and low output resistance, the CC appears to approach the ideal voltage amplifier. Unfortunately, its voltage gain is only

or typically somewhat less than one. The port resistances approach the ideal but the voltage gain is not high enough to be useful. The current gain, however, is b + 1.

None of the three single-transistor configurations is ideal as any of the four amplifier types. Amplifiers can better approach the ideal by combining configurations into multi-transistor amplifiers.

Cascade Amplifier

Amplifiers are cascaded when the output of the first is the input to the second. The combined gain is

where v_i2 = v_o1. The total gain is the product of the cascaded amplifier stages.

The complication in calculating the gain of cascaded stages is the non-ideal coupling between stages due to loading. Two cascaded CE stages are shown below.

Because the input resistance of the second stage forms a voltage divider with the output resistance of the first stage, the total gain is not the product of the individual (separated) stages.

The total voltage gain can be calculated in either of two ways. First way: the gain of the first stage is calculated including the loading of r_i2. Then the second-stage gain is calculated from the output of the first stage. Because the loading (output divider) was accounted for in the first-stage gain, the second-stage gain input quantity is the Q₂ base voltage, v_B2 = v_o1.

Second way: the first-stage gain is found by disconnecting the input of the second stage, thereby eliminating output loading. Then the Thevenin-equivalent output of the first stage is connected to the input of the second stage and its gain is calculated, including the input divider formed by the first-stage output resistance and second-stage input resistance. In this case, the first-stage gain output quantity is the Thevenin-equivalent voltage, not the actual collector voltage of the stage-connected amplifier. The second way includes interstage loading as an input divider in the gain of the second stage while the first way includes it as an output divider in the gain of the first stage.

By cascading a CE stage followed by an emitter-follower (CC) stage, a good voltage amplifier results. The CE input resistance is high and CC output resistance is low. The CC contributes no increase in voltage gain but provides a near voltage-source (low resistance) output so that the gain is nearly independent of load resistance. The high input resistance of the CE stage makes the input voltage nearly independent of input-source resistance. Multiple CE stages can be cascaded and CC stages inserted between them to reduce attenuation due to inter-stage loading.

Darlington Amplifier

A CC stage followed by another CC stage has an input resistance of about (b + 1)² times the emitter resistance of the second stage. More precisely, using the b transform, it is

If R_E1 is removed, the second term is about b² times R_E2. Furthermore, if the collectors are connected together, the result is a Darlington stage, as shown below.

This stage can be viewed as a "Darlington transistor" because it has three terminals and an equivalent b of about b². Darlington BJTs can be used in any of the three BJT configurations.

Differential Amplifier

A differential or emitter-coupled BJT pair is formed, as shown below, by a CC/CE stage driving a CB stage. The first stage is a CE to the first output, v_o- and is a CC to the second stage.

A differential-input amplifier has an input port for which the negative (- ) terminal is not necessarily connected to the common node (usually ground). A differential amplifier (or diff-amp) amplifies the difference between its input terminals:

Amplifiers with differential outputs have two output terminals, neither of which is necessarily common with an input terminal or ground. The output is

x_o = x_o+ - x_o-

The 2-transistor diff-amp has differential inputs and outputs. The voltage gain is found by calculating the gain from each input to each output (4 gains). The differential gain is the ratio of the difference of the outputs over the difference of the inputs. If the gain magnitude (absolute value, neglecting sign) to the output is different for the two inputs, the amplifier is not differential.

The above amplifier gain can be calculated using the transresistance method. The current-source resistor R_EE forms a divider between stages. Ideally, R_EE is a current source. The diff-amp circuit is also symmetrical if corresponding components have equal values:

R_L1 = R_L2 = R_L

R_E1 = R_E2 = R_E

R_B1 = R_B2 = R_B

and

R_EE >> R_E

then the voltage gain is

For non-negligible R_EE, a divider is formed between stages consisting of the source-transistor R_E and R_EE. Apply Thevenin’s theorem for a Thevenin equivalent source driving R_E of the other stage.

Complementary Stages

Not only can npn BJTs or n-channel FETs be used in stages, so can their complementary devices, pnp BJTs and p-channel FETs. Having both polarities of transistors allows for more kinds of amplifiers and makes biasing easier.

For example, a complementary cascade amplifier is shown below. The second (CE) stage uses a pnp BJT. In a representation similar to the power-supply voltage sources, the input voltage source is implicit by the v_i label at the input node. Also, the + terminal of v_o is labeled by "v_o" and is understood to be taken with respect to ground (as the - terminal).

Its gain equation is the same as the all-npn cascade. The advantage of the complementary cascade amplifier is that the CB-stage collector supply (ground) must be at a lower voltage than that of the base, allowing a ground-referenced output. For the all-npn cascade, +V_CC adds to the output voltage developed across R_L instead.

A complementary cascode (CE followed by CB) is shown below, with a JFET input stage for high input resistance. Except for the FET (with r_s instead of r_e) and the addition of R_L1, the gain and port resistance formulas are the same as the all-npn cascode. The Q₂ base-biasing divider resistors form the equivalent R_B.

Many other amplifiers of two or more transistor stages can perform better than the three one-transistor configurations. The ones described here are among the most common and should be recognizable as "components" of larger circuits.