We discussed that a grid voltage, negative with respect to the cathode voltage, controls the current that traverses a vacuum tube and that the grid bias voltage determines the quiescent vacuum tube operating point. The input voltage signal is added to the grid bias voltage and is amplified.

There are two largely used techniques to provide a vacuum tube with a grid bias voltage, negative with respect to the cathode. The *fixed* *bias* technique requires a separate power supply that provides the wanted negative voltage. The *cathode bias* or *self-bias* technique connects the grid to ground and elevate the cathode voltage above ground. In this way the grid voltage is negative with respect to the cathode voltage. Let us discuss these two biasing techniques more in details.

### 3.6.1 Fixed bias

The *fixed bias* schema is given in Figure 11. Two different power supplies are used. PS_{1} gives the high-tension *V+* to the anode of the vacuum tube, through the load. The negative of PS_{1} is connected to ground. PS_{2} produces the needed grid bias voltage *V _{g}*. The negative grid bias

*-V*is obtained by connecting the positive of PS

_{g}_{2}to ground. In this way, the positive of PS

_{2}is at ground level and the negative is at

*–V*with respect to ground. The grid receives the

_{g}*-V*bias voltage using a resistor

_{g}*R*. Given that no current goes through the grid, in normal operations, the resistor

_{l}*R*does not affect the voltage seen by the grid. The cathode is also connected to ground so that the grid is at

_{l}*–V*with respect to the cathode, as needed.

_{g}The usage of resistor *R _{l}* will be better discussed in Section 4.1.1. For the moment, we just mention that one of the purposes of

*R*(also called the

_{l}*grid leak*) is to provide the input signal, received from previous stage, with a high impedance path to ground.

The capacitor *C _{d}*, from the grid leak resistor to ground, decouples the residual input signal, that traverses

*R*, from the bias voltage supply. Consider that, generally the bias power supply provides bias voltage to several vacuum tubes in the amplifier. For instance, in a stereo amplifier, both left and right channels are sometimes biased by the same power supply. The residual input signal, which traverses

_{l}*R*, is added to the bias voltage and goes also to the other channels, where it is amplified by the other vacuum tubes, creating problems of cross-talk. In order to avoid that, the capacitor

_{l}*C*forms, with the resistor

_{d}*R*, a low-pass filter that shorts to ground the residual input signal. The value of this capacitor should be large, so that even very low frequencies are shorted to ground and do not go to the grid of the other vacuum tubes.

_{l}### 3.6.2 Cathode bias or self-bias

Negative voltage between grid and cathode can also be obtained by connecting the grid to ground voltage and by elevating the cathode voltage. This technique is generally referred as *cathode bias* or *self-bias*. The cathode voltage is elevated by connecting it to ground through the resistor *R _{k}*, generally called the

*cathode resistor*, as shown in Figure 12. Given that, generally, there is an anode current also at the quiescent state, the resistor

*R*produces a voltage drop from the cathode to ground so that the cathode voltage is above ground. The grid, being at ground voltage, is negative with respect to the cathode.

_{k}Note that, also in this case, the grid is not directly connected to ground. Rather, a grid leak resistor *R _{l}* is used to provide the input signal with a high impedance path to ground, as we already discussed for the fixed bias. Since there is no current flowing through the grid, it is at ground voltage.

The value of *R _{k}* can be computed using the Ohm law by knowing the

*bias current*, that is the cathode current at the operating point (quiescent state), as shown in next Example.

It is important to mention that the cathode resistor introduces a form of local negative feedback. In fact, when the current increases, the cathode voltage increases as well. In this case, the grid becomes more negative, with respect to the cathode, and tends to reduce the vacuum tube conduction. When current decrease, we have the opposite effect and the grid becomes less negative, increasing vacuum tube conduction. In other words, the cathode resistor tends to oppose the amplification of the signal and reduces the gain of the vacuum tube. In order to mitigate and almost eliminate this effect, a *bypass or decoupling capacitor* *C _{k}*, is generally introduced in the circuit, as shown in Figure 12. The bypass capacitor compensates the cathode voltage variation trying to maintain it as stable as possible, when amplifying a signal. In this way, local negative feedback is significantly reduced and gain significantly increased, as discussed in next Section.

### 3.6.3 Gain of the voltage amplifier with self-bias

The cathode resistor, used for self-bias, introduces a form of local negative feedback: when the current increases, the cathode voltage increases as well reducing the grid to cathode voltage, and vice versa. The result is that the cathode resistor reduces the gain of the vacuum tube. This local negative feedback can be significantly neutralized by using a bypass capacitor connected in parallel to the cathode resistor.

The gain of the voltage amplifier with self-bias can be determined using the equivalent circuit shown in Figure 13. Let us first suppose that no bypass capacitor *C _{k}* is used. Similarly, to what we discussed in section 3.4, when we estimated the gain of the voltage amplifier, the vacuum tube is represented by an AC power supply, in series with the anode resistance

*r*. Let

_{a}*V*be the voltage of the input signal measured from the grid to the ground, that is from the grid to the terminal of the cathode resistor

_{in}*R*opposite to the cathode. The voltage

_{k}*V*measured between the two terminals of

_{fb}*R*, corresponding to the voltage drop introduced by

_{k}*R*itself, is the feedback voltage. The grid to cathode voltage , resulting from the combined action of the input and feedback voltages, is:

_{k}.

The above equation can be better understood using the equivalent circuit in Figure 13, as we already did in Section 3.4. In the circuit, power is supplied by the AC power supply, replacing the vacuum tube. Voltage drops, through the resistors in the circuit, proceeding clockwise. Suppose highest voltage is at the top of the AC power supply (the anode of the vacuum tube), and lowest voltage is at the bottom of the AC power supply (the cathode of the vacuum tube). *V _{in}* is the voltage difference between the input

*in*and the cathode resistor end, opposite to the voltage source (opposite to the cathode). The voltage difference between the input

*in*and the other cathode resistor end (between the grid and the cathode) is higher than

*V*, because of the voltage drop introduced by

_{in}*R*. Given that the voltage drop is

_{k}*V*, will be equal to

_{fb}*V*plus

_{in}*V*.

_{fb}The AC power supply produces a voltage equal to . As we said in Section 3.4, the minus sign here indicates that the phase of the AC power supply is reversed with respect to that of . The output voltage is taken between the two ends of the load *R _{L}* and can be computed using the voltage divider equation as

.

Using the voltage divider equation again, the feedback voltage *V _{fb}* is

.

Replacing *V _{fb}* into the equation for and simplifying we have:

.

Now we can replace this into the equation for and we obtain:

.

Finally, we can express the gain of the voltage amplifier, with local feedback introduced by a non-bypassed cathode resistor, as:

.

As we said in Section 3.4, we do not consider the phase so we omit the minus sign.

Let us now consider the case where a capacitor *C _{k}* is used to bypass the cathode resistor

*R*, as shown in Figure 13. The impedance of the capacitor depends on the signal frequency

_{k}*f*and is

.

The impedance of the cathode resistor in parallel with the bypass capacitor is

.

The gain of the voltage amplifier, with local feedback introduced by a cathode resistor and bypassed by a capacitor, is obtained by replacing *R _{k}* with

*Z*in the equation for

_{k}*A*. We obtain:

^{fb}.

The value of the gain depends on the frequency and on the capacitor. It ranges between these two extremes:

.

Minimum gain, equal to the non-bypassed cathode resistor, occurs when the capacitor impedance is maximum (at very low frequencies and/or very low capacitances). Maximum gain, similar to the circuit without cathode resistor, occurs when the capacitor impedance is minimum (at high frequencies and/or large capacitance).

The value of the capacitor should be chosen so that gain is maximum even at very low audible frequencies, as discussed in next example.