In analog design, establishing the right operating point for active components is crucial to achieving the desired performance. In the case of JFETs, appropriate biasing techniques can make the circuit less sensitive to variations in device parameters. In this chapter, we’ll explain different biasing methods for JFET devices using an example.
Consider an n-channel JFET with I_{DSS} = 8 mA and V_{P} = -2 V. The transfer characteristic of the JFET in the saturation region is shown in Figure 1.
With that in mind, let's assume that the goal is to bias the JFET at point Q corresponding to i_{D }= 2 mA and v_{GS }= -1 V. There are a few basic biasing methods that we’ll explore below.
The simplest biasing scheme, known as constant voltage bias, is shown in Figure 2.
Under normal conditions, the gate current of a JFET is essentially zero, and no voltage drops across R_{G}. Thus, the voltage source, V_{GG,} directly appears across the source-gate junction—i.e., v_{GS }= -V_{GG}. By choosing V_{GG} = 1, we can set the bias point to v_{GS }= -1 V. To find the drain current, we can simply substitute the following into Shockley's equation, shown in Equation 1:
$$i_D=I_{DSS} \big ( 1- \frac{v_{GS}}{V_P} \big )^2$$
Which produces i_{D }= 2 mA.
In addition, a graphical approach can be employed to determine the DC operating point of the above circuit. To do this, we find the intersection point of the line v_{GS }= -1 with the device characteristic curve (Figure 3).
The graphical approach can help us develop a more intuitive understanding of the circuit operation. As an example, consider the following circuit (Figure 4), where the input AC signal is applied through the coupling capacitor, C_{1}.
Let v_{in} be a sinusoidal input with amplitude, A, and assume that the coupling capacitor is sufficiently large so the excursions of v_{in} appear at the gate terminal with no attenuation. In this case, the AC signal adds algebraically to DC bias voltage. Therefore, v_{GS} varies from -1 -A to -1 +A volts. This is illustrated in Figure 5.
By finding the intersection point of the lines corresponding to the maximum and minimum of v_{GS} with the characteristic curve, we can determine the variation in the drain current. In the above example, i_{D} varies from just above 1 mA to slightly higher than 3 mA.
Another advantage of the graphical method is that it helps you visualize how distortion can increase at certain points of the curve. Recall that the slope of the characteristic curve specifies the transconductance of the circuit, which is related to the circuit’s gain. Therefore, a larger curvature in the i_{D} - v_{GS} plot corresponds to a larger distortion. To clarify this, consider the example depicted in Figure 5. Assume that the DC operating point of the circuit is changed to Q_{1}, as depicted below in Figure 6.
The slope of the curve approaches zero as we get closer and closer to the pinch-off voltage. Therefore, in the above example, the transconductance changes from almost zero to a positive value. Thus, we expect the above circuit to have a larger distortion. Also, note that with the new bias point, the variation in i_{D} is much smaller than in the previous case. This means that the new bias point provides a smaller gain.
To avoid the need for two DC supplies, the self-bias configuration shown in Figure 7 can be used.
In this case, the gate is held at zero volts by means of R_{G}; however, the source is at a positive potential because the drain current flows through R_{S}. As a result, a negative v_{GS} is produced without employing a negative voltage supply. Applying Kirchhoff’s voltage law to the input circuit, we get Equation 2.
$$v_{GS}=-R_S i_D$$
In the case of our example, the bias point is given—i_{D} = 2 mA and v_{GS} = -1 V. Substituting these values into the above equation, we obtain R_{S} = 0.5 kΩ. Usually, R_{S} is given, and we need to find the DC operating point. In this case, Equations 2 and 1 should be solved simultaneously to determine the JFET’s DC operating point. Alternatively, we can use the graphical approach and find the intersection point of the bias line (Equation 2) with JFET's characteristic curve. The graphical solution for R_{S }= 0.5 kΩ is shown in Figure 8.
Note that the slope of the line is -1/R_{S}. Also, observe that the value of R_{S} is fixed for a given DC operating point. Now consider applying an AC signal to the self-bias configuration (Figure 9).
Again, assume that the sinusoidal input with amplitude, A, is applied to the input, and the whole signal excursions appear at the gate terminal with no attenuation. Let’s see how the graphical method can be used to find the amplitude of variations in v_{GS} and i_{D}. When the input is at its maximum, the gate voltage is raised from 0 to A volts. Therefore, the bias line is modified to:
$$A=v_{GS}+R_S i_D$$
In the i_{D }- v_{GS} plane, the above equation corresponds to a line with a slope of -1/R_{S} that intercepts the horizontal axis at v_{GS }= A (the orange line in Figure 10).
Similarly, when the AC input is at its minimum, the bias line is a line with a slope of -1/R_{S} that intercepts the horizontal axis at v_{GS }= -A (the pink line in the above figure). The intersection points of these lines with the characteristic curve specify the variation in v_{GS} and i_{D}.
Use LTspice to find the variation in v_{GS} for the following circuit and compare the results with those of the graphical method. The JFET parameters are:
Figure 11 shows the schematic for the self-biasing JFET with an AC input.
Alternatively, the LTspice schematic is shown below in Figure 12.
Figure 13 provides the simulated v_{GS} waveform.
As can be seen, the peak-to-peak variation of v_{GS} is 333 mV. The graphical representation of this problem is shown below in Figure 14.
From the graphical solution, the v_{GS} variation is found to be -0.85 -(-1.18) = 0.33 V, which is consistent with the simulation results.
A disadvantage of the self-bias circuit is that the slope of the bias line, which is equal to -1/R_{S}, is fixed for a given DC operating point (Figure 8). In the following chapters of this textbook, we’ll discuss how the ability to choose the slope of the bias line can reduce the sensitivity of the DC operating point to device parameter variations. The voltage divider biasing arrangement, depicted in Figure 15, enables us to choose the slope of the bias line arbitrarily.
To find the bias line, note that the gate current is essentially zero, and thus, the gate voltage is specified by the resistive voltage divider made of R_{1} and R_{2}. Therefore, the gate voltage can be expressed as Equation 3.
$$v_{G}=\frac{R_2}{R_1 + R_2}V_{DD}$$
Now, applying Kirchhoff’s voltage law to the input circuit produces:
$$\frac{R_2}{R_1 + R_2}V_{DD}=v_{GS}+R_S i_D$$
In the i_{D }- v_{GS} plane, the above equation corresponds to a line with a slope of -1/R_{S} that intercepts the horizontal axis at a v_{GS} value given by Equation 3. Returning to our example, we obtain the graphical representation below in Figure 16.
The above figure shows two possible bias lines that have different slopes and intercept points, A_{1} and A_{2}. In fact, if we aim to have a particular bias line, we can choose the appropriate R_{S} value to provide the required slope; and then choose parameters, R_{1} and R_{2}, to adjust the x-axis intercept point. In the next section of this series, we’ll continue this discussion and investigate bias sensitivity to parameter variations.
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