Coupling and Bypass Capacitors and Frequency Response

Coupling and Bypass Capacitors and Frequency Response

As noted in Unit 1, due to the external capacitors that are attached to the circuit for configuring the signal circuit, the gain of the amplifiers will tend to fall off at the low-f 616t194g requency end of the frequency spectrum. The design basis for choosing the capacitors for a given application is considered in the following. The gain-frequency dependence due to the input coupling capacitor, the source-resistor bypass capacitor, and the load coupling capacitor is discussed. A discussion of response roll-off on the high end of the spectrum is deferred to Unit 11

6.1 Grounded-Source Amplifier: Coupling Capacitor

A version of the circuits of Fig. 5.2, as given in Fig. 6.1, is generalized to include a resistance, R_s, as part of the input signal source. The input signal at the gate (and thus the output) will fall off for decreasing frequencies as the magnitude of reactance of the capacitor increases. The value of the capacitor is selected such that the reactance will be small compared to the value of the sum of the resistors, at the lowest operating frequency.

Figure 6.1. Circuit with coupling capacitor. Capacitor blocks the signal at low frequencies. C_g and R_G = R_G1 || R_G2 must be selected for | X_Cg | <<R_s + R_G at the gain measurement frequency.

In this amplifier, all the gain-frequency dependence is between V_g and V_s. The relation between the gate signal, V_g, and source signal, V_s, based on a voltage-divider relation is

Equation 6.1

with

Equation 6.2

where R_G = R_G1 || R_G2.

With R_s included, the f_3dB expression applies to an actual amplifier. In Project 6, the signal-source voltage is applied directly to the gate. We select a combination of R_G and C_g to obtain a suitably low f_3dB. This must be more than a factor of 10 lower than the gain measurement frequency assure that the capacitor is not influencing the measurements.

In Projects 5 and 7, no capacitor is used at the input as the signal is superimposed on the dc gate voltage. This is to facilitate the need for both DAQ output channels to provide the bias. However, Project 6, one channel provides the bias, as in Figs. 5.1(a) and , and the other channel is used for the signal with connection to the input through a coupling capacitor.

6.2 Current-Source Bias Amplifier: Bypass Capacitor

The PMOS common-source amplifier circuit with the bypass capacitor is shown in Fig. 6.2. At some low frequency, the effect of the capacitor is absent, and the gain is computed from ( ), which is

Figure 6.2. Dual-power-supply common-source amplifier stage with bypass capacitor. The object is to determine the frequency for which the source is effectively at signal ground.

Over a range of frequencies, the gain evolves from ( ) to ( ), a_v = -g_mR_D (neglecting the output resistance of the PMOS). The frequency-dependent transition region is determined by replacing R_S in ( ) with the impedance of R_S in parallel with the reactance of C_s. This is

Equation 6.3

or

Equation 6.4

with

Equation 6.5

and

Equation 6.6

Note that the "RC" factor has an "R," which is R_S in parallel with 1/g_m, the latter being the output resistance looking into the source of the transistor. The two frequencies f_s and f_z are technically the pole and zero of the response function.

We note that g_mR_S = 2I_DR_S/V_effp. Since V_effp could be as low as V_effp 0.2 V, then g_mR_S >> 1 in some cases. The corner frequency, f_3dB, occurs at

Equation 6.7

which gives

Equation 6.8

such that

Equation 6.9

where the far-right-hand term applies for g_mR_S >> 1. This is typically only marginally satisfied in MOSFET circuits.

The derivation carried out here was initiated from ( ), which neglects the output resistance of the transistor. In the case of the MOSFET devices of our projects, the simplification is valid for the NMOS transistor but marginal for the PMOS transistor. This is because l_p >> l_n. The result ( ) still serves to estimate the required value for C_s, even for the case of the PMOS. Nevertheless, a more detailed derivation is carried out in the next unit. This permits comparisons, in a project, of SPICE and Mathcad solutions with amplifier gains and frequency response.

6.3 Precision Formulation of the Low-Frequency Gain and Characteristic Frequencies

In Unit 8, the gain of a common-source stage with source resistor, which includes the effect of the output resistance, is shown to be [(

This is designated here as a_volo to emphasize that it is the constant low-frequency asymptotic gain. If the source-branch impedance is substituted for R_S, this becomes

Equation 6.10

where f_z is ( ). This can be rearranged in the form of ( ), which is

Equation 6.11

where the new pole frequency is

Equation 6.12

and a_volo is (

6.4 Load Coupling Capacitor

The complete practical amplifier includes a load, R_L, at the output. This requires an additional coupling capacitor. The amplifier with an attached load is shown in Fig. 6.3. We will assume that the effect of the output resistance, r_ds, can be neglected. The transistor appears as a current source of magnitude I_d. The equivalent signal circuit at the output can be represented as shown in Fig. 6.4(a) and (b). In Figure 6.4(b), the current source and drain resistor are replaced with a voltage-source equivalent. From Fig. 6.4(b), the output voltage is

Equation 6.13

Figure 6.3. Amplifier with capacitively coupled load resistor, R_L. The capacitor must be large enough not to attenuate the output between the drain and load resistor.

The expression with the high-frequency asymptotic value and frequency dependence is

Equation 6.14

which is

Equation 6.15

where

Equation 6.16

Figure 6.4. (a) Equivalent circuit consisting of a current source and the load components. (b) Conversion of current source to voltage source.

Current source I_d is not dependent on frequency (at the low end of the spectrum) when no other capacitors are considered. However, in general, with both C_d and C_s attached, the combined response is

Equation 6.17

where f_d is ( ), f_z is ( ), f_s is ( ), and is R_D in parallel with R_L. With C_d = C_s, usually f_d << f_s such that ( ) is the approximate f_3dB for the circuit with both capacitors connected.

Combining the frequency-dependent terms to obtain ( ) assumes that current source I_d is independent of components C_d, R_D, and R_L. This is a correct assumption as long as the output resistance of the transistor is neglected. When better precision is required, the derivation starts with the general form of the circuit transconductance, as deducible from ( ), and replaces both R_D and R_S with the impedance in those respective branches, Z_D and Z_S. This gives

Equation 6.18

The gain follows from multiplying the transconductance by the load at the drain, Z_D, and multiplying by the voltage divider, V_o/V_d = R_L/(R_L + 1/j2pfC_d). The result is

Equation 6.19

6.5 Summary of Equations

6.6 Exercises and Projects