Chapter 4. CMOS Cascode Amplifiers. 4.1 Introduction. 4.2 CMOS Cascode Amplifiers

Chapter 4 CMOS Cascode Amplifiers 4.1 Introduction A single stage CMOS amplifier cannot give desired dc voltage gain, output resistance and transconductance. The voltage gain can be made to attain higher value by using active load like current source. A single stage CS amplifier can offer infinite input resistance, moderate output resistance and moderate dc gain even with current source load. To achieve desired performance two or more stages of MOS devices are cascaded to form a single device called cascode amplifier [49]. This cascode amplifier can give desired performance parameters as per stages selected for cascading. 4.2 CMOS Cascode Amplifiers Amplification is an essential function in many analog circuits. The cascode amplifier consists of common source (CS) and common gate (CG) configuration to achieve higher gain. The analysis and design of cascode amplifier hence will start with the MOS device physics, analysis and design CS stage. 4.2.1 MOS Device Physics Assuming the basic concepts of doping, mobility, and pn junctions, the physics of MOSFETs at elementary level is dealt here. Fig, 4.1 shows a physical structure of an n- type MOS (NMOS) device. The device consists of two heavily doped n-type regions indicated in the figure as source (S) and drain (D) terminals, a heavily doped (conductive) piece of polysilicon indicated as a gate (G) and a thin layer of silicon dioxide(sio 2 ) of thickness t ox (2-50nm) insulating gate from p-type substrate on which the device is fabricated. Metal contacts are made to the gate on the top of oxide layer, the source region, the drain region, and the substrate; which is also known as the body (B). The structure is symmetric with respect to S and D and the useful action occurs in the substrate region under the gate oxide [44]. The length L is a dimension of the gate along the source-drain path and perpendicular to that is width W. The actual distance between source and drain (L eff ) is slightly less as S/D 30

Figure 4.1. Physical structure of NMOS device junctions side diffuse. These L eff and t ox play important role in the performance of the device. Hereafter L denotes effective length. Actually, the substrate potential also greatly affects the device characteristics. As in typical MOS operation, the S/D junctions must be reverse- biased; the substrate of NMOS is connected to the most negative supply in the system. In CMOS technologies, both NMOS and PMOS devices are available. Usually NMOS and PMOS devices must be fabricated on the same substrate and for this reason PMOS device is placed in a local substrate called a well as shown in Fig.4.2. Here to keep the S/D junctions reverse-biased, the n-well must be connected to the most positive voltage. An interesting fact can be seen from Fig.4.2, all NMOS devices share same substrate but PMOS devices can have an independent n-well. Figure 4.2. PMOS device inside n-well MOS I/V Characteristics To raise the abstraction from physics level to circuit level the equations for I/V characteristics need to be studied. I/V characteristics, some of the terms and secondary effects will be dealt in this section. 31

Threshold Voltage When a positive voltage is applied to the drain with respect to source, as the voltage at gate with respect to source increases, the width of depletion region and the potential at oxide silicon interface also increase. As soon as the gate potential reaches a sufficient high value, electron flow from source to drain starts. The value of gate potential for which this current flow occurs is called the threshold voltage V TH. The choice of V TH depends on the device performance in typical circuit applications. The upper limit is roughly equal to V DD /4 and lower bound determined by the variation with temperature and process, and the channel length. Typical value of V TH is adjusted by implantation of dopants into the channel area during fabrication. I/V characteristics The NMOS with gate and drain voltages applied with the normal directions of the currents is indicated in Fig.4.3 (a). The I D versus V DS characteristics are plotted as a family of curves, each measured at a constant V GS as shown in Fig. 4.3 (b). The shape of I D -V DS characteristics can be expected from physical operation of the device [43]. The characteristic curves indicate clearly three distinct regions of operation: the cut off region, the triode region, and the saturation region. The MOSFET operates as an amplifier in the saturation region and it operates as a switch in the cut off and triode regions. The device is in cut off when V GS <V TH whereas to operate in triode region V DS <(V GS -V TH ). The I D -V DS characteristic in the triode region can be described using physics and the process of the device as, Figure 4.3. (a) An n-channel enhancement type MOSFET. (b) Output characteristics (4.1) 32

where, µ n is the mobility of charge carriers of N-MOSFET, C ox the gate oxide capacitance, W the width and L the length of channel of the MOSFET. If V DS is sufficiently small, the term of its square can be neglected from (4.1) and we obtain, (4.2) The (4.2) shows that I D is a linear function of V DS, which is also seen from the characteristic curves. This linear relationship implies that the path from source to drain can be represented by a linear resistor as, (4.3) The term in the bracket is called as gate to source overdrive voltage V OD and given as, (4.4) When the drain-source voltage exceeds V OD, as seen I D becomes relatively constant and we say the device operates in the saturation region. The boundary between the triode and saturation region is characterized by, (4.5) Substituting this in (4.1) gives the saturation value of I D as, (4.6) A saturated MOSFET can be used as a current source connected between the drain and the source. Second-Order Effects The analysis of MOSFET carried out so far involved various assumptions, which are not valid for many analog circuits. Three such essential second-order effects are discussed here. Body Effect In the analysis so far, it was assumed that the body and source of MOSFET were tied to the ground. If body voltage of NMOS drops below the source voltage, the S/D junctions remain reverse-biased but certain characteristics may change. To elaborate this effect, assume V S =V D =0, and V G is somewhat less than V TH such that a depletion region is formed under the gate but no inversion layer exists. Since V B becomes more negative, more holes are attracted to the substrate, leaving behind a large negative charge widening 33

the depletion region. The threshold voltage is a function of the total charge in the depletion region because the enough gate charge must be supplied to form an inversion layer. Hence, as V B drops, V TH increases and this is called the body effect or the backgate effect. It is proved that with body effect: (4.7) Where V SB is the source-body potential difference, V TH0 the threshold voltage without body effect that is with V SB =0, γ denotes the body effect coefficient and Φ F the difference between the work functions of the polysilicon gate and the silicon substrate. The value of γ typically lies in the range of 0.3 to 0.4 V 1/2. Body effect is generally unwanted. The change in the threshold voltage complicates the design of analog as well as digital circuits. Channel-Length Modulation In the analysis of I/V characteristics, it was noted that the actual length of the inverted channel gradually decreases with increase in the potential difference between gate and drain. Thus, effective length L` is actually a function of V DS. This effect is known as channel-length modulation. Let us write, Then, Figure 4.4. Effect of channel length on characteristics in saturation region 34

Assuming a first order relationship between ΔL/L and V DS, that is ΔL/L=λV DS, the equation for drain current in saturation gets modified as, (4.8) where, λ is the channel-length modulation coefficient. As shown in Fig. 4.4, this phenomenon results in a nonzero slope in I/V characteristic and hence non-ideal current source between drain and source in saturation. The parameter λ represents the relative variation in length for a given increment in V DS. Hence, for longer length channels, λ is smaller. In short channel MOSFETs, the linear approximation ΔL/L α V DS becomes less accurate resulting into a variable slope in the saturated I/V characteristics. The V DS can be used to set I D offering more flexibility to V OD. However as dependence of I D on V DS is much weaker, it is not used to set I D. The variation of I D with V DS is usually considered an error. Subthreshold Conduction In analysis of the MOSFET, the assumption was that the device turns off abruptly as V GS drops below V TH. Actually, for V GS V TH, a weak inversion layer still exists and some current flows from D to S. Even I D is finite for V GS <V TH, but the dependence is exponential. This effect known as subthreshold conduction can be expressed by formula for V DS greater than roughly 0.2 volts as, (4.9) Where δ > 1 is a nonlinearity factor and V T the thermal voltage given by V T = kt/q with k the Boltzmann s constant, T the absolute temperature in kelvins and q the magnitude of electronic charge. With V GS <V TH, it is said that the device operates in a week inversion. Except for δ, the (4.9) is similar to the I C /V BE characteristics of the bipolar transistor. The important point here is that as V GS falls below V TH, the I D drops at finite rate. With typical values of δ and at room temperature, V GS decreases by approximately 80 mv for onedecade decrease in I D. Thus for example if a threshold voltage is selected 0.3V to allow 35

low voltage operation, then as V GS is reduced to zero, the I D decreases by a factor of 10 3.75. Subthreshold conduction can result in significant power dissipation, especially in large circuits like memories. The exponential variation of I D with V GS in subthreshold region can lead to achieve a higher gain. Since only a large device width or low I D meets these conditions, the speed of circuit is severely limited. 4.2.2 Common Source (CS) Amplifier Consider the single stage CS amplifier with an ideal current source load shown in Fig. 4.5. A small signal input voltage (v in ) is applied to the gate and output is taken at drain (v o ). The drain current of MOSFET depends on its gate voltage (V g ) and drain voltage (V d ). Thus, the drain current can be expressed as, Figure 4.5. A single stage CS amplifier (4.10) As per the definition is nothing but the transconductance (g m ) of the MOSFET and its output conductance. For small signals, we can write the quantities in above equation as small case v g, v d and i d. Here v g is same as the input signal v i and v d is the output of 36

amplifier v o. As the MOSFET is fed by a current source, it does not allow any variation in the drain current. Hence,. Therefore, (4.11) The voltage gain (A v ) of CS stage becomes (4.12) Where r o, the output resistance of MOSFET is defined to be 1/g o. Assuming MOSFET to be in saturation and using approximate model, drain current ( ) can be given by (4.6) as, As per definition g m can be expressed as, (4.13) (4.14) Substituting I d from (4.13) for right hand side, we get, (4.15) Substituting for V g - V TH from (4.15) in (4.14) (4.16) The above three equations show different proportions of g m with parameters (V g V TH ), W/L, and I d. This is because three parameters are not independent of each other. Which equation should be used for g m depends on which parameters have been fixed by design and biased conditions. For example, if size and gate bias is fixed, the (4.14) should be used for g m. On the other hand, if the current bias is fixed (4.16) is appropriate. Finally, if both gate voltage and drain current have been fixed by the design, (4.15) should be used to evaluate g m. The output conductance (g o ) is evaluated as the slope of drain characteristics in the saturation region. With simple Early effect like model and using (4.8), g o comes out to be, 37

(4.17) Where, λ is the channel-length modulation coefficient. In terms of geometry and (V g V TH ) we can write, (4.18) The voltage gain can be written in terms of geometry and (V g V TH ) as, (4.19) In terms of drain current and geometry the voltage gain is given by (4.20) Thus if MOSFET is biased at constant current, the DC gain is determined by the square root of gate area, as λ α 1/L. AC behavior of CS stage The AC equivalent circuit of MOSFET can be drawn as shown in Fig. 4.6, including the number of inter electrode capacitances. Figure 4.6. AC equivalent circuit Applying KCL at drain terminal (D), we get, (4.21) Separating the terms of v i and v o, we get, (4.22) 38

AC gain A vf can now be written as, (4.23) Let, C tot = C gd + C o also recognizing ωc gd /g m << 1 Then high frequency gain of CS amplifier is given by (4.24) This describes the frequency response of CS amplifier with one dominant pole. The bandwidth is given by 1/r o C tot. The gain bandwidth product is given by, (4.25) It is seen that the gain bandwidth product is independent of r o. The maximum possible cutoff frequency occurs at unloaded amplifier, where capacitive load is device capacitance itself. The total capacitance C tot is directly proportional to W and technological parameter λ, and g m is proportional to W. Thus, gain-bandwidth product depends only on technological parameter and is independent of geometry and operating point of MOSFET. Thus, DC gain and cutoff frequency cannot be set independently. The above CS amplifier circuit is analyzed with an ideal current source load. In practice, different types of loads are possible, even a practical current source is implemented by operating MOSFET in saturation and will not offer infinite output resistance as assumed earlier. Thus, it is needed to analyze CS stage with different loads. CS Stage with Resistive Load With transconductance, a MOSFET converts variations in its gate-source voltage to a small signal current, which after passing through a resistor generates an output voltage. Now the output current will pass through not only the output resistance of MOSFET r o, but also externally connected load resistance R D, coming in parallel. Thus, output voltage can be given as, Normally r o is larger compared to external load resistor R D, 39

(4.26) The voltage gain depends on device parameter g m which itself is a function of input voltage and aspect ratio W/L. The important observation here is that gain varies with the signal swing, and this leads to nonlinearity. The gain can be increased by increasing aspect ratio W/L and/or external load resistor R D. By increasing W/L, the device becomes larger leading to greater device capacitances thus limiting high frequency response. If R D is increased, the maximum voltage swing at the output gets limited. As seen in the analog design octagon Fig. 1.3, the circuit exhibits trade- offs among gain, bandwidth, and voltage swings. CS stage with Diode-connected load It is difficult to fabricate resistors with accurate values in many CMOS technologies. Hence, it is desirable to replace a resistive load with a MOS transistor. A MOSFET operates as a small signal resistor if gate and drain are shorted, called a diode-connected Figure 4.7 (a) Diode connected NMOS and PMOS devices (b) Small signal equivalent circuit. device as depicted in Fig.4.7(a). Here MOSFET is always in saturation as the drain and gate have same potential. With small-signal equivalent circuit shown in Fig. 4.7(b), the output impedance can be obtained as 1/g m r o 1/g m. Considering body effect, it becomes 1/(g m +g mb ) where g mb is the body transconductance. The impedance offered by diodeconnected load is lower when body effect is included. Now let us consider CS stage with diode-connected load shown in Fig. 4.8. Neglecting channel length modulation effect, the voltage gain becomes, 40

Figure 4.8. CS stage with diode-connected load (4.27) (4.28) Where ε=g mb2 /g m2. Using (4.16) to express gm in terms of device dimensions and bias currents, A v becomes, (4.29) But I D1 =I D2, (4.30) This equation reveals an important fact that if variation of ε is negligible with output voltage, the gain is independent of the bias currents and voltages. The diode load can be implemented with a PMOS device also. If body terminal is tied to the source, the circuit is free from body effect, and the small signal voltage gain becomes, (4.31) 41

Equations (4.30) and (4.31) show that the gain of the CS stage with diode-connected load is a relatively weak function of the MOSFET dimensions. The high gain requires a strong input device and a weak load device. This also reduces the allowable output voltage swings. CS stage with practical current source load Equation (4.26) for voltage gain of the CS stage suggests that the voltage gain increases with increase in load impedance. With resistive or diode-connected load, increase in the load resistance limits the output voltage swing. The load replaced with a current source load is a more practical approach as shown in Fig. (4.9). Here the requirement is that both MOSFETs operate in saturation. The total impedance at the output node is equal to r o1 r o2, and the gain becomes, (4.32) The output impedance of MOSFETs at given drain current is directly proportional to L. As the gain is proportional to r o1 r o2, the longer MOSFETs yield a higher voltage gain, but at the same time reducing bandwidth and requiring more overdrive voltage for M 2 to maintain it into saturation. The single stage CS amplifier cannot achieve high gain and other configuration such as cascode amplifier must be used to achieve higher gain. Figure 4.9. CS stage with current-source load 42

4.2.3 Cascode Amplifiers A cascode stage using CS and CG configurations is shown in Fig. 4.10. The input signal goes to the gate of lower MOSFET (M 1 ) arranged in CS configuration. The gate of upper MOSFET (M 2 ) is held at a fixed DC voltage V r, such that both MOSFETs operate in saturation all the time. The M 2 acts as a common gate amplifier. The two MOSFETs connected in series act as a single compound device with gate and source of M 1 and drain of M 2. This compound device is evaluated to find its equivalent g m and g o, denoted as g meq and g oeq in a fashion analogous to the CS stage. Two MOSFETs are in series; hence, their drain currents are equal. (4.33) Thus, with dv d2 = 0 (4.34) with dv g1 = 0 (4.35) Figure 4.10. A CS-CG cascode amplifier 43

g meq of the cascode stage To find g meq of cascode stage, make dv d2 = 0 and evaluate the change of drain current with changing gate voltage of M 1.As the drain as well as the gate of M 2 are held at fixed voltages, (4.36) (4.37) Using these equations, we can write for a small signal changing drain current, (4.38) (4.39) Hence, (4.40) Note that the source of CG stage is not at substrate potential. Thus, as the drain voltage of CS stage changes, the drain current of CG stage changes due to two identified reasons: 1) The gate source voltage of MOSFET changes the drain current. As the drain voltage of M 1 increases, the gate source voltage of M 2 reduces thereby reducing its current. 2) Since the source voltage of M 2 changes, its threshold voltage changes. This changes its effective over voltage, and thus the current also. When the drain voltage of M 1 increases, the threshold voltage of M 2 becomes higher, thereby reducing its current. It is assumed that g m2 is defined to account for both M 1 and M 2. V d1 can be eliminated from (4.38) to get, 44

Hence, (4.41) Commonly g m values are much higher than g o values. Then (4.41) gets reduced to g meq g m1. The value of g meq will be slightly lower than g m1 due to feedback through the drain voltage variation of M 1. g oeq of the cascode stage As seen from (4.33), we must make dv g1 = 0 and solve for the change of drain current with respect to change in drain voltage of M 2 to evaluate g oeq. With these conditions, dv gs1 = 0 (4.42) dv gs2 = 0 dv s2 = - dv d1 (4.43) dv ds2 = dv d2 dv d1 (4.44) Using these equations, we can write small signal change in drain current, (4.45) (4.46) Also from (4.45), Then (4.46) becomes, (4.47) Thus we get equivalent g o as, (4.48) Using the fact that g m values are much higher than g o values, 45

Hence, g oeq is lower than g o1 by a factor equal to the voltage gain of the upper MOSFET M 2.As seen earlier, the equivalent g m is practically the same as that for a single MOSFET and the output resistance is higher (g oeq lower), the DC gain is increased. DC gain of the cascode stage Once evaluated g meq and g oeq, now the DC gain of the cascode stage can be written as, Once again using the fact that g m values are much higher than g o values, we get, It can be written as, (4.49) The individual DC gains of the two MOSFETs in CS and CG can be defined as, and Then, Hence, DC gain of cascode amplifier is equal to the product of DC gain of individual CS and CG stages. AC behavior of cascode amplifier As the g meq of the cascode stage is about same as that of a single stage CS amplifier, the gain bandwidth product also remains same. Due to higher output resistance, the DC gain is higher and bandwidth gets reduced for a cascode stage. 46

Figure 4.11. AC equivalent circuit of the cascode amplifier The Fig 4.11 shows the AC equivalent circuit of the cascode amplifier. It can be seen that v x is quite small. Due to this, the effect of the drain capacitance of M 1 and gate capacitance of M 2 can be neglected. Then r o1 may be replaced by a parallel combination of r o1 and the capacitance seen at the drain of M 1. Applying KCL at output node, we get, (4.50) Simplifying (4.51) As A 2 the gain of CG stage is quite large, v x is very small compared to v o. Applying KCL at the drain of M1 gives, (4.52) Finally the gain becomes as, (4.53) Since sr o1 C dg1 is small, the equation gets simplified as, 47

(4.54) As compared to CS stage the DC gain of cascode stage is multiplied by A 2 and bandwidth reduced roughly by the same factor. Thus, the gain bandwidth product is the same as the single CS amplifier. Even if the DC gain is higher and gain bandwidth product remains unchanged, the design of amplifier, which goes beyond technological limit, is possible. Practical Cascode circuits: The circuit of cascode amplifier analyzed has used an ideal current source as the load. Actually, the current source load is implemented using MOS transistors. Then the output resistance of the cascode stage comes in parallel with the output resistance of this current source load. If single MOSFET current source is used, the output resistance becomes r o, while that of cascode stage is A x r o. The lower resistance of current source load dominates the final output resistance and the advantages of the cascode stage are lost [51]. Therefore, the load also should be a cascode current source with high output resistance. This leads to the circuit with four MOSFETs in series as shown in Fig. 4.12, often called as the telescopic cascode amplifier. For this circuit all four MOSFETs are to be maintained in saturation and hence consume larger voltage overdrive. However, this limits the swing in output voltage for which all MOSFETs will be maintained in saturation. Figure 4.12. Telescopic cascode amplifier 48

The folded cascode amplifier is as shown in Fig. 4.13. Telescopic cascode limits the swing in output voltage. The folded cascode gives increased output voltage swing but at the cost of reduced voltage gain and increased device count [45]. 4.2.4 Design Examples Figure 4.13. Folded cascode amplifier Statement: 1. Design single stage NMOS CS amplifier with load resistor 2 KΩ, small signal gain 10, supply voltage 3 V, and Power dissipation not more than 3 mw. Assume µ n C ox = 60 µa/ V 2 and λ = 0. Design Steps Power dissipation = V DD I D Thus ID = 1 ma As λ = 0, A v = - g m R D Thus, g m = 5 ma/v Now, using relation V od = 400 mv To find dimensions of NMOS using equation of I D 49

Figure 4.14. Schematic of CS amplifier with resistive load Figure 4.15. Frequency and phase response of CS amplifier with resistive load 50

W/L = 208, as λ is assumed zero, L cannot be taken too lower. Let L = 0.5µm, then W = 104 µm. Simulated circuit is shown in Fig. 4.14 with Tanner schematic editor. The frequency and phase response this design example shown in Fig. 4.15 indicates that the voltage gain comes out to be 11.2 and bandwidth 100 MHz due to L taken of lower value 0.5µm. Statement: 2. Repeat above design example with diode connected load and small signal voltage gain of 2.5, neglect body effect, Design Steps The voltage gain with diode connected load neglecting body effect is given as, (W/L) 2 = (104/.5)/(2.5) 2 = (20/0.5) µm/µm Figure 4.16. Schematic of CS amplifier with diode connected load 51

The circuit of diode connected CS amplifier simulated with EDA tool Tanner is shown in Fig. 4.16. The simulation result in Fig. 4.17 shows the closely matching voltage gain with theoretical value 2.2 and bandwidth of 450 MHz is measured. Figure 4.17. Frequency and phase response of CS amplifier with diode connected load Statement: 3. Design current source load CS amplifier to provide output voltage swings 2.2 V with bias current of 1mA and small signal voltage gain 50. Assume µ n C ox = 2µ p C ox = 60 µa/v 2 Design Steps Using circuit of Fig. 4.9 With supply voltage 3 V and output voltage swing 2.2 V, total overdrive available is 0.8 V. Thus, overdrive of 0.4 V is allotted to both NMOS and PMOS device. The dimensions can be calculated as, W/L of NMOS = 104/0.5 µm/µm and W/L of PMOS = 208/0.5 µm/µm Calculated g m of NMOS = 5mA/V and Gain = - g m1 (r o1 r o2 ) = -50. Thus required r o of both devices comes to be 20 kω giving us a λ = 0.05. 52

The circuit is simulated with EDA tool Tanner as shown in Fig. 4.18. The phase and frequency response simulation shown in Fig. 4.19 gives bandwidth 20 MHz much lower Figure 4.18. Schematic of CS amplifier with current source load Figure 4.19. Frequency and phase response of CS amplifier with current source load 53

than that of resistive and diode connected load. This is due to additional junction capacitance offered by current source. The advantage of this kind of load is the higher gain offered by the circuit. 4.3 Current Mirror Sources Current mirrors using MOSFETs have been widely used in analog integrated circuits both as biasing elements and as active loads for amplifier stages. The current mirrors as biasing circuits result in circuit performance insensitive to variations in power supply, temperature and presence of common mode noise [36]. In MOS circuits the bias current is small, thus the use of current mirrors become economical than resistors in terms of the die area required. For low power supply amplifiers, high voltage gain is obtained using current mirror as a load by offering high incremental resistance. Figure 4.20. Current mirror block diagram with reference to ground and power supply. A current mirror is an element with at least three terminals (IN, OUT, COMMON) as shown in Fig.4.20. The power supply is applied to a common terminal and the input current source to the input terminal. The output current comes out to be equal to the input current multiplied by current gain. If this current gain is unity, the input current reflects at output terminal and the circuit is named as current mirror. Ideally, the current mirror gain is independent of input frequency, and the output current is independent of potential between the output and common terminals. The voltage between input and common 54

terminal is ideally required to be zero, as this allows the entire supply voltage to drop across the input current source. Some times more than one input and or output terminals are used. Practical MOS current mirrors deviate from this ideal behavior. The deviations from ideal conditions are as follows. 1. The variation of the current mirror output current with changes in voltage at the output terminal is one of the most important deviations. The small signal output resistance, R o of the current mirror characterizes this effect. This resistance is included in Nortonequivalent model at the output. The R o directly affects the performance of circuits that use current mirrors. Higher value of R o is achieved when the output current decreases. This also decreases the maximum operating speed. 2. The gain error, which is deviation of gain of current mirror from its desired value, is another error source. The gain error comprises of the systematic gain error and the random gain error. The systematic gain error ε is the gain error caused even if all matched elements in the mirror are perfectly matched. The unintended mismatches between matched elements cause the random gain error. 3. The voltage available across the input current source is reduced due to a positive voltage drop (V IN ) created by current source connected to input. Low V IN simplifies the design of input current source. 4. The output current depends mainly on the input current if the output voltage V OUT is positive. This output voltage is limited to a minimum value (V OUTmin ) that allows the output device to operate in saturation. Minimizing V OUTmin maximizes the range of output voltages for which the current mirror output resistance is almost constant. The performances of various current mirrors are compared to each other for above four parameters: R o, ε, V IN and V OUTmin. 4.3.1 A Simple MOS Current Mirror Figure 4.15 shows a simple current mirror using MOSFETs. The drain of M 1 is shorted with gate; therefore, the channel does not exist at the drain. Thus, M 1 operates in the saturation if threshold voltage is positive [7]. The M 1 is said to diode connected. Let us assume that M 2 also operates in saturation region and both M 1 and M 2 have infinite output 55

resistance. The drain current of M 2 (I D2 ) is controlled by V GS2 that is equal to V GS1. The gate source voltage of MOSFET is usually separated into the threshold voltage V TH and the overdrive voltage V od. The overdrive for M 2 assuming square law behavior of MOSFET becomes, (4.55) The mobility falls with increasing temperature. Hence, the overdrive rises with rising temperature. Rearranging (4.55) and equating gate voltages of M 1 and M 2, Figure 4.21. Simple current mirror (4.56) Thus, the overdrive of M 2 is equal to that of M 1. (4.57) If MOSFETs are identical, (W/L) 2 = (W/L) 1 and therefore, (4.58) This shows that the current flowing in the drain of M 1 is mirrored to the current flowing in the drain of M 2. As for MOSFETs the forward current gain (β F ) tends to, applying KCL at the drain of M 1, (4.58) becomes, 56

(4.59) Hence, the gain of the current mirror is unity, for identical devices operating in saturation region with infinite output resistance. This result assumes that the gate currents are zero. The result in (4.59) holds good for dc and low frequency ac currents. The gate currents of M 1 and M 2 increase with increase in input frequency due to finite gate-source capacitance. The part of the input current that flows through the gate does not flow into the drain of M 1. This decreases the gain of current mirror as the frequency of input increases [10]. If the devices are not identical, then from (4.56) and (4.59), (4.60) The aspect ratio of MOSFETs changes the gain of the current mirror. To change the gain of current mirror, width (W) is changed keeping length (L) constant. The drain currents of MOSFETs are assumed independent of their drain-source voltages in (4.56). In saturation region, the drain current actually increases slowly with drain-source voltage. The output characteristic of M 2 is as shown in Fig.4.16. The output resistance of the current mirror at given operating point is the reciprocal of the slope of the output characteristic at that point. Here, (4.61) where, V A is the early voltage and λ is the channel-length modulation parameter. At the point shown on the characteristic curve V DS2 =V DS1 and V GS2 =V GS1. If the slope of characteristic curve in saturation region is constant, a straight line passing through the operating point predicts the variation in I D2 with V DS2. Therefore, (4.62) The ideal gain of current mirror is (W/L) 2 /(W/L) 1. Then from (4.62) the systematic gain error ε, is given as, (4.63) 57

Thus the current mirror currents can differ by more than ε percent from values calculated by assuming MOSFET output resistance to be infinite. The input voltage for MOS current mirror (V IN ) is given as, (4.64) According to square law behavior of MOSFET, the overdrive voltage is proportional to square root of the input current. The minimum output voltage required to keep M 2 in saturation is, (4.65) V OUTmin can be made arbitrarily small as it depends on device geometry. If MOSFETs operate in weak inversion, then V OUTmin becomes (4.66) Here V T is thermal voltage and at room temperature equals to 26 mv. 4.3.2 Cascode Current Mirror One of the desirable characteristics for a current mirror is a very high output resistance that cascode connection achieves. A MOSFET current mirror based on cascode connection is shown in Fig. 4.17. The MOSFETs M 1 and M 3 form a simple current mirror. M 2 acts as the common gate part of the cascode and transfers the drain current of M 1 to the output presenting a high output resistance. M 4 acts as a diode level shifter and assures that M 2 and M 1 always remain in saturation. The small signal output resistance of M 2 in common gate configuration, with r o1 as a source resistance is given in terms of small signal parameters as, (4.67) For MOS cascode as dc current gain β 0, the number of stacked cascode devices can be increased. This will be limited by the MOS substrate leakage current that creates a resistive shunt dominating output resistance for V OUT >V OUTmin. 58

Applying KVL in Fig. 4.17, (4.68) As V DS3 =V GS3, (4.68) shows that V DS1 =V DS3 when V GS2 =V GS4. With this condition, the systematic current gain of the cascode current mirror is zero due to identical bias on M 1 Figure 4.22. MOS cascode current mirror and M 2 and β F. Actually V GS2 is not exactly equal to V GS4 even with perfectly matched MOSFETs unless V OUT =V IN. Thus, V DS1 V DS3 and leads to, (4.69) The input voltage of this MOS cascode mirror is, (4.70) The input voltage here is composed of two gate-source drops, each including threshold and overdrive components. Neglecting body effect and assuming that MOSFETs have got equal overdrives, (4.71) 59

Addition of further cascode levels increases the input voltage by another threshold and another overdrive component per cascode. This fact increases the difficulty of designing the input current source with low power supply voltages. When both M 1 and M 2 operate in saturation region, V DS1 V DS3 =V GS3.To operate M 2 in saturation it is required that V DS2 >V OV2. Hence, minimum output voltage for which M 1 and M 2 operate in saturation region is, If all MOSFETs have identical overdrives, (4.72) If V OUT < V OUTmin, M 2 operates in the triode region and if V OUT < V od1, both M 1 and M 2 operate in the triode region. Even if the overdrive is made small by using larger W, the threshold term remains and represents significant loss of voltage swing. Thus the cascode current mirror gives better output resistance but at cost of increased minimum output voltage. 4.2.4 Wilson Current Mirror The Wilson current mirror is as shown in Fig. 4.18. Let us consider the circuit without M 4.This circuit uses negative feedback through M 1 and activates M 3 to raise the output resistance. A feedback path is formed that regulates I D3 so that it equals the input current. To find the output resistance of the Wilson current mirror when all MOSFETs operate in saturation, the small-signal model is used [43]. Applying a test mirror comes out to be, (4.73) 60

The calculation of R o ignores body effect. The body effect of M 2 has little effect on R o because M 1 is diode connected; the voltage from source of M 2 to ground is almost constant. The systematic gain error here is not zero even if β F, as the drain source voltage of M 3 differs from that of M 1 by the gate source voltage of M 2. Thus without M 4, Figure 4.23. MOS Wilson current mirror (4.74) When M 4 is inserted in series with M 3, it equalizes the drain source voltages of M 3 and M 1. Then, ε 0. The output resistance is still given by (4.73), if all MOSFETs operate in saturation. Insertion of M 4 does not change the minimum output voltage or input voltage. The minimum output voltage ignoring body effect and assuming equal overdrives is, (4.75) With this condition the input voltage is 61

I OUT (µa) (4.76) 4.3.5 Design Example MOS current mirror The proposed circuit of MOS current mirror is built selecting NMOS devices M 1 and M 2 of same dimension of aspect ratio 10/0.5 µm/µm as shown in Fig. 4.24 (a). The transconductance (g m ) and output resistance (r o ) of the MOSFETs used are found out using simulation results as 1.2mA/V and 45.43kΩ. The response of all the current sources discussed above depends on the current source I IN. The current source I IN used here is a PMOS device maintained in saturation for all operating conditions, with the help of bias V b, applied on the gate. The current supplied by I IN is maintained 365µA with the help of bias V b on the gate of PMOS device M 3. The PMOS device M 3 is selected with aspect ratio 5/0.5 µm/µm and gate bias V b required is 0.6639V. The supply voltage selected is 2.5V. The gate bias 0.6639V is obtained from supply voltage 2.5V using voltage reference circuit made up of one PMOS and one NMOS device of aspect ratio 3/0.5µm/µm and 18.85/0.5µm/µm respectively [60]. The similar reference voltage circuit is used in all proposed circuits. 450 400 350 300 250 200 150 100 50 0 0 1 2 3 V OUT (V) Figure 4.24. (a) Schematic of CMOS current mirror source and (b) Output characteristics The output characteristic of MOS current mirror source is as shown in Fig. 4.24 (b). The MOSFET M 2 is seen to be in triode region for V DS2 < 0.7V and enters saturation region thereafter. The output resistance R o calculated from output characteristic curve comes out 62

I OUT (µa) to be 47.34kΩ. The minimum output voltage for which M 2 remains in saturation is 0.7V approximately equal to overdrive of MOSFET. The load resistance R L is connected at the drain of M 2. The output voltage and output current readings are obtained by varying R L. Cascode current source The proposed cascode current mirror source is as shown in Fig. 4.25 (a). The aspect ratio dimensions of NMOS devices M 1 -M 4 is kept same to be 10/0.5µm/µm. The dimension of current source PMOS device M 5 is also kept same 5/0.5 µm/µm and to get the current of 365µA, now the required V b comes out to be 0.1996V. This required voltage reference of 0.1996V is derived from 2.5V V DD using a voltage reference circuit as shown in Fig. 4.25 (a). As the reference voltage is too lower, two PMOS and one NMOS device is required. The PMOS device dimensions come out to be W/L=0.05/2 µm/µm and NMOS device dimension comes out to be W/L = 61.12/0.5µm/µm. The PMOS devices with small W and large L offer higher resistance and NMOS device with large W and small L offer lower resistance required for generation of small V b from large value of V DD. 400 350 300 250 200 150 100 50 0 0 1 2 3 V OUT (V) Figure 4.25. (a) Schematic of CMOS cascode current mirror source and (b) Output characteristics The output resistance measured from the output characteristics curve in Fig. 4.25 (b) comes out to be 1.622MΩ. When output voltage starts to built up, initially M 1 and M 2 operate in triode region. As output voltage increases, M 1 enters into saturation first. With further increase in output voltage, both the devices operate in saturation region. The 63

I OUT (µa) minimum output voltage required to maintain both devices in saturation comes out to be 1.4V, which is approximately double of the overdrive required for single device. Wilson current mirror source If the three current sources are to be compared, they must be fed from identical current source of 365µA. The aspect ratio dimensions of NMOS devices M 1 -M 4 are selected to be same 10/0.5µm/µm and the aspect ratio of current source PMOS device M 5 also selected same 5/0.5 µm/µm. To get the current of 365µA from M 5 the required gate drive voltage V b is 0.2135V. This once again is a lower reference voltage derived from higher V DD. 400 350 300 250 200 150 100 50 0 0 1 2 3 V OUT (V) Figure 4.26. (a) Schematic of CMOS Wilson current mirror source and (b) Output characteristics The reference voltage circuit comes out to be similar as that of cascode current mirror source with dimension of PMOS devices W/L = 0.05/2 µm/µm and dimension of NMOS device W/L = 39.7/0.5 µm/µm as shown in Fig. 4.26 (a). The output I-V characteristics of Wilson current mirror source is as shown in Fig. 4.26 (b). As one of the NMOS device comes across output in diode configuration, the output does not built from zero. The output starts to build from 0.42V and M 2 reaches saturation at 1.94V approximately. The output resistance measured from I-V characteristics curve comes out to be 1.71MΩ. The parameters of three current mirror sources are summarized in Table 1. 64

Table 4.1: Summary of parameters measured from current source circuits Circuits/ Simple current Cascode current Wilson current Parameters mirror mirror mirror R o 47.34 kω 1.622 MΩ 1.71 MΩ ε 1.093 1.0045 1.0045 V IN 0.8644 1.8813 1.8734 V OUTmin 0.7V 1.4V 1.94V The table 4.1 shows comparison of three current mirror sources based on their parameters. From the table it is seen that highest output resistance is offered by Wilson current mirror source but its V OUTmin is too larger which reduces its operating range. The simple current mirror source offers comparatively low output resistance but operating range is wider than other two sources. Thus simple current mirror source is suited as an active load whereas other two would be suitable as tail current sources. To conclude, the basic building block of IA, a gain stage is treated in detail in this chapter with possible high gain stages; telescopic cascode and folded cascode. The behavior of MOSFET with its physics is accomplished in detail. The basic circuit of CS amplifier is designed and simulated with passive and active loads. The other building blocks required frequently in monolithic form of IA are the current mirror sources. The current mirror sources are analyzed and also the designing is carried in detail with simulation for comparison. ***** 65