Analog Integrated Circuit Design --- Problem Answers

Textbook by David Johns and Ken Martin, Wiley, 1997.

Solutions by Khoman Phang and Ali Sheikholeslami

		====================
		ANSWERS TO PROBLEMS 
		====================

Chapter 1:
==========
1) n:10e25 carriers/m^3, p:3.6e8 carriers/m^3
2) 0.87V, decreases
3) 140fC
4) t_fall=0.37ns, t_rise=0.48ns
5) t_fall=0.37ns, t_rise=0.44ns
7) 143uA
8) r_ds=167kohms, lambda=0.3
9) r_ds=182kohms, gm=230uA/V, gs=44uA/V
10) Cgs=86fF, Cgd=10fF, Cdb=60fF, Csb=74fF
11) 0.976V
12) 1.25ns, 3.33ns
13) 1.35ns, 6.1ns
14) gm=3.8mA/V, r_pi=26kohms, r_e=260ohms, r_o=800kohms, gmr_o=3000
15) t_fall=690ps, t_rise=7.8ns
16) t_fall=850ps, t_rise=10.3ns

Chapter 2:
==========
8) 0 lambda
12) Cdb=0.019pF, Csb=0.034pF
13) Cdb=0.016pF, Csb=0.44pF
14) y2=2.415x1, x2=0.631x1
15) -0.6%, +0.15%
16) 350 Ohm
17) 379.26 Ohm, height=24um, width=152um

Chapter 3:
==========
1) I_out=20uA, R_out=640kohms, Max. Vout=170mV
2) gain = -4800*sqrt(2*un*Cox*W*L/Ibias)
3) w_3dB = Ibias/(4800*L*C_l)
5) 2*PI*1.05MHz
6) 1/R_out = gm1 + gs + g_ds1 + g_ds2
7) 1/R_out = gm1 + gs
8) a) w_o=2*PI*35MHz, Q=0.332, w_z=2*PI*844MHz 
   b) w_o=2*PI*33MHz, Q=0.570, w_z=2*PI*844MHz 
9) a) C_1=0.125pF, R_1=4780ohms, p_1=2*PI*3.61MHz, p_2=2*PI*281MHz
   b) C_1=0.144pF, R_1=4780ohms, p_1=2*PI*3.61MHz, p_2=2*PI*244MHz
10) 2*PI*1.3MHz
11) R_out = 2*r_ds1
12) V_out >= 1.33V
13) R_out = r_ds4*[1+ r_ds2(gm4+gs4+g_ds4)]
16) 2*PI*57kHz
17) 2*PI*80MHz
22) R=22kohms, R_out=400kohms
23) gain=-V_a/[2*V_t]
25) gain=0.9993, R_out=52ohms
26) gain=beta*R_e*(R_l//R_c)/[(beta+1)*(re*Rs+Re*Rs+re*Re)], Rin=re//Re 
31) gain = 116, w_3dB=2*PI*10kHz

Chapter 4:
==========
1) dBm(50 Ohm)=dBm(75 Ohm)+1.76
2) a)-18.24dBm, b)-17.16dBm, c)-15.35dBm, d)-30.67dBm
3) -44.77dBm, 0.058mV/root_Hz
4) -31.6dBm
6) they are equal.
7) 34.2uV rms
8) twice the noise power or, equivalently, 3dB
10) b)0.32uV rms, c)infinity
11) 10.2uV rms
12) 125uV rms
13) 1uV rms
14) 0.18uV rms, 1/f rule: 0.17uV rms
16) a: 41.94dB, b: 57.7dB
17) 44.24uV rms, 67.08dB
18) 5.75 nV/root_Hz, 4.84 nV/root_Hz

Chapter 5:
==========
1) a)2*PI*1.1kHz, b)2*PI*14MHz, c)10V/us
2) 25V/us
3) Width of Q7 must be half that of Q6
4) 0.7mV
5) Maximum: Vout=3.9V, V_incm=3.8V
   Minimum: Vout=-4.9V, V_incm=-4.99V
7) Maximum: Vout=2.9V, V_incm=3.7V
   Minimum: Vout=-4.9V, V_incm=-6.8V
9) 2*PI*41MHz and 2*PI*51MHz
10) 0.4pF
11) 63fF
12) w1=1.8e4 rads/s, wt=1.43e8 rads/s
13) A(s)=Ao*(1+s*Tz)/[T1*T2*[s^2+s*(1/T2+Ao*beta*Tz/(T1*T2))+Ao*beta/(T1*T2)]]
    wo= sqrt(Ao*beta/(T1*T2)), Q= sqrt(wo*T1*T2/(T1+Ao*beta*Tz)
15) Rb=5.2kohms, Veff=0.31V

Chapter 6:
==========
1) 1/gm
2) W1=W2=W3=W4=43.5um, W5=5.76um
3) VDS3=0.39V, VDS4=0.57V
4) RB=5.43KOhm
5) 0.29V
6) 0.23V
8) upper limit r0(1+beta), independent of A.
9) 71MOhm vs. 2.25MOhm 
10) 30.2MHz, 12.5 V/us, 19.8V/us (with clamp xtors)
11) f2=306.7MHz, 2.71pF, 46.1V/us, 73V/us (with clamp xtors)
12) 0.69pF, 526 Ohm, 438MHz, 181V/us, 287V/us (with clap xtors)
13) wt=(K*Itotal)/((K+1)(Veff*CL)) 
14) Av0=K*(2*a^2*L^2)/(Veff1*Veff8)
15) wt(folded_cas.)/wt(current_mir.)=root((K+3)/(2K(K+1)))
16) 19.5ns
17) yes, 98%, 30.6ns, 24ns
18) +SR=8V/us, -SR=4V/us, for current_mir., +SR=-SR=16V/us
19) KIbias/CL
20) wt=(K/CL)*root((2/(2+K))*Itotal*un*Cox*(W/L)4)
21) Veff=0.78V, Vout+(max)=1.1V, Vout+(min)=-1.1V
22) Veff=0.63V, Vout+(max)=0.9V, Vout+(min)=-0.9V
23) (1+R2/R1)+R2(gm1+gm2)
25) Tmin=(1/Kgm)*2.5ps

Chapter 7:
==========
1) 99mV
2) -7.2uV
3) reset: 0.17ns, comparison: 0.77ns
5) V_ioff=(Verr3+Voff3/(1-A3))/(A1*A2)
6) 54MHz at 71dB
7) L=0.8um, W1=W2=25um, W3=W4=100um, W5=W6=30um, W7=W8=5um, W9=W10=2.4um
   Wtrk=25um, Wcmfb=12um
8) tau=1ns, settling time is 3ns
9) 3.65ns
10) 1.7V
11) 2.6V

Chapter 8:
==========
2) 0.63ns
3) 13mV, -13mV
4) 0.03ns
6) Vo=(A/(A+1))Vin+(A/(A+1)^2)Vos
7) Vo[nT]=(1/(C1+C2))*{C2Vo[nT-T]+C1Vin[nT-T]}
9) 57.2mV, 0.178mV/K
10) 10.35
12) R1=5.5kOhm, R2=1kOhm, R3=900Ohm, R4=10kOhm
14) 50uA < I1 < 400uA, io1(peak) < 30uA
15) I1: delta_io=-0.05io, I2: delta_io=0
16) delta_io=(0.05/(2*Beta))*i2

Chapter 9:
==========
2) 6
6) poles: 0.8+i0.1, 0.8-i0.1, all zeros at infinity
7) 0.544+i0
8) DC gain=0dB, gain at w=PI is -31.8dB, w_3dB=0.051rads/sample
9) y(n)= 0 when n=0, y(n)=(-0.3)^(n-1) for n>0
10) H(z)=0.095*(z+1)/(z-0.8098)
11) H(z)=9.45e-4*(z+1)^2/(z^2-1.91*z+0.915)
12) output sequence: 1 1 0 0 1 1 0 0 1 1...
    - 3dB attenuation with a phase shift of 45 degrees.
13) 100Hz:1V, 300Hz:0.33V, 500Hz:0.2V, 700Hz:0.14V
14) 100Hz:1V, 9.9kHz:10.1mV, 10.1kHz:9.9mV
15) 100Hz:998mV, 99.9kHz:0.999mV, 100.1kHz:0.997mV
16) -20dB is provided, an additional 60dB is required

Chapter 10:
===========
2) Vo(z)/Vi(z)=(-C1/C2)/(1-1/z)
3) Vo(z)/Vi(z)=(C1/C2+Cp1/(C2z))/(1-1/z)
5) C1=0, C2=-8.75pF, C3=8.75pF, the new gain=-0.304
6) C1=0, C2=-6.683pF, C3=6.683pF, -24.1dB
7) H(z)=0.1z/(1.1z-1)
8) H(z)=-(C1-(C1+C2)/z)/((CA+C3)-CA/z)
10) low_Q: 318, high_Q: 20

Chapter 11:
===========
1) 1.023V
2) SNR=64.5dB, Vin=0.6mV(peak to peak) 
3) Vout = Vref[-b1*2^-1 + b2*2^-2 + b3*2^-3 + ... + bn*2^-N
5) Extend word by copying MSB to new bit locations.
7) offset=-0.01LSB, gain_err=0.09LSB, max. INL_err=-0.091LSB, 
   max. DNL_err=-0.073LSB
8) max abs error = 0.08LSB (6.6bits accuracy)
   max rel error = -0.091LSB (6.5bits accuracy)
9) 200uV/C
10) offset=0.01LSB, gain_err=0.01LSB, max abs err=-0.03LSB
    max rel err=-0.047LSB(6.4bits accuracy)
11) 1.22mV
12) less than 0.24ns

Chapter 12:
===========
1) 2(2^N-1)
2) tau=1.7ns, t99.9%=12ns
3) 2^N+2^(N/2)
4) output opamps: 2.5mV, middle opamps: 80mV
5) 512
6) b2: 2 times, b3: 4 times, b4: 8 times
7) 0.75LSB
8) resistance_ratio improvement=2^(N/2)
9) 8, 2
11) 0.08LSB, 0.005LSB
12) w(3dB)=2*pi*17.9MHz
13) 0.0375V, 2.0625V, 3.8375V
14) 31mV
17) 0.027, 5nA
18) 0.27, 50nA

Chapter 13:
===========
1) 105ms
2) 2620Mohms
3) Bout= (vin+err)/(Vref-Voff1) where err=R1*C1*Voff2/T1
4) 0.002LSB offset
5) multiples of 15.3Hz are completely attenuated. Attenuation at 60Hz is 34dB.
6) multiples of 1.53Hz are completely attenuated. Attenuation at 60Hz is 46dB.
7) Bout=0110 binary
8) Bout=0110 binary
9) Bout=0110 binary. Parasitics attenuate gain by 1/3.
11) Invert the MSB
12) Vx1=-1.35e-2*Vref
13) Vx2=-2.7e-3*Vref
14) 3.44usec
17) a differential input of 67mV is required. The resolution is 5bits.
18) 0.3125V < Vin < 0.4375V
20) 2^N
21) 2^(N-F)
22) A reduction factor of 8, and 8 resistors needed between input comparators   

Chapter 14:
===========
1) 33.345GHz!
2) fs/10
5) 512MHz
6) 18.88MHz, 10.98MHz
7) 760kHz, 100.8kHz
8) 1/7 = -8.5dB
9) G(z)-1=(1/z)(1/z-2)
12) time-invariant versus a time-varying system

Chapter 15:
===========
2) Gm1=4.71mA/V, Gm2=0.47mA/V
3) C_A=C_B=2pF, Gm1=0.13mA/V, Gm2=0.13mA/V, 
   Gm3=0.13mA/V, Gm4=0.63mA/V, Gm5=0mA/V
4) Cx = k2*Cb, Gm1=k1*Cb, Gm2=k0*Ca/k1, Gm3=k1*Cb*wo^2/k0, wo*Q=k0/k1
5) Gm = beta/[(beta+1)*(Re+(2*re+Re)/A)]
6) 2.1V, 2.8V
7) 4/Re
8) I1=625uA, I2=156uA
9) -5.3% error
10) a)Gm=0.46mA/V, b)io1=46uA and 230uA, 
    c)io1=45uA(2% error) and 160uA(-30% error)
11) L=1um, W1=W2=67um, W3=10um
12) Gm=36uA/V
14) 36mV
16) W1=0.39*L, W2=0, W3=0.49*L, W4=0.2*L, W5=0.31*L
17) 0.12%
18) -14dBm
19) OIP3 = 22dBm, IIP3=16dBm, Id1=-3dBm, 
    SFDR=55dB, Id1=-5dBm if No=-60dBm

Chapter 16:
===========
1) C1=100nF, R2=1kOhm, R1=2.5MOhm, delta_phi=1.19rad, 
   2.5MHz < f(osc) < 17.5 MHz, C2=10nF
2) A=2/3, C1=10nF, R2=847Ohm, R1=2333Ohm
   the lock range will increase if not limited by the vco!
3) w0=0.93w0(original), Q=1.07Q(original)
5) C1=62.5pF, C2=6.25pF, R=20kOhm
7) Cosc=0.5pF, I1=14uA, f(fr at Tnom+20)=10.6MHz, 
8) 0.1Tosc(5V)
9) f(osc)=Ib/(n*Vref*CL*ln(2))  n is the # of stages.
10) f(osc)=Vcntl/(n*Vref*R*CL*ln(2)), f(osc)=36MHz, Kosc=2pi*36MHz/s.V
11) f(osc)=RC1C2(VDD-Vref)/(Vcntl*(C1+C2))