Question

2.34 the circuit in fig. p2.34 utilizes an ideal op amp.
(a) find ( i_1, i_2, i_3, i_l ), and ( v_x ).
(b) if ( v_o ) is not to be lower than ( -8 , \text{v} ), find the maximum allowed value for ( r_l ).
(c) if ( r_l ) is varied in the range ( 100 , omega ) to ( 500 , \text{k}omega ), what is the corresponding change in ( i_l ) and in ( v_o )?

Explanation:

Step1: Calculate $I_1$ (ideal op-amp virtual ground)

The inverting terminal is at 0V (virtual ground). $I_1 = \frac{1\ \text{V} - 0\ \text{V}}{10\ \text{k}\Omega}$
$I_1 = \frac{1}{10 \times 10^3} = 100\ \mu\text{A}$

Step2: Find $V_x$ (Ohm's law for 100Ω)

$V_x = I_1 \times 100\ \Omega$
$V_x = 100 \times 10^{-6} \times 100 = 0.01\ \text{V} = 10\ \text{mV}$

Step3: Calculate $I_3$ (Ohm's law for 100Ω)

$I_3 = \frac{V_x - 0\ \text{V}}{100\ \Omega}$
$I_3 = \frac{0.01}{100} = 100\ \mu\text{A}$

Step4: Calculate $I_2$ (Kirchhoff's current law)

At node $V_x$, $I_2 = I_1 + I_3$
$I_2 = 100\ \mu\text{A} + 100\ \mu\text{A} = 200\ \mu\text{A}$

Step5: Calculate $I_L$ (Kirchhoff's current law)

$I_L = I_2$ (ideal op-amp draws no input current)
$I_L = 200\ \mu\text{A}$

Step6: Part (b): Relate $V_O$ to $R_L$

$V_O = V_x - I_L R_L$
Rearrange to solve for $R_L$ when $V_O = -8\ \text{V}$:
$R_L = \frac{V_x - V_O}{I_L}$
$R_L = \frac{0.01 - (-8)}{200 \times 10^{-6}} = \frac{8.01}{200 \times 10^{-6}} = 40.05\ \text{k}\Omega$

Step7: Part (c): $I_L$ change analysis

$I_L$ is determined by $I_2$, which depends only on fixed resistors and input voltage. So $\Delta I_L = 0$

Step8: Part (c): $V_O$ range calculation

For $R_L = 100\ \Omega$:
$V_{O1} = 0.01 - (200 \times 10^{-6})(100) = 0.01 - 0.02 = -0.01\ \text{V}$
For $R_L = 500\ \text{k}\Omega$:
$V_{O2} = 0.01 - (200 \times 10^{-6})(500 \times 10^3) = 0.01 - 100 = -99.99\ \text{V}$
Change in $V_O$: $\Delta V_O = V_{O2} - V_{O1} = -99.99 - (-0.01) = -99.98\ \text{V}$

Answer:

Part (a)

$I_1 = 100\ \mu\text{A}$, $I_2 = 200\ \mu\text{A}$, $I_3 = 100\ \mu\text{A}$, $I_L = 200\ \mu\text{A}$, $V_x = 10\ \text{mV}$

Part (b)

Maximum $R_L = 40.05\ \text{k}\Omega$

Part (c)

Change in $I_L = 0\ \mu\text{A}$, Change in $V_O = -99.98\ \text{V}$ (ranges from -0.01 V to -99.99 V)