E84 Homework 2

The calculations of all problems in this problem set are straight forward. However, the concepts involving source, load, internal (output) impedance of the source and the input impedance of the load are very important. After finding the numerical solutions of the problem, pause and reflect what they mean.

1. (a) In reality, an ammeter can be modeled by an ideal meter with zero impedance in series with an internal impedance $r_a$. To minimize the influnce of the ammeter on the circuit during the measurement, should $r_a$ be minimized or maximized?

   (b) A voltmeter can be modeled by an ideal meter with infinite impedance in parallel with an internal (or input) impedance $r_v$. To minimize the influnce of the voltmeter on the circuit during the measurement, should $r_v$ be minimized or maximized?

   (c) Calculate current $I$ and voltage $V$ in the simple circuit below, assuming $V_0=10V$, $R=500\Omega$ and the measuring meters are not connected, i.e., the ammeter is short circuit, the voltmeter is open circuit.

   (d) What is the reading on the ammeter with $r_a=10\Omega$ inserted in the circuit as shown? Assume the voltmeter is not connected.

   (e) What is the reading on the voltmeter with $r_v=10^4 \Omega = 10 K\Omega$, connected to the circuit as shown? Assume the ammeter is not connected.

   

   Solution:

   (a) $r_a$ should be minimized, ideally, zero.

   (b) $r_v$ should be maximized, ideally, infinity.

   (c)
   

   $$I=10V/(500+500)\Omega=10 mA,\;\;\;\;V=10V\frac{500}{500+500}=5 V$$
   
   

   (d)
   

   $$I=10V/(500+500+10)\Omega=9.9 mA$$
   
   

   (e)
   

   $$V=10V\frac{500\vert\vert 10000}{500\vert\vert 10000+500}=4.878 V$$
   
   

2. Ideally, a voltage source, such as a battery, should output constant voltage, independent of how much current will be drawn by its load, e.g., a resistor. However, in reality, all voltage sources have some internal (output) impedance, modeled by an ideal voltage source $V_0$ in series with an impedance $R_0$. Assume $V_0=10V$ and $R_0=10\Omega$ in the following circuit, find the voltage $V_L$ the load resistor $R_L$ gets when (a) $R_L=100 \Omega$ and then (b) $R_L=10K\Omage$. (c) In general, if you want the load to get maximal voltage from the source, do you want $R_L$ and $R_0$ to be minimized or maximized, individually?

   

   Solution:

   (a) When $R_L=100 \Omega$,
   

   $$V_L=V_0 \frac{R_L}{R_L+r_0}=10 \frac{100}{100+10}=9 V$$
   
   

   (b) $R_L=10K\Omage$.
   

   $$V_L=V_0 \frac{R_L}{R_L+r_0}=10 \frac{10000}{10000+10}=9.99 V$$
   
   

   (c) To maximize $V_L$, we want maximize $R_L$ and minimize $R_0$.

3. The circuit below with $R=10 \Omega$ can be used to measure the internal impedance $R_0$ of a battery.
   • When the battery is fresh, it is found $V_{out}=1.6V$ when the switch is open and $V_{out}=1.4V$ when the switch is closed. Find the internal impedance $R_0$.
   • The same battery is tested one year later, and $V_{out}=1.2V$ when the switch is open and $V_{out}=0.3V$ when the switch is closed. Find the internal impedance $R_0$.
   (Hint: assume the input impedance of the voltmeter is infinity, i.e., the voltmeter does not draw any current.)

   

   Solution:
   

   $$V_{out}=1.6\times \frac{10}{10+R_0}=1.5, \longrightarrow R_0=1.43 \Omega$$
   
   
   
   
   $$V_{out}=1.2\times \frac{10}{10+R_0}=0.3, \longrightarrow R_0=30 \Omega$$
   
   

4. The circuit below can be used to find the input impedance $r_v=R_{in}$ of a voltmeter (or an oscilloscope that also measures voltages). Assume $V_0=9V$ and $R_1=R_2=10 M\Omega$, and the reading of the voltmeter is $V_L=3V$. What is the input impedance of the voltmeter?

   

   Solution:

   
   

   $$V_L=V_0\frac{R_2\vert\vert R_{in}}{R_2\vert\vert R_{in}+R_1} =9 \frac{R_2 \vert\vert R_{in}}{R_2 \vert\vert R_{in}+R_1}=3$$
   
   
   Solving this equation for $R_{in}$ we get $R_{in}=10 M\Omega$.