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To determine the minimum and maximum Zener currents for the given Zener diode circuit, let's walk through the steps in a detailed, step-by-step manner.
### Given:
- Zener voltage ([tex]\( V_Z \)[/tex]) = [tex]\( 10 \)[/tex] volts
- Series resistance ([tex]\( R \)[/tex]) = [tex]\( 820 \)[/tex] ohms
- Source voltage varies between:
- Minimum ([tex]\( V_{S_{min}} \)[/tex]) = [tex]\( 20 \)[/tex] volts
- Maximum ([tex]\( V_{S_{max}} \)[/tex]) = [tex]\( 40 \)[/tex] volts
### Zener Diode Operation
A Zener diode in breakdown maintains a constant voltage across it (the Zener voltage [tex]\( V_Z \)[/tex]) regardless of changes in the current flowing through it, provided it remains in the breakdown region.
### Step-by-Step Solution:
1. Calculate the circuit current:
For a series circuit containing the Zener diode and resistor, the current [tex]\( I \)[/tex] can be found using Ohm's law:
[tex]\[ I = \frac{V_{S} - V_{Z}}{R} \][/tex]
where [tex]\( V_{S} \)[/tex] is the source voltage, [tex]\( V_{Z} \)[/tex] is the Zener breakdown voltage, and [tex]\( R \)[/tex] is the series resistance.
2. Minimum Zener Current ([tex]\( I_{Z_{min}} \)[/tex]) calculation:
- When the source voltage is at its minimum ([tex]\( V_{S_{min}} = 20 \)[/tex] volts):
[tex]\[ I_{Z_{min}} = \frac{V_{S_{min}} - V_{Z}}{R} \][/tex]
Plugging in the values:
[tex]\[ I_{Z_{min}} = \frac{20 \text{ V} - 10 \text{ V}}{820 \text{ Ω}} = \frac{10 \text{ V}}{820 \text{ Ω}} \][/tex]
[tex]\[ I_{Z_{min}} \approx 0.0122 \text{ A} \][/tex]
This corresponds to 12.20 mA.
3. Maximum Zener Current ([tex]\( I_{Z_{max}} \)[/tex]) calculation:
- When the source voltage is at its maximum ([tex]\( V_{S_{max}} = 40 \)[/tex] volts):
[tex]\[ I_{Z_{max}} = \frac{V_{S_{max}} - V_{Z}}{R} \][/tex]
Plugging in the values:
[tex]\[ I_{Z_{max}} = \frac{40 \text{ V} - 10 \text{ V}}{820 \text{ Ω}} = \frac{30 \text{ V}}{820 \text{ Ω}} \][/tex]
[tex]\[ I_{Z_{max}} \approx 0.0366 \text{ A} \][/tex]
This corresponds to 36.60 mA.
### Summary:
- Minimum Zener current: [tex]\( I_{Z_{min}} \approx 0.0122 \text{ A} \)[/tex] (12.20 mA)
- Maximum Zener current: [tex]\( I_{Z_{max}} \approx 0.0366 \text{ A} \)[/tex] (36.60 mA)
Thus, the minimum Zener current is approximately 12.20 mA and the maximum Zener current is approximately 36.60 mA.
### Given:
- Zener voltage ([tex]\( V_Z \)[/tex]) = [tex]\( 10 \)[/tex] volts
- Series resistance ([tex]\( R \)[/tex]) = [tex]\( 820 \)[/tex] ohms
- Source voltage varies between:
- Minimum ([tex]\( V_{S_{min}} \)[/tex]) = [tex]\( 20 \)[/tex] volts
- Maximum ([tex]\( V_{S_{max}} \)[/tex]) = [tex]\( 40 \)[/tex] volts
### Zener Diode Operation
A Zener diode in breakdown maintains a constant voltage across it (the Zener voltage [tex]\( V_Z \)[/tex]) regardless of changes in the current flowing through it, provided it remains in the breakdown region.
### Step-by-Step Solution:
1. Calculate the circuit current:
For a series circuit containing the Zener diode and resistor, the current [tex]\( I \)[/tex] can be found using Ohm's law:
[tex]\[ I = \frac{V_{S} - V_{Z}}{R} \][/tex]
where [tex]\( V_{S} \)[/tex] is the source voltage, [tex]\( V_{Z} \)[/tex] is the Zener breakdown voltage, and [tex]\( R \)[/tex] is the series resistance.
2. Minimum Zener Current ([tex]\( I_{Z_{min}} \)[/tex]) calculation:
- When the source voltage is at its minimum ([tex]\( V_{S_{min}} = 20 \)[/tex] volts):
[tex]\[ I_{Z_{min}} = \frac{V_{S_{min}} - V_{Z}}{R} \][/tex]
Plugging in the values:
[tex]\[ I_{Z_{min}} = \frac{20 \text{ V} - 10 \text{ V}}{820 \text{ Ω}} = \frac{10 \text{ V}}{820 \text{ Ω}} \][/tex]
[tex]\[ I_{Z_{min}} \approx 0.0122 \text{ A} \][/tex]
This corresponds to 12.20 mA.
3. Maximum Zener Current ([tex]\( I_{Z_{max}} \)[/tex]) calculation:
- When the source voltage is at its maximum ([tex]\( V_{S_{max}} = 40 \)[/tex] volts):
[tex]\[ I_{Z_{max}} = \frac{V_{S_{max}} - V_{Z}}{R} \][/tex]
Plugging in the values:
[tex]\[ I_{Z_{max}} = \frac{40 \text{ V} - 10 \text{ V}}{820 \text{ Ω}} = \frac{30 \text{ V}}{820 \text{ Ω}} \][/tex]
[tex]\[ I_{Z_{max}} \approx 0.0366 \text{ A} \][/tex]
This corresponds to 36.60 mA.
### Summary:
- Minimum Zener current: [tex]\( I_{Z_{min}} \approx 0.0122 \text{ A} \)[/tex] (12.20 mA)
- Maximum Zener current: [tex]\( I_{Z_{max}} \approx 0.0366 \text{ A} \)[/tex] (36.60 mA)
Thus, the minimum Zener current is approximately 12.20 mA and the maximum Zener current is approximately 36.60 mA.
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