Practical Voltage and Current Sources, equivalent circuit diagram

Voltage source

An ideal voltage source provides constant voltage at its terminal to the circuit regardless of the load current.

Practical voltage sources are real and used in daily life. A practical voltage source has internal resistance that causes a drop in terminal voltage due to the current flow.

Ideal Voltage Source and Practical Voltage Source - equivalent circuit diagram

(Index S stands for source, L for load)

Note: The equivalent circuit diagram is a representation that can be used to calculate the true terminal voltage of a voltage source when current is drawn from it. Note that internal resistance is an inherent property of a source, it is not a discrete component that can be measured with an ohmmeter.

Regarding the Practical Voltage Source:

• If no load, the voltage at the terminals is the same as the internal voltage V₀.
• With a load you get:  V_out = V_S – R_S * I
• In case of short circuit applies:  I_max = V_S / R_S

Current Sources

Current sources are active network elements that should provide the same current to any load connected across it.

Ideal current sources are providing the exact same current to any resistance connected to it. Where practical current sources may vary current resistance.

Ideal Current Source and Practical Current Source and their equivalent circuit diagram

Regarding the Practical Current Source:

• With no load you get the voltage at the terminals:  V = R_s * I
• In case of short circuit applies: I_max = I_q

Let´s go into detail

Characteristics of a Real Voltage Source

Due to the internal resistance R_s, the voltage at the terminals decreases with increasing current:

Real voltage source - dependence of terminal voltage on load

The maximum terminal voltage can be obtained when no current is flowing. As the current increases, the terminal voltage decreases because a voltage drops across the internal resistor R_s:

V_out = V_s – V_Rs

V_out = V_s – R_s* I     … Eq. (1)

The short-circuit current is calculated as:     I_max = V₀ / R_s

In the short-circuit case, i.e. when the terminals are short-circuited, the terminal voltage is of course 0 V. However, this means that the total source voltage V_s drops across the internal resistor.

Determine the internal resistance:

It is not useful or often not possible to determine the short-circuit current. It is better to determine one point of the load characteristic curve, i.e. to connect a defined resistor to the terminals.

The second point can then be the case, when no load is connected to the terminals, that means I = 0A.

If a current measurement is to be dispensed with, this equation can be transformed.

With I = V_out / R_L you get:

Exercise - Practical voltage source

A car battery has a terminal voltage of 12 V. The maximum short-circuit current (at 20 °C) is 1 kA. To start the engine, the battery has to supply a current up to approx. 100 A.

Calculate the maximum internal resistance of the starter, the internal resistance of the battery and the minimum terminal voltage that is reached when starting.

The minimum terminal voltage required for a start is 9.8 V. At what minimum temperature is a start still possible if the battery has a temperature coefficient of α = - 2.78 × 10 -2 1/K? To simplify matters, the temperature coefficient of the starter can be neglected.

Exercise Practical Voltage Source

internal resistance Rs by regarding the short-circuit case:

R_s = V_s  =   12 V      =  0,012 Ω
.       I_max    1000 A

Terminal voltage during the starting, d.h. I = 100 A:

V_Kl = V_s – R_s x I  =  12 V – 0,012 Ω x 100 A  = 10,8 V

resistance of the starter:

R_Starter =  V_out  =  10,8 V   =  0,108 Ω
.                  I         100 A

---

Minimum possible temperature:

Minimum current for successful start:

I_min = V_{out min}     =  9,8 V       = 90,74 A
.          R_Starter        0,108 Ω

so that R_{s max} = Vs - V_{out min} = 12 V - 9,8 V = 0,0242 Ω
.                                I_min               90,74 A

The linear temperature dependence of a resistor is determined by:

R(T) = R₂₀ + R₂₀ × α × ΔT 

ΔT  = R(T)  - R₂₀    =  0,0242 Ω - 0,012 Ω                 
.            R20 × α         0,012 Ω  × - 2,78 × 10 ^-2 1/K

ΔT  =  - 36,57 °C