user69337: We have an Arduino Nano connected to 4 small (0.5") common-anode 7-segment displays and 2 large (3") common-anode 7-segment displays. We are trying to multiplex the displays. The larger one requires more power than the smaller one (9V) which the shift register cannot supply.
Therefore, we have supplied 9V to the open drain outputs of the shift register that are connected to the larger displays with a pull-up resistor (390ohm). However, more power (7.5V) goes to the resistor than the display (1.5V).
They are connected like in the ShiftDisplay library's schematic, with a pull-up resistor added according to this link on open drain.
How do we use the open drain outputs logic to supply power to the display?
The data sheet of the big 7-segment display can be found here. Each display requires 160mA of continuous current.

user4143: You are getting confused. The pullup resistor is only needed on open drain when you are using it to generate logic levels. It turns the OD output into an RTL output.
You don't want that.
Instead you want your open drain outputs to act like a low-side switch transistor - which is infact what they are.
You can do that by simply placing your LEDs in series with your "pullup" resistor, thus:

simulate this circuit – Schematic created using CircuitLab
I have only shown two elements there (everything inside the dashed box is inside the shift register), but you can see now how when an output is "on" the transistor conducts, and the current flows through the LED, through the resistor, through the transistor, and down to ground.
Your resistor value is then calculated as simply
    9 - Vfled
R = ---------
      Iled

For 20mA and a 2V forward voltage that comes out as 350Ω - so your 390Ω resistors are perfectly fine.

69337: <p>We have an Arduino Nano connected to 4 small (0.5&quot;) common-anode 7-segment displays and 2 large (3&quot;) common-anode 7-segment displays. We are trying to multiplex the displays. The larger one requires more power than the smaller one (9V) which the shift register cannot supply.</p>
<p>Therefore, we have supplied 9V to the open drain outputs of the shift register that are connected to the larger displays with a pull-up resistor (390ohm). However, more power (7.5V) goes to the resistor than the display (1.5V).</p>
<p>They are connected like in the <a href="https://github.com/MiguelPynto/ShiftDisplay" rel="nofollow noreferrer">ShiftDisplay library's schematic</a>, with a pull-up resistor added according to <a href="https://forum.digikey.com/t/what-does-open-drain-mean/716" rel="nofollow noreferrer">this link on open drain</a>.</p>
<p>How do we use the open drain outputs logic to supply power to the display?</p>
<p>The data sheet of the big 7-segment display can be <a href="https://cdn.reignofcomputer.com/images/1637378503422487352_5cab8653-2a2f-4dfb-8d5a-1ee1585e736e.png" rel="nofollow noreferrer">found here</a>. Each display requires 160mA of continuous current.</p>
 4143: <p>You are getting confused. The pullup resistor is only needed on open drain when you are using it to generate logic levels. It turns the OD output into an RTL output.</p>
<p>You don't want that.</p>
<p>Instead you want your open drain outputs to act like a low-side switch transistor - which is infact what they are.</p>
<p>You can do that by simply placing your LEDs in <em>series</em> with your &quot;pullup&quot; resistor, thus:</p>

<p><img src="https://i.stack.imgur.com/gVu8N.png" alt="schematic" /></p>

<p><sup><a href="/plugins/schematics?image=http%3a%2f%2fi.stack.imgur.com%2fgVu8N.png">simulate this circuit</a> &ndash; Schematic created using <a href="https://www.circuitlab.com/" rel="nofollow">CircuitLab</a></sup></p>
<p>I have only shown two elements there (everything inside the dashed box is inside the shift register), but you can see now how when an output is &quot;on&quot; the transistor conducts, and the current flows through the LED, through the resistor, through the transistor, and down to ground.</p>
<p>Your resistor value is then calculated as simply</p>
<pre><code>    9 - Vfled
R = ---------
      Iled
</code></pre>
<p>For 20mA and a 2V forward voltage that comes out as 350Ω - so your 390Ω resistors are perfectly fine.</p>
