OK, I've gone for a circuit that's permanently powered from 110VAC, doesn't use a transformer (uses a "capacitor fed" power supply), and gets its timing from the mains.
It uses two ICs, but it's pretty accurate and doesn't need any adjustment. I hope you don't think it's too complicated to build.
A constant 110VAC comes in on wires W1 and W2.
D1~4 are a bridge rectifier (not drawn the usual way, but that's what it is) that is fed from C1 in a circuit called a "capacitor-fed power supply". C1 has a characteristic called capacitive reactance, which allows a limited amount of current to flow from the mains input into the bridge rectifier; most of the voltage is lost in C1, and no actual power is dissipated in C1, due to the magic of reactance.
R1 limits the surge current when power is applied, limits the current due to noise spikes on the mains supply, and acts as a fast-acting fuse if C1 fails short.
R2 discharges C1 if mains power is removed, so the circuit does not store a charge that could bite you if you touched the wires.
The incoming 110VAC also feeds further along into other parts of the circuit.
The bridge rectifier output is clamped at about 50V by D5 and smoothed by C2. When power is initially applied, because of the limited current available, C2 does not reach 50V immediately. It takes about a second to get within a few volts of the target voltage. This is normal.
Because the circuit is powered directly from the mains via a bridge rectifier, the 0V rail should be considered potentially live at all times. The whole circuit (everything shown on the diagram) must be properly insulated - for example, enclosed in a fully plastic box, with only the insulated wires emerging, and with markings on the inside of the box lid, and on the circuit board, saying "All circuitry is LIVE" or similar.
Q1 is a Darlington transistor - a combination of two transistors that has a high current gain. It is used as an emitter follower (aka common collector amplifier) to buffer the 12V reference voltage set by D6 and provide a (crudely) regulated 11V supply to the rest of the circuit.
The 220VAC supply from the other motor is fed into wires W5 and W6. It goes through a circuit similar to the capacitor-fed power supply, which provides a low-current AC signal into U1, a 4N33 optocoupler. The optocoupler provides electrical isolation and propagates the mains-frequency signal through to its output on pin 4. This signal is pulled down to 0V by R8 when there is no AC on W5/W6, but when that AC is present, pin 4 is high for every positive half-cycle of the AC waveform.
This signal is filtered by R9 and C6 to filter out any disturbances due to noise or interference on the 220VAC feed, and drives the RESET input of U2.
U2 is a CD4040BE CMOS "12-bit ripple carry binary counter". This means it counts transitions at its CLOCK input (pin 10) and reports the count on twelve binary outputs, Q1~Q12. When RESET is high, the count is held reset, and all the Q outputs are low (0V). When RESET is low, each falling edge (transition from high to low) on the CLOCK input causes the count to advance by 1.
The CLOCK input is driven with a signal derived from the 110VAC mains frequency via R11 and one gate in U3 which I will explain later. Its frequency is equal to the 110VAC mains frequency, which in this case is 60 Hz.
So 60 times per second, the count in U2 increments by 1. After one second, the count will be 60, which is 111100 binary, where those six digits correspond to Q6~Q1 in that order.
But if AC is present on W5/W6, U1 will reset U2 on every mains cycle, so the count in U2 will never advance very far. So although the count might advance as far as 1 or 2, all of the higher Q outputs will remain low while AC is present on W5/W6. Once the other motor is turned OFF and the AC disappears, U2's RESET input will remain low, and U2 will start to count upwards.
The Q9, Q10 and Q11 outputs of U2 are combined using one gate in U3 and Q2. This circuit detects when Q9, Q10 and Q11 are all high simultaneously, and when this occurs, brings the signal marked "RUN" low. (RUN is normally high.) This corresponds to a count of 11100000000 in U2; in decimal, that number is 1792. Dividing 1792 by 60 (the number of CLOCK transitions per second) gives 29.8667. So this count will be reached, and RUN will go low, about 30 seconds after the other motor is switched OFF.
U3 is a "quad 2-input NAND gate with Schmitt trigger inputs". It contains four independent gates. Each gate has two inputs and one output, and performs a logical NAND function - it drives its output to the logical opposite of the AND of its inputs, which is true only if input A AND input B are high. So its output will be high unless both inputs are high, in which case its output will be low.
The gate connected to pins 1~3 creates a logical NAND of the Q11 and Q10 outputs from U2. Normally, its output (U3 pin 3) is high; it only goes low when Q11 and Q10 are both high. Also, transistor Q2 is driven from the Q9 output, and it pulls its collector high unless Q9 is also high. R11 combines the two signals, producing a combined signal called RUN that goes low when U2's count reaches 11100000000, i.e. after 30 seconds with the other motor switched off.
When RUN goes low, the clock input to U2 is disabled via the second NAND gate in U3, because when the gate input on U3 pin 5 goes low, the output on pin 4 is forced high regardless of the signal on pin 6 (the mains frequency feed that provides the clock source for U2).
This stops U2 from counting, so it remains frozen at that count, and can only be un-frozen by an active signal at RESET, which occurs the next time the other motor is turned ON.
The signal at RUN is also fed through the remaining gates in U3, which are cascaded so they behave like a buffer. The final output on U3 pin 10 drives Q3 via R12. While RUN is high, Q3 is turned ON and keeps relay K1 energised. When RUN goes low, Q3 turns OFF and K1's contacts open, removing power to the 110V motor that is powered from wires W3 and W4. D8 protects Q3 from the back EMF from K1's coil when current through it is interrupted by Q3 turning OFF. Without D8, when Q3 turned OFF, K1's coil would generate a large "flyback" voltage that would pull Q3's collector voltage well above its rated maximum collector-emitter voltage and damage it.
When the circuit is powered up, U2's output states are undetermined. If there is no AC present at K5/K6, the circuit may initially start up with the motor enabled, and will switch it off after an undermined amount of time (less than 30 seconds). Once the other motor has been run, U2 is forced into a known state by its RESET input, and the circuit will then give the specified run-on period.
R13 and C9 are an "RC snubber" network that protects K1's contacts from arcing due to back EMF generated by the motor on W3/W4 when power to it is interrupted. Because of its inductive nature, the motor can generate a high voltage, which can cause arcing in the relay contacts. The RC snubber suppresses this voltage and protects the relay contacts.
The components connected to U3 pin 6 provide a clock source that's derived from the 110VAC supply frequency. From the point of view of the circuit, i.e. relative to the 0V rail, the voltage at the bottom end of R11 swings between about 0V and about 50V at mains frequency. R11 has a high resistance so it limits the current available, to prevent damage to U3. C7 provides a small amount of smoothing on the signal to prevent possible bogus clocking due to noise on the mains supply.
U3's inputs all have a "Schmitt trigger" feature which gives them a very clean switching behaviour. Instead of having a single voltage threshold, like normal logic ICs, they have two thresholds. When the input voltage is low, it must rise above the higher threshold before U3 will recognise it as high; once it has been recognised as high, it must drop below the lower threshold before U3 will recognise it as low. This characteristic is called hysteresis, and it creates a deadband between the low and high thresholds, where input variations will be ignored. This feature is only needed on this one input; all of the other signals going into inputs on U3 are well-defined logic levels.
C5 and C8 are decoupling (aka bypass or reservoir) capacitors for U2 and U3 respectively. Logic ICs, especially counters (U2), draw brief spikes of significant current through their supply pins, but for reliable operation, they need a clean supply voltage. These capacitors provide a tightly coupled local reservoir to ensure reliable operation. They must be connected as directly as possible (with leads as short as possible) between the VDD and VSS pins of their respective ICs - pins 16 and 8 for U2, and pins 14 and 7 for U3.
For operation from 50 Hz mains, the count detection logic needs to be changed, so RUN goes low at a lower count (as U2 is clocked more slowly). This can be done by deleting R10 and Q2, and replacing R11 with a link. This causes RUN to go low at a count of 11000000000 binary, which is 1536 decimal. Dividing 1536 by 50 gives the timeout period, 30.72 seconds.
Capacitor-fed power supplies cannot supply much current, and the current budget for this circuit is around 15 mA, most of which is used by the relay coil. Therefore, not all 48V-coil relays are suitable. See below.
Notes on components
I have tried to use a limited number of different types of components to make the bill of materials a bit simpler. For example, D7 and D8 don't need to be rated for 1A but it's simpler to keep them the same as D1~4 instead of adding a new, different component.
The three 100 ohm fusible resistors should be rated at 0.5W. I suggest
http://www.digikey.com/product-detail/en/FRM-50JR-52-100R/100DTCT-ND/2813208.
If you decide to fully encapsulate the circuit, you should ensure an air gap around the fusible resistors; if they are fully enclosed, they may not be able to fuse properly (there will be nowhere for the magic smoke to go). I'm not sure how to do that, but I have an idea, so let me know if you want to encapsulate it.
D1~4 and D7 and D8 can be any member of the 1N400x family rated for at least 100V. That is, 1N4002~4007 are all suitable.
C1, C4 and C9 must all be "X2 rated" or better. X1, Y2 and Y1 ratings are all better than X2. These ratings mean that the capacitor is designed to have mains-frequency AC voltage across it permanently, without failing. These capacitors have a special construction. It is NOT safe to use capacitors that are not thus rated, or are rated for lower AC voltages.
C1: 0.33 µF 150VAC:
http://www.digikey.com/product-detail/en/B32922C3334M/495-2321-ND/778983
C4: 22 nF 275VAC:
http://www.digikey.com/product-detail/en/PHE850EB5220MB06R17/399-5420-ND/1927365
C9: 0.1 µF 150VAC:
http://www.digikey.com/product-detail/en/ECQ-UAAF104M/P14779-ND/2674011
Suitable relay types for K1 are listed. These all have 16A contact ratings, with coils rated for 48VDC with coil currents less than 12 mA. That corresponds to a coil resistance of 4000 ohms or higher. Don't use a relay with a coil resistance less than 4000 ohms, because the capacitor-fed power supply cannot supply a lot of current.
These relays are all available from
http://www.digikey.com and are listed in ascending price order.
Panasonic ALE1Px48 ('x'is either B or F for the temperature class: B is up to 85°C; F is up to 105°C)
http://www.digikey.com/product-detail/en/ALE1PB48/255-2373-ND/1680241
Panasonic ALZxyz48 ('x' is either 1 for SPDT contact or 5 for SPST contact (preferred); 'y' is either 1 for flux-resistant or 2 for fully sealed (not important); 'z' is either B or F for temperature class - see above)
http://www.digikey.com/product-detail/en/ALZ51B48/255-3369-ND/2224606
Omron G2RL-1(A)(4)E DC48 ('A' present means SPST contact (preferred); '4' present means fully sealed (not important))
http://www.digikey.com/product-detail/en/G2R-1A-E-DC48/Z2309-ND/368728
TE Connectivity / Schrack RTxyz048 ('x' is either D for "wash tight" (sealed) or 3 for "flux proof" (not fully sealed) (not important); 'y' is either 1 for SPDT contact or 3 for SPST contact (preferred); 'x' is either 4 for standard or 5 for gold plated contact (not important))
http://www.digikey.com/product-detail/en/RT314048/RT314048-ND/1128613
TE Connectivity / Schrack RPxy0048 ('x' is either 3 for "flux proof" or 7 for "wash tight" (not important); 'y' is either 1 for SPDT contact or 3 for SPST contact (preferred)).
http://www.digikey.com/product-detail/en/3-1393230-3/PB1674-ND/2397842
This circuit can be constructed on stripboard. Keep U2 and U3 away from the high-voltage sections of the circuit, and cut the tracks that connect to them, close to the ICs, to reduce the possibility of signal coupling or shorts between those tracks and other higher-voltage parts of the circuit.
I recommend using IC sockets for U1~3. Decoupling capacitors for U2 and U3 can be soldered on the underside of the board (use sleeving on the capacitor leads to prevent shorts) and can be soldered directly between the corner pins of the IC socket.
Heh! Just as well I can type fast, eh
